This chapter covers determinants, the specific values used to identify singular matrices and solve linear systems. Mastering these is vital for calculating inverses and analysing matrix properties.

You will master evaluating determinants of any order; applying minors and cofactors; using algebraic properties to simplify calculations; and solving linear systems using Cramer's rule.

Determinants

This chapter covers matrix transposition — the swapping of rows and columns. Mastery of these patterns is essential for simplifying calculations in engineering and science.

You will define transpose notation; identify symmetric and orthogonal types; compute conjugate transposes; and classify Hermitian and unitary matrices.

Transposition of Matrices

This chapter covers elementary row and column operations to simplify complex matrices. Mastering these transformations is vital for finding the rank, nullity, and reduced row echelon form of any matrix.

You will master executing elementary row and column transformations; identifying equivalent and elementary matrices; reducing matrices to row echelon and reduced row echelon forms; and calculating the rank and nullity of a matrix.

Elementary Transformations

This chapter moves matrix theory from paper to software. Mastering spreadsheet tools is essential for automating calculations and handling large data sets in professional environments.

You will perform matrix arithmetic and transposes; calculate determinants and inverses; and solve square systems of linear equations using built-in spreadsheet functions.

Computer-Aided Handling

This chapter establishes the basic building blocks of matrix theory. Recognising these structures is essential for solving complex linear systems and performing matrix transformations in advanced mathematics.

You will define matrix order and notation; identify row, column, and null matrices; distinguish between square, scalar, and identity types; and classify triangular, banded, and sparse matrices.

Introduction

This chapter covers solving multiple equations using matrix algebra. Mastering these methods is essential for identifying solution types and modelling complex systems in physics and engineering.

You will master converting equations to matrix forms; testing consistency with ranks; solving homogeneous and nonhomogeneous systems; and using pivots to handle infinite solutions.

Systems of Linear Equations

This chapter covers matrix inversion and its core uses. Mastering the inverse is vital for solving square systems of equations and reversing matrix operations.

You will compute matrix adjoints; determine invertibility; apply inverse properties to products and transposes; solve square linear systems; and define similar matrices.

Matrix Inverses

This chapter establishes the core arithmetic of matrices. These operations provide the necessary tools for solving linear equations and handling large data sets in professional fields.

You will define matrix equality; perform addition and subtraction; apply scalar multiplication; execute matrix multiplication; and identify key properties like non-commutativity and associativity.

Algebra of Matrices

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.

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

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.

OAU, Ife

Advanced engineering mathematics covering solid analytical geometry, integrals, scalar and vector fields, matrices and determinants and complex variables.

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

Engineering Analysis I

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

NUC CCMAS

Master the mathematical language of engineering. This programme delivers the complete analytical toolkit required for a successful engineering career, covering single-variable calculus, multivariable calculus, linear algebra, and vector analysis. It provides the essential foundation for all subsequent engineering courses.

This programme is for second-year undergraduate students across all engineering disciplines. It delivers the official NUC CCMAS curriculum for Engineering Mathematics, providing the core training required for advanced modules in mechanics, thermodynamics, and circuit theory.

Model and analyse complex physical systems using calculus, linear algebra, and vector analysis. You will be equipped to solve problems in dynamics, statics, and field theory, providing the quantitative proficiency required for advanced engineering study and professional practice.

Engineering Mathematics I

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

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

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science.

This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions.

Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Elementary Mathematics I

Algebra and Trigonometry

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

Linear Algebra

Definition of a space, linear vector space, with examples.

Linear combinations of two or more vectors and linear spans of vector spaces.

Linear Combinations and Spans

Linear dependence and independence of vectors in a vector space.

Linear Dependence and Independence

Meaning, bases and dimensions of vector spaces derived from matrices - row, column and null spaces.

Matrix Spaces

Sums and intersections of subspaces of a vector space - their basis, dimension and other properties.

Sums and Intersections

Meaning of coordinates of a vector in a vector space with respect to a given basis, and how to find the coordinates.

Coordinates

Definition and examples of linear vector subspaces.

Vector Subspaces

Basis and dimension of linear vector spaces.

Basis and Dimension

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

This chapter builds the logic of differentiation from scratch. You will learn why derivatives work before using shortcuts. Mastering these foundations prevents errors in complex calculations later.

You will define the derivative via limits, distinguish continuity from differentiability, and prove the power and sine rules. Practical examples cover quadratics, reciprocals, roots, and trig functions.

