NUC CCMAS

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.

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Linear Algebra

Matrices, Determinants, and Systems of Linear Equations - 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)

Learn to analyze the three-dimensional Euclidean space using coordinates and equations with pin-point precision.

Designed for students, researchers, and professionals in fields such as engineering, computer graphics, physics, astronomy, medicine, biology, architecture, and building technology, this course provides a thorough understanding of coordinate systems, equations, and quadric surfaces.

On completing this course, you will have mastered:

- One-dimensional coordinate system and locating points

- Two-dimensional coordinate system, lines, and curves

- Conic sections and their equations

- Three-dimensional coordinate system and quadric surfaces

- Computer-aided visualization using Geogebra

- Coordinate transformation and rotation

- Equations in polar, cylindrical, and spherical coordinates

Throughout the course, you'll engage with high-quality video lessons, completely-solved examples, free-hand sketching techniques, and computer-aided visualizations. Multiple-choice quizzes will keep you focused and help reinforce your learning.

Prerequisites:

This course is designed for first-year undergraduate students in engineering and sciences. Prior knowledge of basic trigonometric ratios, determinants, and elementary row operations on matrices is helpful but not required, as the course is designed to be accessible to learners with varying backgrounds.

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.

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Analytic Geometry

3D Coordinates and Quadric Surfaces - Analytic Geometry (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.

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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.

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Applications of Partial Differentiation - Multivariable Calculus (Undergraduate Advanced)

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

Vectors

Differentiation and Integration of Vectors (Undergraduate Foundation)

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.

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Calculus

Multiple Integration - Calculus (Undergraduate Advanced)

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)

Vector calculus connects spatial shapes with their rates of change. This course explains the foundation of advanced calculus using vector integral theorems. We study conservative vector fields, independence of path, and the fundamental theorem of line integrals. We also cover the theorem of Green, the theorem of Gauss, and the theorem of Stokes alongside the calculation of flux integrals over oriented surfaces.

Mastering these concepts is necessary for solving spatial problems in engineering. Students use this knowledge to simplify the calculation of surface integrals in structural analysis. By understanding the physical interpretation of flux out of a closed surface and circulation along a boundary, learners can model spatial forces accurately. This mathematical framework allows engineers to build precise models for physical structures and mechanical projects.

By the end of this programme, learners will evaluate line integrals via double integrals over an enclosed region using the theorem of Green. You will learn to apply the fundamental theorem of line integrals, analyse conservative vector fields, and compute flux integrals across oriented surfaces. Students will also master the two-dimensional Divergence theorem and the theorem of Stokes to translate complex surface integrals into simple boundary calculations.

This course is primarily for early university students and Nigerian secondary school leavers preparing for engineering degrees. It also provides foundational value for aspiring spatial data analysts requiring a strict mathematical background. The curriculum directly benefits these learners by translating complex university calculus into simple logic. Completing this programme gives students the exact technical competence required to pass advanced examinations and solve spatial engineering problems.

vector-integral-theorems-calculus-undergraduate-advanced

Vector Integral Theorems - Calculus (Undergraduate Advanced)

Complex numbers extend the standard number system to solve equations that lack real solutions. This course covers the core theory of functions of a complex variable. We study the definition of complex numbers, their graphical representation, basic arithmetic, and roots. We also examine point sets, limits, and continuity for complex functions. The material includes differentiability, the complex derivative, the Cauchy-Riemann equations, and regular functions. We exclude topics like contour integration to focus strictly on foundational analysis.

Understanding complex functions is essential for advanced engineering and physics. Students apply this knowledge to model physical systems like the flow of fluids, alternating currents, and quantum mechanics. These principles dictate how engineers analyse signals and build control systems for machines. A firm grasp of these mathematical tools allows learners to translate difficult physical problems into straightforward algebraic steps.

You will learn to evaluate limits and verify the continuity of complex functions. You will apply the Cauchy-Riemann equations to determine if a function is analytic and connect analytic functions to harmonic functions. You will compute results using elementary functions of a complex variable, including exponential, logarithmic, trigonometric, and hyperbolic extensions. You will also locate the zeroes of a complex function, identify singularities, and state the fundamental theorem of algebra.

