NUC CCMAS

Calculus begins where algebra ends. This course builds the exact foundation you need to understand function behaviour. You will study function types, graph transformations, and arithmetic operations before moving to limit evaluation and continuity analysis. The lessons progress from basic definitions to advanced techniques, including factorisation, rationalisation, the squeeze theorem, and the intermediate value theorem. Each topic pairs instruction with targeted practice problems to confirm your grasp.

These concepts drive engineering calculations, economic modelling, physics simulations, and algorithm design. You will manipulate functions to predict system responses, evaluate limits to handle sudden data changes, and apply continuity principles when modelling physical or financial processes. Mastering this material replaces formula memorisation with reliable problem-solving for technical work.

You will identify the domain and range of complex expressions, sketch transformed graphs, and calculate composite and inverse functions. You will evaluate limits through direct substitution, algebraic manipulation, trigonometric identities, and the squeeze theorem. You will test for continuity at specific points, classify discontinuity types, and apply the intermediate value theorem across intervals.

This course serves university students preparing for first-year mathematics, secondary school graduates entering science programmes, and developers or analysts needing quantitative skills. Professionals returning to mathematics will rebuild speed and accuracy through structured exercises. Anyone processing technical data will apply these methods daily, regardless of degree title.

functions-limits-and-continuity-calculus-undergraduate

Calculus

Functions, Limits and Continuity - Calculus (Undergraduate Foundation)

Differentiation measures how quantities change at exact moments. This course builds your complete calculus foundation from first principles to advanced methods. We start with rates of change and formal notation, then cover definitions, continuity, differentiability, and direct proofs. The sequence moves through core rules, composite functions, implicit techniques, parametric systems, and higher-order operations. Each concept links to targeted exercises that test your accuracy before you proceed.

These methods solve real problems in physics, engineering, finance, and software development. You will track speed, predict growth, calculate structural stress, and measure market sensitivity. Software developers apply these calculations to optimise machine learning models. Business analysts use them to adjust variables for target profits. The techniques here convert abstract equations into working tools for daily technical tasks.

You will calculate derivatives for polynomial, trigonometric, exponential, logarithmic, and hyperbolic functions without reference material. You will apply the product rule, quotient rule, and chain rule to complex expressions. You will perform implicit, parametric, and logarithmic differentiation. You will compute second-order and higher-order derivatives for standard forms. You will verify differential equations through systematic steps. Every lesson forces active practice and precise calculation.

The material suits first-year university students, polytechnic learners, and independent learners who require a solid calculus base. Secondary school graduates will gain the exact preparation needed for advanced science and mathematics exams. Technical professionals can use the content to refresh forgotten methods or build reliable calculation habits. The explanations skip heavy theory and focus on working steps. Anyone who handles data, graphs, or mathematical models will apply the methods immediately. You will solve standard calculus problems independently.

fundamentals-and-techniques-of-differentiation-calculus

Fundamentals and Techniques of Differentiation - Calculus (Undergraduate Foundation)

Physical laws are often expressed as equations containing rates of change. This course explains the fundamental properties of ordinary differential equations without computing explicit solutions. You will study the basic definitions, the order, and the degree. We cover the derivation of an equation from a primitive function by differentiating and eliminating arbitrary constants. You will then classify these mathematical models by linearity and type to build a strict foundation for advanced analysis.

Understanding the structure of these models is essential for representing real systems in engineering. Professionals use these rules to define dynamic events like fluid flow before attempting to find numerical answers. By evaluating the existence and the uniqueness of solutions qualitatively, you will confirm if a model has a valid theoretical framework before starting physical projects. This step prevents resource waste in applied mechanics and technical project design.

You will master the logical frameworks required to set up these equations. You will determine the order of the equation and the degree of the highest derivative. You will state the conditions for the existence and the uniqueness of solutions. You will also sketch direction fields, interpret families of solutions visually, and extract the equilibrium solutions of a system from its slope field. Analytical methods for calculating exact mathematical answers are excluded from this material.

This material targets Nigerian secondary school leavers entering university science and engineering programmes. It also serves current undergraduates who must fix gaps in their knowledge of mathematical modelling. The content connects basic calculus to advanced university mathematics. Mastering these concepts gives learners the precise analytical skills required to pass strict university examinations and handle technical demands in early career engineering roles across the globe.

introduction-to-ordinary-differential-equations

Differential Equations

Introduction to Ordinary Differential Equations (Undergraduate Advanced)

Mathematics links quantities to their rates of change through differential equations. This course teaches core techniques to solve first-order and reducible higher-order ordinary differential equations. You will classify equations and apply analytical methods to find exact solutions. We focus strictly on pure mathematical methods; numerical approximations and physical applications are excluded. Topics include separation of variables, transformation of homogeneous equations, exact differential equations, linear first-order equations, and Bernoulli equations.

These methods provide the mathematical foundation required for advanced university studies in engineering. You will use these skills to solve complex problems in subsequent academic modules. Knowing these analytical techniques allows you to verify the output of computer simulations in your future career. The course builds the logical framework necessary for theoretical projects. You will learn to manipulate complex equations accurately and efficiently.

By the end of this programme, you will master techniques to solve ordinary differential equations. You will apply separation of variables and transformation rules for homogeneous equations. You will test for exactness and find integrating factors by inspection or by groups. You will solve linear first-order and Bernoulli equations using standard substitutions. You will also reduce higher-order equations to first-order forms, handling cases with the independent or dependent variable absent.

This course targets early university students in Nigeria pursuing science and engineering degrees. It also benefits bright secondary school leavers preparing for rigorous university mathematics. The direct explanations bridge the gap between secondary school calculus and advanced university requirements. Students seeking a strict treatment of analytical methods will gain the exact skills needed to pass difficult examinations and understand higher-level mathematics.

first-order-and-reducible-higher-order-ordinary-differential

First-Order and Reducible Higher-Order Ordinary Differential Equations (Undergraduate Advanced)

Vectors define physical quantities with magnitude and direction. This course delivers the core principles of vector algebra and geometry for university science. You will study vector classifications and fundamental algebraic laws. The curriculum covers addition, scalar multiplication, and position vectors. We then examine Cartesian components, direction cosines, line division, vector projections, and centroids. Every section includes practical worked examples.

A firm grasp of vectors is mandatory for engineering, physics, and programming. Civil engineers resolve vector components to calculate structural forces. Software developers apply vector geometry to simulate motion in digital environments. Academically, this knowledge forms the strict basis for mechanics and electromagnetism. Mastering these concepts lets you convert physical problems into precise mathematical models to find accurate solutions.

You will master the algebraic manipulation and geometric application of vectors. You will compute the magnitude of a vector and determine direction cosines using Cartesian coordinates. The instruction explains how to apply addition laws, perform scalar multiplication, and establish collinearity. You will also calculate the internal and external division of a line, evaluate the projection of one vector onto another, and find the centroid of a system using weighted means.

This course is for Nigerian secondary school leavers and first-year university students taking foundation science modules. It provides a direct transition from secondary mathematics to university engineering. Individuals outside these groups, including self-taught programmers and professionals entering technical roles, will also gain essential mathematical skills. The straightforward instruction guarantees you build practical competence quickly and securely.

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Vectors

Vector Algebra and Geometry - Vectors (Undergraduate Foundation)

Vectors multiply in ways entirely different from ordinary numbers, producing either a single numerical value or a completely new direction. This course covers the core mechanics of vector multiplication. You will learn the operations of the dot product, the cross product, the scalar triple product, and the vector triple product. The material details the strict definitions, the mathematical properties, and the exact geometric interpretations of each operation. You will also study the core principles for manipulating and solving complex vector equations.

Understanding how vectors interact is crucial for solving real physical problems in engineering and physics. When engineers calculate the work done by a force or determine the turning effect of a spanner, they use these exact vector products. You will apply this knowledge to find the area of a parallelogram, calculate the volume of a parallelepiped, and test if multiple forces act on the same flat surface. These tools are the foundation for analysing structural loads, programming spatial graphics, and building physical mechanical systems.

By the end of this course, you will master specific mathematical techniques and logical frameworks. You will learn to calculate geometric projections, express the cross product as a determinant, and expand the vector triple product. You will acquire the skills to test for perpendicularity, parallelism, and coplanarity between multiple vectors. Furthermore, you will understand exactly why you cannot divide an equation by a vector, learning instead how to solve for unknown quantities by dividing by a scalar, using scalar products, or eliminating a vector and the coefficient of the vector.