This chapter consolidates your knowledge of differentiation techniques. It ensures you have the technical foundation needed for the upcoming applications course.

You will review the hierarchy of rules, identify the best technique for any function, and confirm mastery of successive differentiation.

Conclusion

Stop using first principles for every sum. This chapter gives you the standard rules to find gradients fast. You will learn to handle constants, powers and roots without stress. These shortcuts save time and reduce errors in complex calculations. They are the basic tools for all future calculus work.

You will master the power rule for positive and negative indices; convert roots and fractions into easy powers; apply linearity to split long sums; and simplify algebraic terms before differentiating. These skills let you solve polynomial problems quickly and accurately.

Fundamental Rules

Functions multiplied or divided need special rules. The power rule fails here, so you must learn new techniques to find gradients accurately.

Master the product and quotient rules; handle triple products and trigonometric fractions; apply shortcuts for reciprocals; evaluate derivatives at specific points.

Product and Quotient

This chapter provides standard derivatives for trigonometric, exponential, logarithmic, hyperbolic and absolute value functions. These fixed results allow you to solve routine physics and engineering problems without manual proofs.

You will derive key rates from first principles; apply rules to reciprocals and inverses; and handle mixed products. You will memorise standard forms for quick recall during calculations.

Common Derivatives

Functions inside functions require the chain rule. This chapter teaches you to differentiate composite expressions without expanding them, a critical skill for handling complex real-world models in engineering and physics.

Master the general formula and derived shortcuts for powers, roots, exponentials, and logs. Learn to combine this with product rules and solve multi-layer trigonometric and inverse trigonometric problems efficiently.

Chain Rule

Standard rules fail for complex curves. This chapter teaches methods for equations where variables are mixed, defined by parameters, or have variable powers. These techniques handle real-world engineering problems that basic algebra cannot solve.

You will master implicit differentiation for hidden relationships, parametric gradients for motion, and logarithmic differentiation for variable exponents. Learn to simplify tough expressions using log laws before calculating derivatives.

Advanced Techniques

This chapter measures the rate of change of the gradient. It is vital for calculating acceleration and analysing curvature in engineering and physics. You will move beyond simple slopes to understand how functions behave over successive changes.

Master second-order derivatives, successive differentiation, and nth derivative patterns. Learn to handle implicit and parametric forms. Apply these skills to verify solutions for differential equations.

Higher-Order Derivatives

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.

Calculus is the mathematical tool for measuring change and finding the best results in any system. This track follows the official NUC CCMAS MTH 102 curriculum to teach you how functions behave, how to calculate exact rates of change, and how to sum up tiny movements to find total areas or volumes. You will move from basic limits to complex integration techniques used to solve practical problems in the physical world.

This track is for first-year university students in engineering, science, and economics departments across Nigeria. It also serves secondary school leavers preparing for technical degrees or university entrance exams. Anyone needing to build a strong foundation in mathematical logic for data analysis or professional licensing will find these lessons essential.

After finishing this programme, you will calculate derivatives using first principles and shortcuts, find turning points for optimisation, and solve definite integrals for land measurement or tank design. You will use the trapezium and Simpson's rules to handle experimental data accurately. These skills ensure you pass your university exams and excel in technical careers like civil engineering, physics research, or financial forecasting.

Elementary Mathematics II

Calculus

Management and administration require precise numerical logic for decision-making. This track follows the official NUC CCMAS syllabus for AMS 102, covering real numbers, set theory, and complex systems. You will master algebraic operations, permutations, and combinations alongside the foundations of trigonometry and calculus. It provides the mathematical tools to model financial trends, handle business data, and solve optimisation problems in corporate settings.

This programme is built for first-year university students in Accounting, Business Administration, Banking, Finance, and Public Administration. It also serves secondary school leavers preparing for management degrees and professionals needing a refresher on business mathematics. It is suitable for anyone moving into roles that demand accurate quantitative analysis and logical deduction.

You will gain the ability to simplify complex business formulas, calculate probabilities for risk assessment, and find the exact points where profits are highest. You will be able to handle sequences for interest calculations and use differentiation to manage production costs. Completing this track ensures success in university examinations and builds the analytical strength required for careers in financial analysis, auditing, and strategic management.

Basic Mathematics

fundamentals-and-techniques-of-differentiation-calculus

Meaning, operations and properties of eigenvalues and eigenvectors of matrices.

Eigenvalues and Eigenvectors

Diagonalization of matrices; evaluating polynomials and transcendentals of matrices.