This course targets early university students and recent Nigerian secondary school leavers preparing for advanced mathematics. It also benefits practising engineers and programmers who need to review the theory of complex analysis for technical projects. The direct language makes difficult concepts clear for young learners while maintaining the strict standard required for global university programmes.

functions-of-complex-variables-complex-analysis

Complex Analysis

Functions of Complex Variables - Complex Analysis (Undergraduate Advanced)

Complex mapping transforms geometric shapes into simpler forms to solve difficult problems. We start with the mapping of a straight line under a transformation. You will examine linear transformations of the form w = az + b and nonlinear transformations. The course covers the mapping of regions, the conditions for conformal transformation, and the analysis of critical points. We conclude with the Schwarz-Christoffel transformation and the mapping of open polygons.

Engineers use conformal mapping to solve problems in fluid dynamics and heat conduction. When complex boundaries make calculation impossible, transforming the space simplifies the task. You will apply these techniques to analyse airflow or heat flow through irregular plates. Converting complicated shapes into simpler geometries gives you tools to solve physical problems. This knowledge supports advanced studies, allowing you to model systems accurately.

You will master the conditions required for a transformation to be conformal by maintaining angles between intersecting curves. You will identify critical points where conformality fails because the value of the derivative is zero. You will apply formulas to map regions between the z-plane and the w-plane. You will use the Schwarz-Christoffel formula to map the upper half of the complex plane onto the interior of an open polygon to solve boundary problems.

This course targets Nigerian secondary school leavers transitioning into university physics or engineering programmes. Early university students will find this approach helpful for moving from basic calculus to mathematical modelling. People outside this group, like working engineers reviewing spatial mathematics, will also benefit from the clear explanations. Mastering these transformations builds a strong foundation for future technical coursework and practical projects.

conformal-mapping-complex-analysis-undergraduate-advanced

Conformal Mapping - Complex Analysis (Undergraduate Advanced)

Complex integration extends standard calculus to the complex plane to turn difficult real integrals into simple algebraic sums. This course covers the core mechanics of line integrals, contour integration, and the theorem of Cauchy. You will learn to deform contours around singularities and expand functions using the series of Maclaurin, Taylor, and Laurent. The material addresses poles, removable singularities, the circle of convergence, multivalued functions, and branch cuts. It strictly excludes conformal mapping and harmonic functions to focus entirely on integration methods.

The methods taught here provide exact solutions for complex physics and engineering problems. Engineers and scientists use contour integration to evaluate stubborn real definite integrals that appear in signal processing, fluid dynamics, and electromagnetism. By mastering these techniques, you acquire the mathematical tools necessary to solve practical equations in advanced university projects and professional scientific analysis.

You will master specific analytical tools by the end of the classes. You will evaluate line integrals over complex contours; apply the theorem of Cauchy to simplify calculation; compute residues at simple and multiple poles; and use the residue theorem to evaluate real definite integrals. You will also determine the radius of convergence for power series, classify the types of singular points, and manage multivalued functions by defining accurate branch cuts.

This course is primarily for university students in science and engineering programmes. A dedicated Nigerian secondary school leaver preparing for a university degree in mathematics or physics will also gain a massive head start here. Even if you are slightly outside this core group, such as a self-taught programmer building engineering software, the strict logic of complex analysis will sharpen your ability to break down and solve hard technical problems.

complex-integration-and-residues-complex-analysis

Complex Integration and Residues - Complex Analysis (Undergraduate Advanced)

Engineering systems rely on advanced mathematics to model physical forces and structural behaviours. Engineering Mathematics III covers linear algebra, multivariable calculus, vector calculus, and complex analysis across twelve structured courses. You will master matrices, eigenvalues, three-dimensional coordinates, partial differentiation, vector integral theorems, and complex integration to solve multi-variable engineering problems.

This learning track targets three hundred level undergraduate students in engineering and technology following the NUC CCMAS. It is built for learners who have completed foundational calculus and linear algebra and need rigorous mathematical training to handle complex technical calculations in their core professional examinations and academic projects.

Upon completion, you will possess the analytical skills to evaluate determinants, compute eigenvalues, solve partial differential equations, apply vector integral theorems, and perform contour integration. You will be fully prepared to tackle advanced engineering coursework, solve spatial mechanics problems, and build accurate mathematical models for practical technical applications.

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 III

GET 301: Engineering Mathematics III

Tutorials for 300-level students, based on the general NUC-approved curriculum (NUC CCMAS) for all Nigerian universities.

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