Limits explain function behaviour near specific points, especially where they are undefined. This concept is the mandatory starting point for derivatives and integrals. You will learn to predict output values by observing trends from both sides of an input.

You will master calculating limits using algebraic laws; evaluating one-sided and infinite limits; verifying limit existence; and identifying horizontal and vertical asymptotes to determine end behaviour.

Introduction to Limits

This chapter teaches precise limit evaluation. Direct substitution works first. Algebraic methods then clear undefined forms. These skills build the foundation for all future calculus topics in this course.

You will learn to use direct substitution, factorise expressions to clear zero divisions, rationalise radicals to remove roots, and compare polynomial degrees for infinite limits.

Evaluation of Limits (1)

A function is continuous if its graph can be drawn without lifting your pen. This chapter defines the mathematical conditions required for continuity and introduces existence theorems like the Intermediate Value Theorem. Identifying discontinuities is critical for ensuring the stability of engineering and physical systems.

You will learn the three conditions for continuity; identify different types of discontinuity; solve for constants that make a function continuous; and use theorems to prove the existence of roots.

Continuity

Functions can be combined and reversed to create complex mathematical relationships. This chapter teaches you how to manipulate these models, which is a vital skill for solving multi-step engineering and science problems.

You will master calculating arithmetic combinations of functions; solving and decomposing composite equations; deriving inverses for linear and rational functions; and applying the reverse order rule to invert multi-step operations.

Operations

Graphs convert equations into visual tools for engineering and science. This chapter teaches you to plot functions accurately on coordinate axes to predict trends and solve problems.

Master sketching polynomial, trigonometric, and rational graphs; apply shifts and reflections; identify intercepts and asymptotes; and draw piecewise functions with breaks.

Graphs

This chapter defines functions and rules for mapping inputs to outputs. Accurate modelling requires mastering these basics to prevent errors in advanced calculus. These concepts form the essential bedrock for limits and derivatives.

You will learn to define functions; determine domain and range; and use interval notation. You will also evaluate piecewise functions and simplify the difference quotient.

Introduction

This chapter handles limit problems that break standard substitution. Mastering these edge cases prevents calculation dead ends and builds reliable analytical skills.

You will apply the squeeze theorem to bounded functions, calculate special trigonometric limits, transform shifted angle identities, and verify limit existence at dense piecewise junctions.

Evaluation of Limits (2)

This final chapter consolidates everything you have learned about functions, limits, and continuity. It serves as the final bridge to the next stage of your calculus journey. Reviewing these foundations ensures you are ready for differentiation and integration.

You will summarise the core principles of functions; review limit evaluation techniques; and confirm the conditions for mathematical continuity across different function types.

Conclusion

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.

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.

NUC CCMAS

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

functions-limits-and-continuity-calculus-undergraduate

Formal and informal definitions and proofs of limits at infinity and infinite limits.

Proving Limits Involving Infinity

Meaning of the limits of a real-valued functions - graphical illustrations and formal  (epsilon-delta) definitions.

Different methods of evaluating limits of real-valued functions - direct substitution, theorems, graphing, factorization, conjugates.

Evaluating Finite Limits (1)

Different methods of evaluating limits of real-valued functions at infinity and infinite limits.

Evaluating Limits at Infinity

How to prove that a given finite limit as x approaches a finite value is correct, using the formal or epsilon-delta definition.

Proving Finite Limits

Other methods of evaluating finite limits - L???H??pital???s rule, use of known special limits, squeeze theorem, limits of piecewise-defined functions, etc.

Evaluating Finite Limits (2)

Intuitive methods of evaluating finite limits of real-valued functions.

Evaluating Finite Limits (3)

This course provides a rigorous introduction to the concept of the limit, the theoretical bedrock upon which all of calculus is built. We will move from an intuitive understanding of what it means for a function to 'approach' a value to the algebraic techniques required for precise evaluation. The course covers the limit laws, methods for handling indeterminate forms, and the behavior of functions at infinity.

Understanding limits is non-negotiable for any serious study of calculus. The principles developed here are not merely abstract; they are the tools used to formally define the core concepts of calculus that are used to model the mechanics of change that govern engineering, physics, economics, and computer science.