Diagonalization of Matrices

Quadratic and canonical forms; transformations using symmetric matrices and orthogonal diagonalizing matrices.

Quadratic and Canonical Forms

Welcome and course outline. Review of elementary matrix concepts.

Diagonalization of symmetric matrices - orthogonal diagonalizing matrix, the Gram-Schmidt orthogonalization procedure, and related concepts.

Symmetric Matrices

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

Definition, evaluation and properties of the divergence of a vector field, and an introduction to the Laplacian of scalar and vector fields.

Divergence of a Vector Field

Definition, evaluation and properties of the gradient of a scalar function.

Gradient of a Scalar Field

Definition, evaluation and properties of the curl (spin or rotor) of a vector field.

Curl of a Vector Field

Definitions, examples and some visualizations of scalar and vector fields.

Scalar and Vector Fields

Scalar (dot) and vector (cross) products of vectors using sign functions.

Scalar and Vector Products

Definitions and illustrations of coordinates, coordinate lines and coordinate surfaces for the Cartesian coordinate system, general curvilinear coordinates, and general orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (1)

Definitions and operations of gradient of a scalar field and divergence and curl of a vector field using sign functions and conventions such as the Einstein summation, Kronecker delta and Levi-Civita notations.

Use of Sign Functions

The gradient operator and contravariant bases vectors in general orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (3)

Divergence and curl operators in general orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (4)

The Laplacian operator in general orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (5)

Some special curvilinear coordinates - cylindrical and spherical coordinate systems.

Orthogonal Curvilinear Coordinates (6)

Meaning of right-handed coordinate systems, vector products of covariant base vectors in right-handed systems, length and volume elements in general orthogonal curvilinear coordinate systems.

Orthogonal Curvilinear Coordinates (2)

Meaning of scalars, vectors and tensors; Einstein's summation convention, Kronecker delta and Levi-Civita notations; orthogonal and orthonormal vector bases.

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!

Vector analysis is the mathematical backbone of classical mechanics, electromagnetism, and fluid dynamics. This learning track delivers the complete NUC CCMAS MTH 210 curriculum, rigorously progressing from fundamental vector algebra to the advanced differential and integral calculus of scalar and vector fields used in complex engineering and scientific modelling.

This programme is targeted at undergraduates in engineering, physics, mathematics, and computer science. It provides the essential mathematical toolkit for students entering disciplines that rely on applied mathematics and spatial analysis, and serves as a rigorous refresher for professionals needing to solidify their command of vector principles.

You will master the full spectrum of vector operations including dot, cross, and triple products, and apply them to solve geometric problems and vector equations. You will acquire the skills to analyze the differential geometry of curves using the Frenet-Serret framework and apply the powerful gradient, divergence, curl, and Laplacian operators in various coordinate systems. Completion establishes the critical mathematical foundation demanded for advanced studies in continuum mechanics, electrodynamics, and theoretical physics.

Vector Analysis

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

Force-acceleration analysis of rectilinear motion of a particle.

Rectilinear Motion (1)

Meaning of mechanics, dynamics and kinetics, overview of Newton's second law and impulse-and-momentum methods; units of measurement of mass and force.

Force-acceleration analysis of curvilinear motion of particles using radial and transverse components.

Curvilinear Motion (3)

Force-acceleration analysis of curvilinear motion of particles using normal and tangential components.

Curvilinear Motion (2)

Force-acceleration analysis of curvilinear motion of particles in rectangular coordinates.

Curvilinear Motion (1)

Force-acceleration analysis of absolute and relative motion of particles connected by rigid links or light, inextensible strings passed over pulleys.

Rectilinear Motion (3)

Force-acceleration analysis of absolute and relative rectilinear motion of two or more particles in direct contact.

Rectilinear Motion (2)

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.

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

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

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

Engineering Mechanics

Analysis of motion of particles by relating the linear momentum of the body with the impulse of the forces applied on it.

Linear Impulse and Momentum

Analysis of motion of a particle by relating the kinetic energy of the body with work done by the external forces acting on it.

Work and Energy

A study of collision of particles and their motion before and after collision.

Impact

Analysis of motion of particles under conservative forces by considering the conservation of mechanical energy in the system.

Conservation of Energy

Definition of work. Determining the work done by a force under various conditions.

Work

Welcome to the course and course outline. Review of kinematics and kinetics of particles concepts.

Determining instantaneous power, average power, and efficiency of a force by the principle of work and energy.