This course is primarily for Nigerian secondary school leavers preparing for higher education and early university students in science and engineering programmes. It provides a direct transition from basic secondary school mathematics to rigorous university mechanics. It also benefits self-taught programmers and independent learners who need a firm grasp of spatial mathematics for technical projects. By stripping away unnecessary abstractions, the instruction delivers immediate, practical skills that ensure you can confidently handle foundational university coursework and physical structural analysis.

vector-products-and-its-applications-vectors-undergraduate

Vector Products and Its Applications - Vectors (Undergraduate Foundation)

Vectors describe physical quantities possessing both magnitude and direction in space. This course covers the calculus of vector-valued functions of a single variable. You will study how to differentiate and integrate these functions using established mathematical rules. The curriculum details the differentiation of sums and products of vectors. It also explains partial differentiation and the calculation of unit tangent vectors. The focus remains on fundamental calculus applied strictly to spatial dimensions.

The mathematics of vectors forms the basis of physics and engineering mechanics. Engineers use these exact principles to model the motion of a particle along specific curves in space. By mastering these concepts, you can accurately compute position, velocity, and acceleration for moving objects. This knowledge is essential for building structural models, programming physics engines, or analysing spatial forces. You will apply these direct mathematical tools to solve concrete physical problems in kinematics.

By the end of this programme, you will master the derivative and the anti-derivative of vector functions. You will evaluate line integrals and compute triple products accurately. The lessons teach the strict application of differentiation rules to complex vector sums and cross products. You will also use the derivative of a position vector to find exact velocity and acceleration equations. You will develop a rigid logical framework for solving spatial motion problems step by step through intensive worked examples.

This foundation course is primarily for recent Nigerian secondary school leavers preparing for university education in science and engineering fields. First-year undergraduate students struggling with introductory vector calculus will also find this material highly useful. It bridges the gap between basic secondary school mathematics and advanced university mechanics. Self-taught software developers working on physical simulations will similarly benefit from the direct mathematical techniques taught here. It provides the exact theoretical grounding required to pass early university engineering modules.

differentiation-and-integration-of-vector-valued-functions

Differentiation and Integration of Vectors (Undergraduate Foundation)

Master advanced matrix algebra and linear systems through rigorous technical training. This course covers matrix definitions; algebraic operations; transpositions; and elementary row transformations. You will investigate determinants; matrix inverses using the adjoint method; and the structural properties of similar matrices. Computational methods using MS-Excel or Google Sheets are included to handle large-scale data sets efficiently.

These mathematical structures are the engines behind modern engineering and data science. Structural engineers use matrices to calculate stress in bridges; computer scientists use them to render 3D graphics; and economists use linear systems to model market equilibrium. Mastering these tools allows you to solve multi-variable problems that are impossible to handle manually.

By the end, you will possess the skills to execute matrix additions; multiplications; and scalar operations with absolute precision. You will demonstrate competence in reducing matrices to row echelon form; calculating rank and nullity; and evaluating determinants of any order. Furthermore, you will command the ability to solve complex systems of linear equations using Cramer's rule or matrix inversion; identify special matrix types like orthogonal or symmetric; and automate these calculations using software.

This curriculum is built for undergraduate students in engineering; mathematics; and physics requiring a high-level command of linear algebra. Professionals in data science or finance will find the content vital for understanding the algorithms governing their software. Even those in general sciences will benefit from the structured logical thinking and numerical methods required to manage complex variables in any technical field.

matrices-determinants-and-systems-of-linear-equations

Linear Algebra

Matrices, Determinants, and Systems of Linear Equations - Linear Algebra (Undergraduate Advanced)

Physical systems rely on multiple variables interacting simultaneously. This course explains partial differentiation for functions of several variables. It covers domains, level curves, limits, continuity, and partial derivatives. You will study higher-order derivatives, mixed partial derivatives, and the chain rule for composite functions. The syllabus includes implicit differentiation, systems of equations, Jacobian determinants, the implicit function theorem, inverse functions, homogeneous functions, the theorem of Euler, and the theorem of Taylor for linear approximations and differentials.

Engineers use these mathematical tools to model complex systems. You will apply this knowledge to structural analysis and fluid mechanics in physical projects. Mastery of these rules ensures accurate predictions when multiple factors change at once. The frameworks taught here form the baseline for advanced engineering studies. Understanding the geometry of tangent planes and normal lines to a surface provides direct value for computer graphics and spatial analysis.

You will master precise mathematical skills and logical frameworks. You will calculate partial derivatives, evaluate limits for functions of several variables, and apply the chain rule. You will compute determinants of Jacobians, construct linear approximations using the theorem of Taylor, and solve implicit differentiation problems. You will also apply the theorem of Euler to homogeneous functions and calculate equations of tangent planes and normal lines.

This course targets early university undergraduates and secondary school leavers in Nigeria preparing for STEM programmes. It provides a mandatory foundation for students entering engineering or mathematics. Learners from other disciplines needing strong quantitative skills will benefit from the strict logic taught here. Programmers and analysts slightly outside the main audience will gain the mathematical rigour necessary for advanced computation.

partial-differentiation-multivariable-calculus

Multivariable Calculus

Partial Differentiation - Multivariable Calculus (Undergraduate Advanced)

Multivariable calculus models real systems using multiple inputs. This course covers applications of partial differentiation. You will study extreme values of functions of several variables, the second-derivative test, and saddle points. The content covers extrema on restricted domains and Lagrange multipliers for single or multiple constraints. We examine gradients, directional derivatives, steepest ascent, tangent planes, and the distance from a point to a surface.

Engineers use these tools to optimise physical projects. Understanding steepest ascent helps you achieve maximum efficiency in structural designs. The gradient vector allows civil planners to map slopes, while Lagrange multipliers maximise output under strict mathematical constraints. Mastering these techniques builds a foundation for advanced studies in physics and engineering. You will apply this knowledge directly to solve geometric optimisation problems in your academic work and technical career.

You will compute directional derivatives and locate critical points of mathematical models. You will apply the second-derivative test to classify local extrema and saddle points. You will use Lagrange multipliers to optimise functions subject to constraints. You will construct equations of tangent planes and calculate the shortest distance from a point to a surface. You will integrate these logical frameworks to solve geometric problems without using parametric methods or linear regression.

This course targets early university students and recent Nigerian secondary school leavers preparing for engineering and science degrees. It provides the rigour needed to transition from basic calculus to multivariable analysis. Working professionals in physical engineering slightly outside this core group will benefit from refreshing their mathematics. The direct instruction ensures anyone with basic calculus knowledge can master these spatial concepts for immediate application.

applications-of-partial-differentiation-multivariable

Applications of Partial Differentiation - Multivariable Calculus (Undergraduate Advanced)

Engineering requires precise mathematical models to solve physical problems. This track provides the exact foundation of university mathematics required for technical disciplines. You will transition from basic calculus to multivariable analysis, vector geometry, and linear algebra. The programme replaces formula memorisation with strict analytical frameworks to ensure you calculate derivatives, differential equations, and matrix operations accurately.

This curriculum serves 200-level undergraduates in engineering and technology programmes. The content conforms strictly to the standards of the NUC CCMAS for engineering mathematics. It bridges the gap between first-year foundation courses and advanced technical requirements. You will find the exact preparation necessary to pass complex examinations and handle the strict demands of upper-level engineering coursework.

Upon completion, you will solve reducible higher-order differential equations, execute complex matrix operations, and perform vector differentiation. You will apply partial derivatives to optimise physical systems and determine exact values across multiple dimensions using the theorem of Taylor and Lagrange multipliers. Mastering these tools provides the precise analytical skills required to succeed in difficult university examinations and handle physical modelling in early career engineering roles globally.

Our mission is to simplify complex concepts by teaching from first principles. We go beyond simple problem-solving to build a deep, foundational understanding of the "why" behind the "how."

Our content is meticulously designed for undergraduates navigating the rigorous curricula of top engineering and physical science programmes in Nigeria and globally.

As a core academic partner on UniDrills, we are committed to upholding the highest standards of clarity, precision, and professional excellence in education.

Max Engineering Tutorials

Olowojaye Rufus

An academic and graduate engineer dedicated to explaining complex concepts in mathematics, computer science and engineering. I founded UniDrills to bring this vision to life, creating a world-class e-learning marketplace with a global outlook. I apply a first-principles approach to software architecture to build a professional and intuitive learning experience.

Engineering Mathematics I

GET 209: Engineering Mathematics I

OAU, Ife

Do you want to learn how to describe and analyze the motion of particles in one and two dimensions?