By the end of this course, you will be able to: evaluate limits graphically, numerically, and algebraically; apply the Squeeze Theorem and L???H??pital???s Rule; analyse the end-behavior of functions; and identify vertical and horizontal asymptotes.

This course is designed for first-year undergraduates in STEM fields who are beginning their calculus sequence. It is an essential prerequisite for subsequent courses on continuity and differentiability and is also invaluable for any student or professional seeking to rebuild their mathematical foundation from first principles.

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

limits-of-functions-single-variable-calculus

Single-Variable Calculus

Continuity of functions of several variables.

Formal and informal definitions, evaluation of limits of functions of several variables.

Limits

Meaning of homogeneous functions; Euler's theorem for homogeneous functions.

Homogeneous Functions

Partial derivatives of composite functions of several variables - chain rule of differentiation.

Composite Functions

Formal definition of differentiability for functions of several variables; notations and evaluation of first and higher-order partial derivatives.

Partial Derivatives

Getting the stationary points of a function of several variables by application of partial derivatives.

Extreme Values

Taylor's theorem for functions of two variables.

Taylor's Theorem

Differentiation of implicitly-defined functions of several variables and an introduction to the Jacobian determinant.

Implicit Differentiation

Some theorems on Jacobian determinants - the implicit function theorem and its implications.

Theorems on Jacobians

Applications of partial derivatives - gradient of a differentiable function and its relation to the directional derivative.

Gradients and Directional Derivatives

Equations of lines, planes, curves and surfaces in three-dimensional Cartesian coordinates - vector, standard and parametric equations.

Applications to Geometry

Meaning of real-valued functions of several variables - domain, range, graphs, level curves, surfaces and computer-aided visualizations.

Getting the stationary points of a function of several variables subject to some condition by application of partial derivatives and the method of Lagrange multipliers.

Lagrange Multipliers

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.

Mastering advanced calculus is essential for modelling complex systems in science and engineering. This track delivers the rigorous mathematical foundation demanded by the official NUC CCMAS curriculum for MTH 201. It systematically builds your expertise from fundamental single-variable theory to the sophisticated multivariable analysis used to solve critical problems in physics, economics, and technology.

This programme is for undergraduates in engineering, mathematics, physics, and computer science requiring a deep theoretical and practical command of calculus. It also serves economics students needing advanced quantitative tools or professionals in finance and data science seeking a solid mathematical base for technical research.

You will gain the analytical skills to construct formal proofs for differentiation rules and apply cornerstone theorems like Mean Value and Taylor's. You will master multivariable techniques, enabling you to solve constrained optimization problems with Lagrange multipliers and compute multiple integrals across line, surface, and volume domains. This track is the requisite preparation for advanced studies in differential equations, vector analysis, and complex engineering modelling.

Mathematical Methods I

partial-differentiation-and-its-applications-multivariable

Multivariable Calculus

Meaning and identification of reflexive relations.

Reflexive Relations

Meaning and properties of the empty relation.

Empty Relation

Introduction to ordered pairs, Cartesian products and relations.

Meaning and identification of symmetric relations.

Symmetric Relations

Meaning and identification of equivalence relations.

Equivalence Relations

Meaning and identification of transitive relations.

Transitive Relations

Meaning and calculation of equivalence classes for a given equivalence relation.

Equivalence Classes

Meaning and examples of congruence relations and residue classes.

Congruence and Residue Classes

Meaning and properties of partially-ordered sets.

Partially-Ordered Sets

Lattices

This course builds up a key part of the foundation for abstract algebra???relations.

We explore equivalence relations, partitions, orderings, and how they help us organize and understand mathematical structures. These ideas connect directly to how we define and work with algebraic systems like groups and rings.

Clear, focused, and made for first-time learners stepping into abstract algebra.

Abstract algebra is the structural foundation of modern advanced mathematics. This track delivers the complete NUC CCMAS MTH 203 curriculum, rigorously transitioning you from computational arithmetic to abstract mathematical reasoning. It provides the necessary prerequisite framework for understanding complex mathematical systems and their applications.