Power and Efficiency

Introduction to angular momentum, relation between angular momentum and moment of a force, principle of angular impulse and momentum, conservation of angular momentum.

Angular Impulse and Momentum

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

Meaning of the range of a linear map, with worked examples.

Range

Meaning of the kernel of a linear map, with worked examples.

Kernel

Definition, examples and general properties of linear maps.

Linear Maps

Matrix representation of a linear map, with worked examples.

Matrix Representation

Meaning and properties of the transition matrix between two bases of a vector space.

Transition Matrix

Definition of a map (also mapping or transformation) between two sets, notations and some general properties of maps.

Maps

Further operations on linear maps involving their definition and properties.

Operations on Linear Maps

Analysis of the vector space of all linear maps between two given vector spaces. Addition, scalar multiplication, composition and inverse of linear maps.

Algebra of Linear Maps

Welcome and course outline. Review of elementary matrix and linear vector space concepts.

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

linear-maps-transformations-linear-algebra

This chapter sets the foundation for organic chemistry. It explains how electrons drive bonding and molecular behaviour. You will grasp the core principles needed to predict reaction outcomes and understand stability.

You will learn the course structure, define key electronic terms, and understand the role of valence electrons in bond formation.

This chapter covers quantitative analysis methods for determining elemental composition in organic compounds. Accurate percentage values are essential for verifying purity and establishing empirical formulas in laboratory work.

You will master combustion analysis for carbon and hydrogen, Kjeldahl and Dumas methods for nitrogen, Carius technique for halogens and sulphur, and stoichiometric calculations from experimental data to determine exact mass percentages of each element.

Quantitative Analysis

Pure chemicals are the foundation of valid science. This chapter teaches you to isolate target compounds from complex mixtures using physical properties like boiling point, solubility, and polarity.

Master purity checks, distillation methods, solvent extraction calculations, and chromatographic analysis to ensure your laboratory samples are clean and ready for use.

Purification Methods

This chapter covers the final steps in identifying a compound by calculating its empirical and molecular formulas. This process turns raw analytical data into a usable chemical representation.
You will learn to derive the simplest whole-number ratio of atoms and use molar mass to find the true molecular formula of a substance.

Formula Determination

Electron shifts dictate reactivity. This chapter maps how density moves through sigma and pi systems, defining the electronic landscape of organic molecules.

Master inductive and mesomeric effects, steric hindrance, and their impact on acidity, basicity, and resonance stability to predict chemical behaviour.

Electronic Effects

This chapter teaches you to identify elements inside organic compounds. You will learn standard tests that reveal the chemical makeup of unknown samples, a vital skill for accurate laboratory analysis and quality control in industry.

You will detect carbon and hydrogen using copper oxide; prepare sodium fusion extracts via the Lassaigne test; and identify nitrogen, sulfur, and halogens through specific colour changes with nitroprusside, silver nitrate, and iron(III) chloride.

Qualitative Analysis

Organic chemistry starts with how electrons move and hold atoms together. This course explains electronic theory, covering how bonds form and why electrons sit in specific places within a molecule. You will learn to isolate pure substances from mixtures using standard laboratory methods. We also cover how to identify what is inside a compound and calculate its exact chemical formula through qualitative and quantitative analysis.

Knowing how to purify chemicals is vital for making medicines, refining oil, and ensuring food safety. If you cannot separate a target substance from waste, you cannot manufacture a useful product. These techniques are the same ones used by forensic scientists to solve crimes and by engineers to develop new materials. Mastering these basics allows you to work accurately in any professional science lab or industrial setting.

By the end of this course, you will understand inductive effects, resonance, and how electron distribution dictates chemical reactions. You will be able to perform solvent extraction, chromatography, and distillation to get pure samples. You will also know how to test for specific elements like nitrogen or halogens and use mass data to determine empirical and molecular formulas. These skills ensure your laboratory results are reliable and scientifically sound.

This course is for first-year university students and science enthusiasts who want to move beyond textbook theory into practical lab work. It provides a solid base for anyone entering pharmacy, chemical engineering, or medical sciences. Even if you are just starting out, these lessons simplify complex ideas into clear steps, making it easier to grasp how modern science identifies the invisible components of the world around us.

A passionate teacher of Chemistry who is not just interested in teaching but has a strong desire in seeing students enjoying the beauty of the subject 

Excel in your CHM 102 exams and build a professional career in the global science and engineering sectors. This track follows the NUC CCMAS syllabus, covering carbon bonding, molecular structures, and metal chemistry. You will learn how atoms form fuels and medicines, how to identify pure substances, and why molecular shapes affect biological systems. This programme provides the scientific foundation needed for roles in the oil, gas, and pharmaceutical industries.