Do you want to apply your knowledge of kinematics to solve real-world problems in engineering, physics, and sports?

Do you want to master the use of different coordinate systems and reference frames to simplify your calculations and enhance your understanding?

If you answered yes to any of these questions, then this course is for you!

Engineering Mechanics: Kinematics of Particles comprehensively addresses the fundamental concepts and principles of particle kinematics. You will learn how to:

- Define and measure the position, velocity, and acceleration of a particle moving along a straight or curved path.

- Use various methods to determine the motion of a particle, such as equations of motion, graphical solutions, and numerical solutions.

- Apply the concepts of relative and dependent motion to analyze the motion of several particles or connected bodies.

- Choose the most suitable coordinate system for a given problem, such as rectangular, polar, normal-tangential, or radial-transverse coordinates.

- Understand the effects of different reference frames on the observed motion of a particle, such as inertial, non-inertial, or rotating frames.

By the end of this course, you will have a solid foundation in particle kinematics that will enable you to tackle more advanced topics in dynamics, mechanics, and other fields of engineering and science. You will also have the skills and confidence to apply your knowledge to real-world situations, such as the trajectory of a snowboarder, the orbital speed of a satellite, or the accelerations during acrobatic flying.

This course is designed for students, engineers, and enthusiasts who have a basic background in calculus, physics, and vector algebra. 

Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

kinematics-of-particles-engineering-mechanics

Engineering Mechanics

Kinematics of Particles - Engineering Mechanics (Undergraduate Advanced)

This course will equip you with the skills and knowledge to tackle various engineering and scientific problems involving kinetics of particles, which is the branch of mechanics that deals with the relation between the forces acting on a particle and its resulting motion. From the basics of mechanics, dynamics, and kinetics to getting comfortable with Newton's second law and momentum methods, you will clearly understand how forces cause motion.

Boost your problem-solving skills and open up new career opportunities in fields like aerospace engineering, mechanical engineering, civil engineering, biomedical engineering, physics, chemistry, and more. Imagine being able to predict how a ball flies through the air, or understanding why a car behaves the way it does in a crash. This course helps you do these and many more!

By the end of this course, you will be able to:

- Explain the fundamentals of mechanics, dynamics, and kinetics

- Use Newton's second law to analyze how forces make bodies speed up or slow down

- Learn about energy, how forces do work, and how these relate to motion of bodies

- Figure out how collisions work and what motion happens before and after collisions

This course is designed for students, engineers, scientists, and anyone who is interested in learning the basics of kinetics of particles. The course assumes that you have a basic background in calculus, physics, vector algebra, and statics.

Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

kinetics-of-particles-by-newtons-second-law-engineering

Kinetics of Particles by Newton's Second Law - Engineering Mechanics (Undergraduate Advanced)

Analysis of motion of a particle with respect to the forces causing the motion by work-energy and impulse-momentum principles.

kinetics-of-particles-by-energy-and-momentum-methods

Kinetics of Particles by Energy and Momentum Methods - Engineering Mechanics (Undergraduate Advanced)

Analysis of kinetics of generalized systems of particles by Newton's second law, work-energy principle and impulse-and-momentum methods.

kinetics-of-systems-of-particles-engineering-mechanics

Kinetics of Systems of Particles - Engineering Mechanics (Undergraduate Advanced)

In this course, you will learn the fundamentals of two-dimensional kinematics of rigid bodies, which is the branch of mechanics that deals with the geometry of the motion of rigid bodies without considering the forces that cause it. You will learn how to use different coordinate systems and reference frames to express the position, velocity, and acceleration of various points on a rigid body in plane motion.

Empower yourself with the knowledge and skills needed to excel in diverse fields such as mechanical engineering, robotics, aerospace engineering, and beyond. Imagine designing a robotic arm that moves with precision, or optimizing the performance of a rotating machinery. These are some of the countless applications of rigid-body kinematics you will learn in this course.

By the end of this course, you will be able to:

- Understand the rotation of rigid bodies about fixed axes and its applications

- Analyze absolute motion to understand how rigid bodies move through space

- Explore relative velocity analysis to solve complex motion problems

- Master the concept of instantaneous centres of rotation for precise motion analysis

- Understand relative acceleration analysis to predict the behavior of moving bodies

- Analyze motion relative to a rotating frame to unravel the dynamics of rotating systems

Whether you're a seasoned professional looking to expand your knowledge or a curious beginner eager to explore the motion of rigid bodies, this course is for you.

Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

2-d-kinematics-of-rigid-bodies-engineering-mechanics

2D Kinematics of Rigid Bodies - Engineering Mechanics (Undergraduate Advanced)

Do you want to learn how to apply the laws of physics to analyze the motion of rigid bodies in two dimensions?

Do you want to understand the concepts and methods of mass moment of inertia, general equations of motion, work, energy, impulse, and momentum?

Do you want to design and optimize mechanical systems and devices that involve the rotation and translation of rigid bodies?

If you answered yes to any of these questions, then this course is for you!

In this course, you will learn the fundamentals of two-dimensional kinetics of rigid bodies, which is the branch of mechanics that deals with the relation between the forces acting on a rigid body and its resulting motion. The course will equip you with the skills and knowledge to solve various engineering and scientific problems involving the motion of rigid bodies in two dimensions. You will be able to apply the concepts and methods of two-dimensional kinetics of rigid bodies to fields such as mechanical engineering, aerospace engineering, civil engineering, robotics, biomechanics, and more. You will also be able to appreciate the power and elegance of the physical laws that govern the motion of rigid bodies.

By the end of this course, you will be able to:

- Define and explain the meaning of kinetics, rigid body, and plane motion

- Calculate the mass moment of inertia of a rigid body using integration or parallel axis theorem

- Derive and use the general equations of motion for a rigid body in plane motion

- Analyze the motion of a rigid body undergoing translation, rotation about a fixed axis, or general plane motion

- Apply the principle of work and energy to determine the motion of a rigid body or a system of rigid bodies

- Apply the principle of conservation of energy to determine the motion of a rigid body or a system of rigid bodies under conservative forces

- Apply the principle of impulse and momentum to determine the motion of a rigid body or a system of rigid bodies

- Apply the principle of conservation of momentum to determine the motion of a rigid body or a system of rigid bodies

Tailored for students, engineers, scientists, and anyone passionate about learning the basics of 2D kinetics of rigid bodies, this course assumes a basic background in calculus, physics, vector algebra, and statics. 

Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

2-d-kinetics-of-rigid-bodies-by-newtons-second-law

2D Kinetics of Rigid Bodies by Newton's Second Law - Engineering Mechanics (Undergraduate Advanced)

Analysis of motion of a rigid body with respect to the forces causing the motion by work-energy and impulse-momentum principles.

2-d-kinetics-of-rigid-bodies-by-energy-and-momentum-methods

2D Kinetics of Rigid Bodies by Energy and Momentum Methods - Engineering Mechanics (Undergraduate Advanced)

This course teaches the fundamentals of dynamic systems modelling and simulation. You will learn how to:

- Define and classify static and dynamic systems by their mathematical models

- Use differential equations, transfer functions, and state-space models to describe the behavior of dynamic systems

- Analyze the stability, controllability, and observability of dynamic systems

- Model and simulate translational, rotational, electrical, electromechanical, hydraulic, and thermal systems

- Draw analogies between different types of systems and exploit their similarities

- Use popular software tools such as MATLAB/SIMULINK to implement and test your models

By the end of the course, you will have a solid foundation in dynamic systems modelling and simulation, and you will be able to apply it to your own projects and interests. You will also gain a valuable skill that is in high demand in many industries and disciplines, such as:

- Engineering: design, control, and optimization of dynamic systems such as robots, vehicles, machines, power systems, etc.

- Science: understanding and prediction of natural phenomena such as population dynamics, climate change, epidemics, etc.

- Economics: modelling and analysis of economic systems such as markets, supply chains, production, consumption, etc.

- Policy: evaluation and improvement of policy actions and interventions in complex social systems such as health, education, environment, etc.

Whether you are a student, a professional, or a hobbyist, this course will help you master the art and science of dynamic systems modelling and simulation, and open up new possibilities for your learning and career.

Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

introduction-to-dynamic-systems-and-vibration-engineering

Introduction to Dynamic Systems and Vibration - Engineering Mechanics (Undergraduate Advanced)

Master the principles governing the motion of engineering systems. This learning track provides a complete education in dynamics, systematically building from the kinematics of particles to the kinetics of rigid bodies and culminating in an introduction to dynamic systems and vibration. You will learn to analyse and predict how mechanical systems behave under the influence of forces.