This programme is targeted at undergraduate students in mathematics, computer science, and physics requiring a firm grounding in fundamental algebraic structures. It also serves professionals in fields like cryptography or theoretical computer science who need a rigorous theoretical refresher.

You will master the precise definitions, properties, and relations of core algebraic structures, including groups, subgroups, rings, and fields. You will gain competence in constructing formal proofs using set theory and logic, and understand how homomorphisms preserve mathematical structure. Completion establishes the critical theoretical base demanded for advanced studies in algebra, coding theory, and algorithm design.

Sets, Logic and Algebra I

relations-introductory-abstract-algebra-undergraduate

Introductory Abstract Algebra

Understanding the mean-value and Rolle's theorems.

Theorems on Differentiable Functions

Higher-order derivatives of differentiable functions - meaning, proof of Leibniz's formula and its applications.

Higher-Order Derivatives

Meaning of the differentiability of real-valued functions at a point; illustration by slope of a straight line.

Taylor and Maclaurin series expansion of differentiable functions.

Taylor and Maclaurin Series

Differentiability on an interval and the meaning of derivatives; relating differentiability and continuity of a function.

Derivatives

How to find the derivatives of real-valued functions; rules of derivatives of sums, products and quotients of functions and their proofs.

Rules of Differentiation

This course provides a rigorous, theoretical introduction to differential calculus. We will move beyond computation to understand the fundamental nature of the derivative, beginning with its formal definition as the limit of a difference quotient. The curriculum is designed to build a deep conceptual understanding of what it means for a function to be differentiable and the profound consequences of this property.

The primary focus is on formal proofs and theoretical results. We will systematically derive the rules of differentiation, including the product, quotient, and chain rules, directly from first principles. This course emphasizes the "why" behind the mechanics of calculus, establishing the logical framework upon which all applications are built, including the cornerstone theorems of Rolle, the Mean Value Theorem, and Taylor's Theorem.

By the end of this course, you will be able to: explain the derivative from its limit definition; prove the relationship between differentiability and continuity; formally derive all major differentiation rules; and understand the theoretical significance of the Mean Value Theorem and Taylor's Theorem.

This course is designed for first-year undergraduates in mathematics, physics, and engineering who require a deep theoretical foundation for their studies. It is the ideal precursor to subsequent courses on differentiation techniques and applications, providing the essential logical underpinnings for those more practical subjects.


Real Analysis bridges the critical gap between computational calculus and rigorous advanced mathematics. This learning track delivers the complete NUC CCMAS MTH 207 curriculum, transitioning you from intuitive understanding to formal mathematical proof. It establishes the theoretical foundation required for serious modelling in science, engineering, and pure mathematics.

This programme is targeted at mathematics majors and advanced undergraduates in physics and engineering who have completed foundational calculus. It is designed for students requiring the rigorous analytical skills demanded by graduate-level studies and theoretical research.

You will master the construction of rigorous proofs for sequence and series convergence, applying cornerstone theorems like Bolzano-Weierstrass and Cauchy criteria. You will achieve a formal command of continuity and differentiability, deriving major calculus rules from first principles. Completion provides the non-negotiable prerequisite knowledge for advanced studies in functional analysis, differential equations, and theoretical physics.

Real Analysis I

differentiability-and-derivatives-of-functions

This chapter moves from the general definition of a function to a systematic classification of key function families. We will identify and analyse the core algebraic types - including polynomial, rational, and piecewise functions - that form the basis of mathematical modelling across scientific disciplines.

By the end of this chapter, you will be able to: classify functions as polynomial, rational, or algebraic; determine the natural domain for each of these types; interpret and analyse piecewise-defined functions; and test any function for symmetry.

Kinds of Real-Valued Functions

This chapter establishes the foundational concepts of the real number system. A command of these definitions is a prerequisite for defining and analysing the functions that form the core of this course.

By the end of this chapter, you will be able to: differentiate between key number systems from natural to real; define and represent intervals on the real line; and correctly apply the mathematical concept of infinity.

This chapter introduces the function - the central object of study in calculus. We will formally define what constitutes a real-valued function and establish the critical concepts of its valid inputs (domain) and possible outputs (range).

By the end of this chapter, you will be able to: formally define a function; precisely calculate the domain for various real-valued functions; and understand the meaning and importance of a function's range.