This track is for first-year university students in chemistry, engineering, medicine, or pharmacy. It also serves secondary school leavers preparing for university entrance or technical workers needing a refresher on laboratory methods. Anyone starting a science-based degree will find these lessons essential for their academic progress.

You will gain the ability to name organic compounds using IUPAC rules, predict reaction outcomes, and use lab techniques like chromatography and distillation. You will understand how to calculate chemical formulas and explain the behaviour of transition metals. Finishing this programme prepares you for advanced research, quality control roles, and industrial manufacturing.

General Chemistry II

electronic-theory-and-purification-techniques-chemistry

Chemistry

This chapter establishes the course framework. It defines physics as a science of precise measurement and outlines the fundamental approach required for all subsequent topics. Master this section to understand the structure of the course and what is required to succeed.

Key objectives: define the scope and methodology of physics; understand the course structure and assessment; and confirm your command of the required mathematical prerequisites.

Precise measurement is the foundation of physics. This chapter introduces the core concepts of physical quantities and the standards that define them. We will establish the International System of Units (SI), the formal framework required for all subsequent quantitative work.

Key objectives: distinguish between fundamental and derived quantities; identify the seven SI base units; and explain the role of standards in defining physical units.

Measurements

This chapter examines the three fundamental quantities of mechanics: mass, length, and time. We will establish their formal definitions and the SI standards used to quantify them. A precise understanding of these core concepts is the absolute prerequisite for the study of motion.

Key objectives: define mass, length, and time and state their SI units; distinguish clearly between the concepts of mass and weight; and identify the standard instruments used to measure these quantities.

Length, Mass and Time

A physical equation must be dimensionally consistent to be valid. This chapter details the method of dimensional analysis, a critical tool for verifying equations and preventing errors in technical work. Mastery of this technique is a non-negotiable skill for problem-solving.

Key objectives: determine the dimensions of physical quantities; apply the principle of homogeneity to test the validity of equations; and use dimensional analysis to deduce relationships between variables.

Dimensional Analysis

Physical quantities have either magnitude (scalars) or magnitude and direction (vectors). This chapter defines this critical distinction and establishes the complete mathematical framework for vector algebra. Command of vectors is non-negotiable for describing forces, velocity, fields, or any change in physical systems.

You will: distinguish scalars from vectors; resolve vectors into components; add vectors analytically; calculate scalar and vector products; and differentiate vector functions.

Scalars and Vectors

This chapter consolidates the course's core analytical tools. It summarises the complete toolkit ??? dimensional analysis, vector algebra, and vector calculus ??? confirming your command of the non-negotiable foundation for all subsequent study in the physical sciences.

You will: consolidate your understanding of the course's analytical methods; recognise their direct application in mechanics; and verify your readiness for the next physics course.

Physics is not a collection of facts; it is a quantitative language. This course establishes that language. We cover the formal system of measurement and units, the rigorous method of dimensional analysis, the complete framework of vector algebra, and the use of calculus to describe how vector quantities change. This is the essential grammar of science.

The tools in this course have immediate, practical applications. Correct dimensional analysis is a critical technique for verifying equations and preventing catastrophic errors in any technical calculation. Vectors are the required language for describing forces, displacements, velocities, and fields in any engineering or scientific discipline. Mastering this material is the first step to becoming a competent technical professional.

Upon completion, you will command the mathematical toolkit for physics. You will perform dimensional analysis to validate equations. You will resolve vectors into components, calculate vector sums and products, and differentiate vector functions to analyse rates of change, such as deriving velocity from a position vector.

This course is the mandatory starting point for first-year university students of engineering, physics, computer science, and related disciplines. A firm command of secondary school algebra, geometry, and trigonometry is a prerequisite. It is also suitable for professionals who require a rigorous and efficient refresher on the foundational mathematical tools of science.

Courses

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

Linear Vector Spaces - Linear Algebra (Undergraduate Advanced)

Fundamentals and Techniques of Differentiation - Calculus (Undergraduate Foundation)

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

Calculus of Scalar and Vector Fields (Undergraduate Advanced)

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

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

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

Electronic Theory and Purification Techniques - Chemistry (Undergraduate Foundation)

Introduction to Physics (Undergraduate Foundation)

Introduction to Chemistry (Undergraduate Foundation)

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