This programme is for undergraduate students in mechanical, aerospace, civil, or related engineering disciplines. It is also essential for practising engineers and applied scientists who require a rigorous, first-principles command of dynamic analysis for their professional work. A prerequisite knowledge of statics, calculus, and vector algebra is assumed.

Upon completion, you will possess the analytical tools to solve complex dynamics problems for particles and rigid bodies using force, energy, and momentum methods. This provides the essential foundation for advanced study in mechanical design, control systems, and structural analysis, and prepares you for demanding technical roles in the engineering sector.

Engineering Mechanics II

Dynamics

MEE 206: Engineering Mechanics II - Dynamics

Do you want to learn how to manipulate and analyze scalar and vector fields using advanced calculus techniques? Do you want to understand the concepts and methods of scalar and vector products, gradient, divergence, curl, and Laplacian operators? Do you want to apply the principles of vector analysis to solve problems in physics, engineering, and other disciplines?

If you answered yes to any of these questions, then this course is for you!

In this course, you will learn the fundamentals of calculus of scalar and vector fields, which is the branch of mathematics that deals with functions and operators that map points in space to scalars or vectors. You will learn how to use different notations and conventions to express scalar and vector fields, such as Einstein's summation, Kronecker delta, and Levi-Civita symbols. You will learn how to perform scalar and vector products of vectors, which are useful for measuring angles, areas, and volumes. You will learn how to compute and interpret the gradient of a scalar field, which measures the rate and direction of change of a scalar function. You will learn how to compute and interpret the divergence and curl of a vector field, which measure the sources and vortices of a vector field. You will also learn how to use the Laplacian operator, which combines the divergence and the gradient, and is useful for studying heat conduction, electrostatics, and fluid dynamics.

This course will equip you with the skills and knowledge to solve various problems involving scalar and vector fields. You will be able to apply the concepts and methods of calculus of scalar and vector fields to fields such as physics, engineering, chemistry, biology, and more. You will also be able to appreciate the power of vector analysis theorems, such as Gauss', Green's, and Stokes' theorems, which relate the integrals of scalar and vector fields over different domains.

By the end of this course, you will be able to:

- Define and explain the meaning of scalars, vectors, and tensors

- Identify and use different notations and conventions for scalar and vector fields

- Perform scalar and vector products of vectors and calculate angles, areas, and volumes

- Compute and interpret the gradient of a scalar field and its properties

- Compute and interpret the divergence and curl of a vector field and their properties

- Compute and interpret the Laplacian of a scalar or a vector field and its applications

- Use different coordinate systems and bases to express scalar and vector fields and operators

- Compute and interpret the gradient, divergence, curl, and Laplacian in orthogonal curvilinear coordinates

- Recognize and use some special curvilinear coordinates, such as cylindrical and spherical coordinates

- Apply the vector analysis theorems, such as Gauss', Green's, and Stokes' theorems, to relate the integrals of scalar and vector fields over different domains

Created for students, engineers, scientists, and anyone passionate about learning the calculus of scalar and vector fields, this course assumes a basic background in calculus, linear algebra, and vector algebra. With engaging video lessons, interactive quizzes, practice problems, and supportive peer interactions, you'll embark on a journey of discovery and mastery like never before.

Seize the opportunity to advance your mathematical skills and unlock new problem-solving possibilities with an enrolment right away!

calculus-of-scalar-and-vector-fields-undergraduate

Calculus of Scalar and Vector Fields (Undergraduate Advanced)

Standard numbers cannot solve every square root. This course teaches you to use complex numbers for such problems. You will learn number systems, algebra, and how to use the Argand plane. Lessons cover polar forms, De Moivre's theorem, and finding roots. You will also study sine, cosine, and log functions through many worked examples.

These methods are vital for electrical engineering, physics, and signal processing. You need these skills to calculate electricity flow and wave behaviour. Engineers use these tools to build machines and structures. Mastering this allows you to solve the difficult maths used in modern technology and research. It provides the logic needed for all advanced technical work.

You will learn to add, multiply, and divide complex numbers. You will switch between rectangular, polar, and exponential forms using Taylor series. You will gain the skill to find Nth roots of unity and solve hyperbolic equations. You will also learn to draw complex equations on a graph accurately. By the end, you will handle advanced mathematical expressions without errors.

This course is for university students in engineering or maths who must pass advanced modules. It also serves technical workers who need to refresh their skills. The step-by-step examples make the topics easy to follow for anyone starting out. This knowledge is a requirement for anyone pursuing a career in high-level science or technology.

complex-numbers-mathematical-methods-undergraduate

Mathematical Methods

Complex Numbers - Mathematical Methods (Undergraduate Advanced)

Do you want to learn how to work with abstract spaces and transformations that preserve their structure? Do you want to understand the concepts of vector subspaces, linear combinations, linear dependence, basis, dimension, coordinates, and properties of vector spaces? Do you want to master the skills of defining and manipulating linear maps, their kernels, images, matrix representations, and transition matrices?

If you answered yes to any of these questions, then this course is for you!

Linear Algebra: Linear Vector Spaces and Linear Maps is a comprehensive and engaging course that covers the fundamentals of vector spaces and linear maps and their applications in mathematics and science. You will learn how to:

- Define and classify vector spaces and their subspaces over a given scalar field

- Perform operations on vectors using linear combinations and scalar multiplication

- Determine whether a set of vectors is linearly dependent or independent and find a basis and dimension for a vector space or subspace

- Find the coordinates of a vector with respect to a given basis and change the basis using transition matrices

- Define and classify linear maps between vector spaces and find their domains, codomains, ranges, and null spaces

- Find the kernel and image of a linear map and use them to determine whether a linear map is one-to-one or onto

- Represent a linear map using a matrix and perform matrix operations such as addition, multiplication, and inversion

- Use different methods and tools to solve systems of linear equations, such as Gaussian elimination, row reduction, and inverse matrices

This course is suitable for anyone who wants to learn or review the basics of vector spaces and linear maps and their applications. It is especially useful for students and professionals in algebra, geometry, analysis, differential equations, optimization, cryptography, computer graphics, data science, and other related fields.

By the end of this course, you will have a solid foundation of the theory and practice of vector spaces and linear maps and their operations. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve vector spaces and linear maps.

Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

linear-vector-spaces-linear-algebra-undergraduate

Linear Vector Spaces - Linear Algebra (Undergraduate Advanced)

A comprehensive study of linear transformations (or maps) - definition, kernel, range, matrix representation and related matters.

linear-maps-transformations-linear-algebra

Linear Maps (Transformations) - Linear Algebra (Undergraduate Advanced)

This course provides a clear and structured exploration of eigenvalues, eigenvectors, and diagonalization, focusing on both theory and real-world applications. 

You???ll learn how to compute eigenvalues and eigenvectors, understand their geometric significance, and apply diagonalization to simplify complex matrix operations. 

Topics include linear transformations, dynamical systems, and applications in physics, engineering, and machine learning. 

The course is designed for students, engineers, and data scientists seeking a strong foundation in matrix methods. By the end, you'll confidently apply these concepts in problem-solving and computational modeling.

eigenvalues-eigenvectors-and-diagonalization-of-matrices

Eigenvalues, Eigenvectors and Diagonalization of Matrices - Linear Algebra (Undergraduate Advanced)

Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors.

Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Mathematical Methods II

MTH 202: Mathematical Methods II

Real-valued functions form the base of all advanced calculation in engineering and physics. This course maps the complete structure of these functions from first principles. You will cover number systems, the split between rational and irrational values, and exact interval notation. You will examine function behaviour at infinity. The work moves straight to domain and range analysis, supported by worked examples that lock the theory in place. You will classify polynomial, rational and algebraic forms. You will handle piecewise definitions, odd and even symmetry, and transcendental forms covering exponential, logarithmic, trigonometric and hyperbolic families.

Engineers and analysts use these functions daily to build accurate models and predict system results. You will apply domain rules to stop calculation errors in structural design. You will deploy exponential and logarithmic rules for signal processing and financial modelling. Trigonometric and hyperbolic functions govern wave analysis, circuit simulation and mechanical vibration studies. Correct use of piecewise and odd-even rules simplifies computer algorithms and cuts processing time. Mastery of this material removes guesswork and supplies reliable mathematical tools for real technical tasks.

You will leave able to find the exact domain and range of any function without external aid. You will group functions into algebraic, transcendental or piecewise types by quick inspection. You will adjust exponential, logarithmic and trigonometric expressions using standard identities. You will recognise odd and even symmetry to reduce integration workload. You will switch between direct and inverse forms of trigonometric and hyperbolic functions. You will solve combined problems that merge multiple function types into a single analytical framework.