Real-Valued Functions

This chapter covers transcendental functions, the non-algebraic tools used to model natural phenomena like population growth, radioactive decay, and wave mechanics. We will systematically analyse the properties and graphs of exponential, logarithmic, trigonometric, and hyperbolic functions.

By the end of this chapter, you will be able to: differentiate between algebraic and transcendental functions; analyse the domains and graphs of exponential, logarithmic, and trigonometric types; and define their corresponding inverse functions.

Transcendental Functions

This course provides a rigorous foundation in the concept of functions, the building block of calculus. We begin with the real number system - rationals, irrationals, intervals, and infinity - before defining real-valued functions, their domains, and ranges. You will systematically study all key function families: polynomial, rational, algebraic, piecewise-defined, and transcendental, including exponential, logarithmic, trigonometric, and hyperbolic types.

A command of functions is essential for all technical and quantitative disciplines. These principles are directly applied in physics to model motion, in economics for supply-demand curves, in engineering for signal processing, and in computer science to analyse algorithm complexity. This course delivers the core mathematical tools required for effective modelling and problem-solving in these fields and beyond.

By the end of this course, you will be able to: calculate the domain and range of any given function; classify functions and identify their fundamental properties; analyse functions for key characteristics like symmetry (odd and even); and accurately construct graphs for a wide range of standard and piecewise-defined functions directly from their equations.

This course is designed for first-year undergraduates in science, technology, engineering, mathematics (STEM), and economics. It is also critical for any student preparing for advanced calculus or for professionals who require a solid mathematical base for roles in finance, data science, or technical research.

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

This chapter explains the strength of the electric field and the mapping of lines around a charge. It is the basis for calculating invisible forces and predicting the movement of charges in electrical systems.

You will calculate the field of the point charge; apply the principle of superposition to multiple charges; find the location of the null field; and determine the path of the particle in a uniform field.

Point Charge Fields

Gauss's Law links electric flux to enclosed charge. This shortcut replaces hard integration for symmetric shapes. You will learn why field lines matter and how conductors shield internal space. It is the fastest way to solve complex field problems in engineering.

You will calculate flux through closed surfaces; convert charge densities for lines and volumes; apply symmetry to spheres, cylinders, and plates; and determine field distribution inside conductors and insulators using Gaussian surfaces.

Gauss's Law

This chapter ties together Coulomb's law and Gauss's law. It confirms how static charges create fields and how symmetry simplifies calculation. You will see the full picture of electrostatics before moving to advanced topics.

You will recap force between point charges; review electric field intensity; connect flux to enclosed charge; and solidify the link between discrete charges and continuous distributions for accurate problem solving.

This chapter defines the atomic basis of electricity and the fundamental laws governing static charges. These principles are required to calculate electrical forces and understand modern electronic systems.

You will master charge calculations using mass and electron counts, apply conservation and quantisation laws, identify conductors and insulators, and perform charging by induction and conduction.

This chapter quantifies the force between stationary charges. You will master the inverse square law and vector addition to predict electrical interactions. These calculations form the basis for all electrostatic analysis in engineering.

You will calculate net forces using superposition, solve for unknown charges, and analyse equilibrium in systems with springs or pendulums. Practical examples cover charge redistribution and geometric arrangements like triangles.

Coulomb's Law

Real charges occupy space, not points. This chapter uses integration to calculate fields for lines and surfaces. It builds the skills needed to analyse real engineering components like capacitors and wires accurately.

You will integrate linear and surface densities, derive fields for rings and disks, handle non-uniform charge, and apply superposition to solve complex multi-source problems.

Distributed Charge Fields

Opposite charge pairs create unique fields. This chapter links basic rules to devices like capacitors by analysing dipole behaviour in external fields. You will see how separation creates torque and stores energy.

Master the dipole moment vector, inverse-cube field decay, and torque calculation. Learn to compute work done during rotation and distinguish stable from unstable equilibrium positions for precise modelling.

Electric Dipoles

Static charges power every electronic device you use. This course explains how stationary charges create forces and fields without physical contact. You will move from basic atomic structure to advanced field calculations using Coulomb's law and Gauss's law. It covers charge conservation, induction, superposition, and symmetry. You will learn to model invisible forces that act across space. This foundation is critical for understanding all electrical phenomena.