This course targets undergraduates and graduate students who need advanced calculus for examination success. It suits engineering, physics and computer science students who require strict mathematical training. Learners who finished secondary school mathematics can still use the structured examples to build a solid base. Professionals returning to technical work will refresh core concepts rapidly. The clear progression and focused practice allow any serious student to command real-valued functions for academic or industrial use.

functions-and-their-graphs-single-variable-calculus

Advanced Calculus

Real-Valued Functions - Advanced Calculus (Undergraduate Advanced)

Limits dictate how mathematical models predict exact behaviour at critical points. This course covers formal and informal definitions of finite and infinite limits. You will study direct substitution, algebraic factorisation, conjugate multiplication, the squeeze theorem, and piecewise function analysis. The material handles indeterminate forms through derivative ratio techniques and rationalisation. Each method includes repeated worked examples to fix the procedure in your mind.

Engineers and analysts use limit evaluation to calculate instantaneous speed, stress thresholds in structures, and signal decay rates. Financial planners apply these rules to compound interest and market trend forecasting. Computer simulation software relies on accurate limit logic to prevent calculation crashes and rounding errors. Mastery of this material gives you reliable tools for real technical modelling and removes guesswork from system analysis.

You will construct formal proofs for finite and infinite limits using strict logical bounds. You will select the correct evaluation method from substitution, algebra, or derivative rules without hesitation. You will resolve indeterminate forms and apply boundary checks to complex piecewise cases. You will verify every answer through graphical inspection and numerical approximation. You will gain complete control over the transition points of any real function.

This course targets university students and postgraduate researchers in engineering, physics, or mathematics. Advanced secondary school students with strong algebra skills will secure a clear advantage before university entry. Professionals preparing for technical exams will rebuild their analytical foundation. The structured progression and repeated practice ensure that any disciplined learner can apply limit theory to academic or industrial problems.

limits-of-functions-single-variable-calculus

Limits of Real-Valued Functions - Advanced Calculus (Undergraduate Advanced)

Continuity ensures that a function behaves predictably without sudden jumps or breaks. This course defines continuity at interior points, endpoints, and across entire intervals. You will identify standard continuous functions and test them through repeated worked examples. The material classifies discontinuities, with specific focus on removable gaps that can be repaired. You will study the max-min theorem and the intermediate-value theorem to understand how continuous functions guarantee specific outputs within a range.

Engineers rely on continuity to model physical systems where sudden changes cause failure. Structural analysis requires smooth stress distributions to prevent material fracture. Electrical circuit design depends on continuous current flow for stable operation. Computer graphics algorithms use continuity to render smooth curves and surfaces without visual artifacts. Financial models assume continuous price movements to calculate risk and option values accurately. Mastery of these concepts prevents calculation errors in simulation software and real-world design tasks.

You will determine if a function is continuous at any given point or interval. You will distinguish between removable and non-removable discontinuities using limit analysis. You will apply the max-min theorem to find absolute extrema on closed intervals. You will use the intermediate-value theorem to prove the existence of roots and solutions. You will verify continuity for composite functions and piecewise definitions. You will gain the ability to spot and fix breaks in mathematical models.

This course targets undergraduates and graduate students in mathematics, engineering, and physics. It suits students preparing for advanced calculus examinations or research projects. Secondary school leavers with strong algebra skills can use this material to bridge the gap to university-level analysis. Professionals returning to technical work will refresh their understanding of function behaviour. The clear structure and practical examples allow any disciplined learner to master continuity for academic or industrial application.

continuity-of-functions-single-variable-calculus

Continuity of Real-Valued Functions - Advanced Calculus (Undergraduate Advanced)

Differentiation measures how a function changes at any given point. This course moves from the basic slope of a line to advanced derivative theory. You will define differentiability at a point and on an interval. You will link differentiability to continuity to spot where functions break. The material covers standard rules for sums, products, quotients, and composites. You will study Rolle's theorem and the mean-value theorem to understand function behaviour between points. The course extends to higher-order derivatives using Leibniz's formula. You will finish with Taylor and Maclaurin series to approximate complex functions with polynomials.

Engineers use derivatives to calculate velocity, acceleration, and force in moving systems. Electrical engineers apply these rules to analyse current change in circuits. Economists use differentiation to find marginal cost and maximise profit. Computer scientists deploy Taylor series for fast approximation in graphics and machine learning algorithms. Mastery of these tools allows you to model real-world dynamics with precision. You will gain the ability to predict system responses and optimise designs without relying on trial and error.

You will determine if a function is differentiable at any point or interval. You will apply product, quotient, and chain rules to differentiate complex expressions. You will use Rolle's theorem and the mean-value theorem to prove existence of critical points. You will compute higher-order derivatives and apply Leibniz's formula for product terms. You will construct Taylor and Maclaurin polynomials to approximate functions near a centre. You will estimate error bounds for these approximations. You will solve applied problems that merge multiple differentiation techniques.

This course targets undergraduates and graduate students in mathematics, engineering, and physics. It suits students preparing for advanced calculus examinations or research projects. Secondary school leavers with strong algebra skills can use this material to bridge the gap to university-level analysis. Professionals returning to technical work will refresh their understanding of rate of change. The clear structure and repeated practice ensure that any disciplined learner can master differentiation for academic or industrial application.

differentiability-and-derivatives-of-functions

Differentiability of Real-Valued Functions - Advanced Calculus (Undergraduate Advanced)

Multiple integration extends calculus to measure quantities across multiple dimensions. This course covers line integrals, double integrals, surface integrals, and triple integrals. You will evaluate line integrals for scalar and vector fields along parametrised curves and test for dependence on path. The lessons show how to compute double integrals over rectangular and general regions using iteration in Cartesian coordinates. You will also change variables using polar, cylindrical, and spherical coordinates. Improper integrals are excluded to focus on standard evaluation.

Engineers and scientists use these tools to model physical systems. You will apply double and triple integrals to calculate area, volume, mass, the centre of mass, and moments of inertia for solid bodies. Evaluating surface integrals helps you measure fluid flow across parametric surfaces. These methods provide the exact mathematical framework to solve spatial problems in structural design, fluid mechanics, and applied physics.

By completing this course, you will master the evaluation of integrals through Riemann sums and direct calculation. You will find line integrals with respect to arc length and compute the surface area of a graph. You will gain the exact technical skills to set up and solve triple integrals for complex volumes. You will also learn the logical steps to switch coordinate systems when the boundaries of a region complicate Cartesian integration.

This course is primarily for university students studying science or engineering. It is also useful for Nigerian secondary school leavers preparing for university calculus. These primary learners get a direct breakdown of hard topics to pass exams and build real projects. Students in non-technical fields will also benefit from learning how to analyse spatial data using exact mathematical rules.

multiple-integration-calculus-undergraduate-advanced

Multiple Integration - Calculus (Undergraduate Advanced)

Mathematical models dictate the stability of physical structures and the accuracy of software simulations. This learning track provides the exact mathematical methods required for engineering and the physical sciences. You will study real-valued functions, limit evaluation, continuity, and differentiability. The programme then advances to partial differentiation and multiple integration techniques. You will learn to predict system responses and prevent calculation errors in physical design.

This track targets 200-level university students in mathematics and physical sciences following the NUC CCMAS standard. It is mandatory for undergraduates who need exact analytical skills to pass advanced calculus examinations. Nigerian secondary school leavers preparing for engineering degrees will use this material to prepare for university mathematics. Working professionals will also use the direct instruction to refresh their quantitative skills for immediate technical application.

You will calculate limits, determine the continuity of a function, and apply differentiation rules to complex expressions. You will evaluate multiple integrals to measure volume and calculate the centre of mass for solid bodies. You will use Lagrange multipliers to optimise equations subject to strict constraints. These tools prepare you to pass examinations and solve exact spatial problems in engineering, physical research, and programming. You will construct reliable mathematical models without guesswork.

Mathematical Methods I

MTH 201: Mathematical Methods I

University

This course provides a complete guide to the statics of particles, a foundational topic in engineering mechanics. It covers the principles of force vectors, resultants, and components, first in two dimensions and then extending to three-dimensional space. The core of the course is the analysis of particles in equilibrium, establishing the conditions required for a particle to remain at rest under the influence of multiple concurrent forces.

Statics is the essential science behind structural and mechanical design. The principles covered are the basis for designing every stable structure, from bridges and skyscrapers to aircraft frames and engine components. A command of statics is non-negotiable for any engineer, as it provides the fundamental tools required to analyse forces and ensure the safety and integrity of physical systems.

By the end of this course, you will be able to resolve any force into its components and calculate the resultant of any system of concurrent forces in both 2D and 3D. You will also be able to draw a complete free-body diagram for any particle and apply the equations of equilibrium to solve for unknown forces, tensions, and reactions in a variety of engineering problems.

This course is for undergraduate students in any engineering discipline, including civil, mechanical, and aerospace. It is a core component of any engineering mechanics curriculum and assumes a prerequisite knowledge of introductory physics and vector mathematics. It is the essential foundation for subsequent courses in dynamics and the mechanics of materials.

statics-of-particles-engineering-mechanics-undergraduate

Statics of Particles - Engineering Mechanics (Undergraduate Advanced)

This course is a comprehensive guide to the effects of forces on rigid bodies, a critical topic in engineering mechanics. It moves beyond particles to analyse systems where the point of application of a force is crucial, introducing the concepts of moments and couples. The material covers the calculation of moments in two and three dimensions, the properties of couples, and the methods for reducing complex force systems into simpler, statically equivalent systems.

Understanding moments is fundamental to all structural and mechanical analysis. These principles are used to calculate the bending and twisting stresses in beams, shafts, and machine components, and to ensure the rotational stability of structures and vehicles. A command of this subject is non-negotiable for designing any object intended to withstand real-world loads without failing or rotating unintentionally.

By the end of this course, you will be able to calculate the moment of a force about any point or axis using both scalar and vector methods. You will also be able to determine the moment of a couple, and reduce any complex system of forces and couples acting on a rigid body to its simplest equivalent form, such as a single resultant force and couple moment at a specified point.

This course is for undergraduate students in any engineering discipline who have already completed a course on the statics of particles. It is the direct prerequisite for the study of the equilibrium of rigid bodies, and is a necessary foundation for advanced courses in dynamics, mechanics of materials, and structural analysis.

systems-of-forces-rigid-bodies-engineering-mechanics

Systems of Forces on Rigid Bodies - Engineering Mechanics (Undergraduate Advanced)

Equilibrium is the foundation of all structural and mechanical analysis. This course provides a rigorous study of the equilibrium of rigid bodies in two and three dimensions. It covers reactions, equilibrium equations, constraints, and statical determinacy, supported by extensive worked examples.

Engineers, architects, and applied scientists constantly rely on equilibrium principles to design safe, efficient, and reliable systems. From bridges and buildings to machines and vehicles, every stable structure depends on correct modelling of supports, forces, and constraints. Mastery of equilibrium ensures accuracy in analysis and prevents costly or dangerous failures.

By completing this course, you will be able to draw free-body diagrams, apply equilibrium equations in 2D and 3D, and classify and analyse different support and constraint conditions. You will understand when a system is properly constrained, over-constrained, or indeterminate, and you will have practical experience solving a wide range of equilibrium problems.

This course is intended for undergraduate students in mechanical, civil, and structural engineering, but it is also useful to anyone studying applied mechanics at an advanced level. It provides both a theoretical foundation and practical problem-solving skills valuable to practising engineers, technical instructors, or anyone needing precise understanding of equilibrium in rigid body mechanics.

equilibrium-of-rigid-bodies-engineering-mechanics

Equilibrium of Rigid Bodies - Engineering Mechanics (Undergraduate Advanced)

Analysis of trusses, frames and machines.

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Analysis of Structures - Engineering Mechanics (Undergraduate Advanced)

Analysis of equilibrium of rigid bodies subjected to dry (Coulumb) friction, and its applications to screws, wedges, belts and bearings.

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Dry Friction - Engineering Mechanics (Undergraduate Advanced)

Analysis of centroids and moments of inertia.

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Centroids and Centres of Gravity - Engineering Mechanics (Undergraduate Advanced)

Analysis of the second moments of areas and its applications.

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Moments of Inertia - Engineering Mechanics (Undergraduate Advanced)

Analysis of equilibrium of rigid bodies by the method of virtual work.

method-of-virtual-work-engineering-mechanics

Method of Virtual Work - Engineering Mechanics (Undergraduate Advanced)

Statics decides if a structure stands or fails. This track teaches force calculation, support reactions, and equilibrium analysis for stable engineering systems. You will move from particle resolution to full structural modelling. The curriculum delivers the exact mathematical foundation required for real design work.

This programme is for first and second-year engineering undergraduates. It suits mechanical, civil, aerospace, and structural students, plus polytechnic learners. You only need basic physics and vector algebra. The sequence provides the precise calculation methods your degree demands.

You will finish able to resolve two and three dimensional forces, compute moments and couples, and solve rigid body equilibrium problems. You will calculate truss loads, evaluate friction in screws and belts, locate centres of gravity, and verify systems using virtual work. These results secure high grades in advanced mechanics modules and prepare you for structural design, machine engineering, and civil site roles.

Engineering Mechanics I

Statics

Engineering Mechanics I - Statics

This course provides a complete toolkit for the study of infinite sequences and series. It begins with a rigorous treatment of the convergence of sequences and the algebra of limits. The course then covers the foundational theory of infinite series and provides a comprehensive study of all standard tests for convergence, culminating in an introduction to power series.

The concepts of convergence and infinite series are essential in physics, engineering, and computer science. They are used to solve differential equations, analyse signals with Fourier series, calculate probabilities, and determine the accuracy of numerical approximations. This is the mathematical machinery behind precision in scientific modelling.

By the end of this course, you will be able to determine if an infinite sequence converges and find its limit. You will also be able to apply the complete suite of convergence tests???including the Integral, Comparison, Ratio, Root, and Alternating Series tests???to determine if an infinite series converges, and find the radius and interval of convergence for any power series.

This course is for students who have completed a foundational calculus course. It is the standard curriculum for a second-semester calculus (Calculus II) module and is a direct prerequisite for the study of differential equations, complex analysis, and advanced physics.

convergence-of-infinite-sequences-and-series-advanced

Convergence of Infinite Sequences and Series - Advanced Calculus (Undergraduate Advanced)

This course provides the formal, proof-based foundation for the concepts of convergence covered in calculus. It moves beyond computation to explore the deep theoretical questions of why sequences and series converge. The material covers the topological properties of sequences, key convergence theorems like Bolzano-Weierstrass, Cauchy sequences, and the advanced topic of uniform convergence for series of functions.

This subject is the gateway to higher mathematics. The rigorous proof-writing and analytical skills developed here are non-negotiable for any student pursuing a degree in pure or applied mathematics. These principles are the theoretical bedrock upon which advanced concepts in differential equations, functional analysis, and theoretical physics are built.

By the end of this course, you will be able to construct rigorous proofs for statements about the convergence of sequences. You will also be able to apply the Monotone Convergence and Bolzano-Weierstrass theorems, use the Cauchy Criterion to prove convergence, and analyse the uniform convergence of a series of functions using the Weierstrass M-Test.

This course is for mathematics majors and advanced students who have already completed a full Calculus II course. It is the standard curriculum for a first module in Real Analysis and is an essential prerequisite for graduate-level study in mathematics, physics, and theoretical computer science.

analysis-of-infinite-sequences-and-series-real-analysis

Real Analysis

Analysis of Infinite Sequences and Series - Real Analysis (Undergraduate Advanced)

This course provides a complete guide to the calculus of several variables. It builds from the foundational concepts of multivariable functions, limits, and continuity to the core techniques of differentiation, including partial derivatives, the chain rule, and implicit differentiation. The material culminates in advanced topics such as Taylor's theorem for several variables and the use of Jacobians.

Multivariable calculus is the language of modern science, engineering, and economics. Its principles are used to model complex surfaces, analyse thermodynamic systems, create 3D computer graphics, and solve critical optimisation problems in business and finance. This is the mathematical toolkit for working with systems that have multiple interacting variables.

By the end of this course, you will be able to calculate partial derivatives, apply the multivariable chain rule, and find directional derivatives using the gradient vector. You will also be able to solve both unconstrained and constrained optimisation problems by finding extreme values and using the method of Lagrange multipliers, and apply these derivatives to find tangent planes to surfaces.

This course is for students who have completed a full single-variable calculus sequence. It is the standard curriculum for a multivariable calculus (Calculus III) module and is a direct prerequisite for the study of vector calculus, differential equations, and advanced courses in physics, engineering, and economics.

partial-differentiation-and-its-applications-multivariable

Partial Differentiation and Its Applications - Multivariable Calculus (Undergraduate Advanced)

This course covers numerical methods for solving problems that have no exact analytical solution. It is a practical study of iterative approximation techniques. The material is focused on two core areas: finding the roots of single-variable equations using the Bisection and Newton's methods, and approximating definite integrals using the Trapezoidal and Simpson's rules.

Numerical methods are the foundation of all modern computational problem-solving in science, engineering, and finance. These are the algorithms that power computer simulations, engineering design software, and financial modelling tools. The ability to find approximate solutions to complex equations is a fundamental skill for any technical or scientific professional.

By the end of this course, you will be able to apply the Bisection and Newton's methods to find the roots of single-variable equations. You will also be able to use the Trapezoidal and Simpson's rules to calculate the approximate value of definite integrals, providing a core toolkit for numerical approximation.

This course is for undergraduate foundation students in engineering, physics, and computer science. It is an essential topic for anyone who will use computational tools to solve mathematical problems. A solid understanding of single-variable calculus is a prerequisite.

introduction-to-numerical-methods-undergraduate-foundation

Numerical Methods

Introduction to Numerical Methods (Undergraduate Foundation)

This course provides a complete guide to solving ordinary differential equations (ODEs). It covers the classification of differential equations and details the solution methods for first-order equations, including separable, homogeneous, exact, and linear types. The course then moves to second-order linear equations with constant coefficients and covers the methods of undetermined coefficients and variation of parameters.

Differential equations are the mathematical language used to model dynamic systems in science and engineering. They are used to describe the motion of objects, the flow of electric circuits, population growth, radioactive decay, and chemical reaction rates. A command of this subject is a non-negotiable requirement for any serious study in physics, engineering, or applied mathematics.

By the end of this course, you will be able to classify any ordinary differential equation by its order, degree, and linearity. You will be able to solve a wide variety of first-order ODEs and constant-coefficient second-order ODEs. You will also be able to model and solve real-world problems, such as orthogonal trajectories, exponential growth and decay, and simple electric circuits.

This course is for undergraduate students in mathematics, physics, engineering, and chemistry. It is a standard module in any mathematical methods curriculum and assumes a full prerequisite knowledge of single and multivariable calculus. It is an essential foundation for the study of partial differential equations, control theory, and advanced physics.

ordinary-differential-equations-mathematical-methods

Ordinary Differential Equations - Mathematical Methods (Undergraduate Advanced)

This learning track delivers the complete mathematical toolkit required for a university-level science, engineering, or computing degree. It systematically covers the entire MTH 201 curriculum, building from the foundational principles of single-variable calculus - functions, limits, continuity, and differentiability - to the advanced methods of multivariable calculus, infinite series, numerical methods, and ordinary differential equations. This is the definitive preparation for advanced quantitative study.

This programme is designed for second-year students offering MTH 201 at Obafemi Awolowo University, Ile-Ife, Nigeria. It is also helpful for any student in a STEM field - including physics, engineering, and computer science - who requires a rigorous and comprehensive command of calculus and its applications.

This track delivers a full skill set in mathematical analysis and applied problem-solving. Graduates will be able to solve a wide range of problems, from optimising multivariable functions to modelling dynamic systems with differential equations and testing the convergence of infinite series. This programme directly prepares students for success in advanced courses in vector calculus, partial differential equations, and real analysis, providing the necessary foundation for a career in engineering, data science, or theoretical physics.

Master the non-negotiable principles of Engineering Mechanics. This track delivers a rigorous, complete programme in statics and dynamics, built to the NUC GET 207 core curriculum. It moves methodically from force systems and equilibrium to the kinematics of rigid bodies and the kinetics of particles. This is the foundation of all structural and mechanical analysis.

This programme is for first and second-year undergraduate engineering students. It is an essential requirement for students in Mechanical, Civil, Structural, Aerospace, and Mechatronics engineering. A working knowledge of introductory physics and calculus is assumed.

On completion, you will be able to analyse and solve complex problems in statics and particle dynamics. You will draw free-body diagrams, apply equilibrium equations, analyse trusses and frames, and solve motion problems using Newton's laws, work-energy, and momentum methods. This programme prepares you for advanced courses, particularly Mechanics of Materials, and future professional engineering practice.

Applied Mechanics

GET 207: Applied Mechanics

Advanced linear algebra is the mathematical backbone of modern data science, engineering, and physics. This learning track delivers the rigorous MTH 205 curriculum based on NUC CCMAS standards, focusing on sophisticated matrix analysis and practical computational methods critical for solving complex technical problems.

This programme is targeted at undergraduates in mathematics, engineering, and computer science requiring a deep command of advanced matrix theory. It is equally essential for data scientists and engineers seeking a rigorous theoretical foundation for machine learning algorithms, cryptography, and complex system modelling.

You will master matrix manipulations to solve linear systems and compute determinants and inverses efficiently using various methods including software like Python and MATLAB. You will gain competence in determining eigenvalues and eigenvectors, applying diagonalization to analyze the stability of dynamical systems, and working with quadratic and canonical forms. Completion establishes the critical mathematical expertise required for advanced studies in multivariate statistics, differential equations, and algorithmic development.

Linear Algebra II

MTH 205: Linear Algebra II

Master the algebraic structures that underpin modern science and computation. This academic track delivers the complete NUC CCMAS MTH 204 curriculum, moving rigorously from abstract vector spaces to practical matrix theory. It provides the essential mathematical toolkit required for advanced problem-solving in high-demand STEM fields.

This programme is targeted at undergraduates in mathematics, engineering, and computer science requiring a firm grounding in linear structures. It also serves professionals in data science, cryptography, and machine learning needing a rigorous theoretical refresher on foundational concepts.

You will achieve competence in manipulating abstract vector spaces, determining basis and dimension, and analyzing linear transformations through their kernels and images. You will master matrix arithmetic, compute determinants, solve systems of linear equations using advanced methods, and apply techniques of eigenvalues and diagonalization. Completion establishes the critical foundation demanded for advanced studies in multivariate calculus, differential equations, and complex computational algorithms.

Linear Algebra I

MTH 204: Linear Algebra I

Integration forms the base of modern calculus, allowing engineers and scientists to calculate area, volume, and accumulated change. This course covers the core principles and techniques of integral calculus across ten modules. You will study fundamental antiderivatives, the fundamental theorem of calculus, basic power rules, and standard integrals for trigonometric, exponential, logarithmic, and hyperbolic functions. The content moves through systematic solution methods including variable substitution, pattern recognition, integration by parts, partial fraction decomposition, trigonometric substitution, and reduction formulae.

Strong calculus skills are essential for solving practical problems in engineering, physics, economics, and data science. Learners apply these methods to calculate centres of mass, determine work done by variable forces, model population growth, and analyse signal processing systems. Mastery of these integral techniques prepares students for advanced university modules such as differential equations, fluid dynamics, and structural analysis.

By completing this course, you will learn to choose and apply the correct solution method for any integrand. You will master the LIATE framework for integration by parts, decompose rational expressions into partial fractions, and apply the Weierstrass transformation for trigonometric quotients. Furthermore, you will calculate reduction formulae, use the shortcut of Wallis, transform limits during substitution, and apply parity properties of functions to evaluate definite integrals efficiently.

This course is written for Nigerian secondary school leavers preparing for higher education, first-year university undergraduates in STEM fields, and diploma students in polytechnics. Self-taught learners and older students who need to rebuild their mathematical foundation before tackling advanced courses will also profit from this material. The clear structure equips students with the exact techniques required to pass university calculus examinations.

fundamentals-and-techniques-of-integration-calculus

Fundamentals and Techniques of Integration - Calculus (Undergraduate Foundation)

Vectors form the foundation of physics and engineering mechanics. This course covers the fundamental operations required to manipulate and solve vector equations at the university level. We examine scalar products, vector products, scalar triple products, and vector triple products. You will learn the definitions, geometric meanings, properties, and core formulas of these operations. The content includes extensive worked examples to ensure you understand how to apply the theories of vector mathematics to solve complex algebraic equations.

You need this knowledge to understand forces, fields, and spatial geometry in physical sciences. Engineers and physicists use these exact principles to calculate work done by a force, determine magnetic field interactions, and model three-dimensional structures. Without mastering the products of vectors, you will struggle with advanced mechanics and electromagnetism. This course prepares you to apply vector operations directly to physical projects and academic problems, ranging from determining collinearity in structural design to identifying linear dependence in computer graphics.

By the end of this programme, you will master specific technical skills and logical frameworks. You will learn to calculate scalar and vector products, determine the direction cosines of a vector, and test vectors for perpendicular or parallel alignment. You will also compute scalar and vector triple products, evaluate the linear dependence of multiple vectors, and interpret the geometric meaning of these mathematical operations. Finally, you will develop the capacity to solve multi-variable vector equations using structured algebraic techniques and the fundamental properties of vector multiplication.

This course is primarily for Nigerian secondary school leavers and early university students entering science, technology, engineering, or mathematics fields. It provides the exact mathematical foundation required for first-year undergraduate physics and advanced mathematics modules. It also benefits self-taught programmers and industry professionals who need a rigorous refresher on spatial mathematics for game development or engineering modelling. By completing this content on UniDrills, you gain direct, practical competence in university-level vector mathematics.

scalar-vector-and-triple-products-of-vectors-undergraduate

Scalar, Vector and Triple Products of Vectors (Undergraduate Foundation)

Direct calculation of massive equations wastes time and computing memory. This course explains how to solve large systems of linear equations using iterative numerical methods. We cover vector norms, matrix norms, and the spectral radius of a matrix to establish the required mathematics. You will study the general framework for constructing iterative schemes to approximate solutions. The content focuses strictly on these iterative frameworks; direct exact methods are excluded.

Iterative methods drive modern computer simulations. Modelling heat flow or calculating forces in a bridge truss generates thousands of equations. Engineers use these exact numerical techniques to write efficient code for physical analysis. You will learn to build algorithms that solve complex models without exhausting the memory of the system. This knowledge is mandatory for careers in software engineering, applied physics, and civil infrastructure projects.

You will write algorithms for the Jacobi iterative technique and the Gauss-Seidel iterative technique. You will evaluate convergence conditions for these iterative methods using strict mathematical rules. You will check for diagonal dominance in system matrices to predict if a method will succeed. You will also calculate the spectral radius of the iteration matrix to prove convergence definitively. You will master the logic to set an initial guess and iterate until the error is acceptable.

This course is primarily for early university students in Nigeria studying engineering, mathematics, or computer science. Recent Nigerian secondary school leavers with a strong grasp of matrices will also find it highly accessible for university preparation. It provides a direct path for self-taught programmers needing mathematical grounding to handle heavy computations. Any learner aiming to write faster numerical code will gain practical value from mastering these specific methods.

systems-of-linear-equations-numerical-analysis

Numerical Analysis

Systems of Linear Equations - Numerical Analysis (Undergraduate Advanced)

Many physical systems cannot be modelled using standard elementary functions, making infinite series the only viable analytical approach. This course reviews power series before examining exact analytical solutions around ordinary and regular singular points. We examine the method of Frobenius and the resulting indicial equation to solve complex differential models. The curriculum focuses on exact analytical methods for ordinary differential equations, intentionally omitting numerical approximations and dedicated boundary-value problems.

Engineers and scientists use these mathematical structures to analyse mechanical vibrations, heat conduction, and electromagnetic wave propagation. By mastering these techniques, you will possess the foundation necessary to tackle advanced problems in theoretical physics and engineering mechanics. You can apply these principles directly to your physical projects involving fluid dynamics or structural analysis. This knowledge also serves as a strict prerequisite for postgraduate studies in applied mathematics.

You will formulate series solutions around ordinary and regular singular points using the method of Frobenius and the corresponding indicial equation. You will evaluate the Gamma function, graph the functions of Bessel, and solve the equation of Bessel. Furthermore, you will derive analytical solutions for the equation of Legendre, apply the formula of Rodrigue, and utilise the relevant generating functions. Finally, you will analyse the orthogonality conditions of Sturm-Liouville Systems to establish robust proofs.

This course is primarily for university undergraduates in engineering and physics across Nigeria who have completed foundational calculus. It benefits early university students by bridging the gap between basic differential calculus and advanced mathematical modelling. While the concepts are advanced, the logical progression is straightforward enough to help ambitious secondary school leavers grasp the underlying principles of higher mathematics. Working professionals will also find the rigorous curriculum useful to refresh their theoretical knowledge for complex design work.

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Series Solutions of Ordinary Differential Equations (Undergraduate Advanced)

Periodic functions describe processes that repeat over time. This course covers the mathematics to analyse these patterns. You will study the analytic description of a periodic function and evaluate orthogonal functions. We explain the derivation of Fourier series and Dirichlet conditions. You will examine the effect of harmonics, Gibbs phenomenon, and the sum of a Fourier series at a discontinuity. The module includes odd and even functions, their products, and half-range series.

Engineers use these transforms to process signals in physical projects. You will apply these exact tools to telecommunications and audio analysis. Understanding complex exponentials and complex spectra helps you break waves into basic components. This knowledge is required for academic research in control systems and building physical hardware like electrical filters. You will use these methods to evaluate the stability of signal transmission lines.

You will compute Fourier coefficients for functions with an arbitrary period. You will apply the Fourier integral theorem and evaluate special functions, including the top-hat function, the Dirac delta, and the triangle function. You will master the properties of the Fourier transform; these include linearity, time shifting, frequency shifting, time scaling, symmetry, differentiation, and convolution. You will calculate Fourier sine transforms and the Hilbert transform.

This course targets engineering undergraduates but remains accessible to sharp Nigerian secondary school leavers preparing for university. If you are an early university student seeking direct explanations of calculus, this provides the exact foundation required. Professionals needing a strict refresher on complex Fourier series will also benefit. The instruction delivers the exact steps to solve advanced problems in your career.

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Fourier and Hilbert Transforms - Calculus (Undergraduate Advanced)

Physical phenomena like heat transfer and wave propagation are governed by partial differential equations. This course details the formulation and solution of these equations using specific initial and boundary conditions. We focus on direct integration and the method of separation of variables in two and three dimensions. This reduces partial equations into ordinary differential eigenvalue problems. You will examine the wave equation, the heat conduction equation for a uniform finite bar, and the equation of Laplace in standard and plane polar coordinates.

Engineers use these models to predict the behaviour of structures. You will apply this knowledge to evaluate the flexural motion of beams, analysing free and forced vibration for structural assessment. Calculating heat distribution in solid objects or predicting wave patterns allows professionals to build safe mechanical systems. These methods supply the mathematical foundation required for advanced study and practical projects.

You will master the logical frameworks necessary to construct solutions from basic physical principles. You will solve simple partial differential equations using direct integration and evaluate boundary value problems using integral transforms. You will reconstruct a full solution as a series expansion in eigenfunctions. You will formulate wave and heat conduction equations, extract ordinary differential eigenvalue problems, and apply these techniques to continuous physical systems.

This course targets university engineering students needing a strict mathematical foundation. It is highly accessible to Nigerian secondary school leavers preparing for university. Mastering these concepts provides a strong advantage for students entering demanding academic disciplines. It also benefits learners outside the primary target group, such as engineering technicians, by supplying theoretical justification for physical responses observed in field operations.

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Introduction to Partial Differential Equations (Undergraduate Advanced)

Geophysics relies on strict mathematical models to map the physical properties of the Earth. This learning track delivers the exact calculus, vector algebra, and differential equations required to process spatial data and simulate physical systems. You will study the progression from limits and integration to the calculus of vector fields and numerical analysis. The programme translates theoretical mathematics into direct calculations for wave propagation and fluid dynamics.

This programme targets 200-level university students in geophysics and related programmes following the standards of the NUC CCMAS. Ambitious Nigerian secondary school leavers preparing for rigorous STEM degrees will also find the structure accessible. It serves as the required bridge between foundational secondary mathematics and the heavy computational demands of university mechanics. Working professionals needing a direct mathematical refresher will also benefit.

Upon completion, you will solve complex systems of linear equations and manipulate scalar and vector fields accurately. You will compute partial derivatives, evaluate multiple integrals, and formulate series solutions for ordinary and partial differential equations. These skills allow you to analyse seismic waves, model heat transfer, and write efficient algorithms. You will finish fully prepared to pass advanced university modules and tackle computational tasks in the engineering industry.

Geomathematics

GPH 211: Geomathematics

Tutorials for 200-level students, based on NUC CCMAS, institution-specific and general syllabuses.

Find Your Classes

200-Level University Classes

GET 209: Engineering Mathematics I

MEE 206: Engineering Mechanics II - Dynamics

MTH 202: Mathematical Methods II

MTH 201: Mathematical Methods I

Engineering Mechanics I - Statics

MTH 201: Mathematical Methods I

GET 207: Applied Mechanics

MTH 205: Linear Algebra II

MTH 204: Linear Algebra I

GPH 211: Geomathematics

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