These principles explain lightning, photocopiers, and spray painting. In engineering, this knowledge helps design capacitors and manage electromagnetic interference. You will learn to protect circuits from static discharge and predict high-voltage behaviour. Understanding field distribution around conductors and insulators is vital for real-world applications. This skill set allows you to solve practical problems in power systems and electronics. It connects abstract physics to tangible technology used daily.

By the end, you will calculate force between multiple charges using vector addition. You will determine electric field strength at any point in space for various shapes. You will apply Gauss's law to solve complex problems for spheres, cylinders, and plates. You will explain induction, conduction, and dipole behaviour. These skills enable accurate modelling of electrical interactions in academic and industrial settings. You will master the mathematical tools needed for electromagnetism.

This course is for first-year university students and engineering aspirants. It suits science teachers and technical professionals needing a refresher on core physics laws. Even non-engineers will benefit from the analytical thinking developed here. You will gain a clear understanding of the laws governing modern electronic devices. It builds a strong base for further study in physics or electrical engineering. Anyone interested in how electricity works will find value in this structured approach.

Electricity and magnetism run every home, factory, and phone all over the world. This track builds the technical foundation to master laws governing electrical energy and signals. You will progress from stationary charges to alternating current and electromagnetic waves. It simplifies the NUC CCMAS syllabus into actionable knowledge for solving practical technical problems.

The programme is for first-year university and polytechnic students in engineering or physical sciences. It also serves school leavers preparing for university physics or technical entrance exams. Science teachers and technicians who need a solid refresher on core electrical principles will find the material direct and relevant to their work.

You will learn to calculate electrical forces, design functional DC and AC circuits, and predict how magnetic fields drive motors and generators. You will master the use of Gauss's Law, Kirchhoff's rules, and Maxwell's equations to solve engineering challenges. Completing this track ensures success in PHY 102 exams and prepares you for a career in power systems, telecommunications, or renewable energy.

General Physics II

Electricity and Magnetism

static-charges-and-fields-physics-undergraduate-foundation

Physics

Meaning of differential equations, order, degree, solutions, etc., of differential equations.

Applications of first-order ordinary differential equations.

Applications of First-Order Ordinary Differential Equations

Analytical solutions of homogeneous linear second-order ordinary differential equations with constant coefficients.

Solutions of Second-Order Ordinary Differential Equations (1)

Analytical solutions of non-homogeneous linear second-order ordinary differential equations by the methods of undetermined coefficients and variation of parameters.

Solutions of Second-Order Ordinary Differential Equations (2)

Analytical solutions of first-order ordinary differential equations, such as variable-separable equations, homogeneous equations, non-homogeneous equations convertible to homogeneous forms, exact differential equations, inexact differential equations, linear differential equations, Bernoulli equation, and Riccati equation.

Solutions of First-Order Ordinary Differential Equations

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.

Mastering differential equations is essential for modelling dynamic systems in science and engineering. This learning track delivers the complete MTH 202 curriculum based on NUC CCMAS standards, equipping you with the mathematical command to describe motion, analyse electrical circuits, and predict rates of change across physical phenomena.

This programme is targeted at undergraduates in mathematics, physics, engineering, and chemistry who possess a strong background in single and multivariable calculus. It also serves professionals requiring a rigorous, method-focused refresher on fundamental mathematical modelling tools.

You will achieve competence in classifying equations and deploying solution methods for first-order, reducible higher-order, and general linear ordinary differential equations. You will learn to solve systems of linear ODEs and apply these techniques directly to real-world physical and technical problems. Completion establishes the necessary foundation for advanced studies in partial differential equations, control theory, and advanced physics.

Elementary Differential Equations

ordinary-differential-equations-mathematical-methods

Mathematical Methods

Mean centre (geometric centre) of a number of points; weighted mean centres.

Centroids

Definitions of scalars, vectors and tensors; representation of a vector by a directed line segment; kinds of vectors - free, localized, equal, null, unit, like and unlike vectors.

Position Vectors

Ratio Division of a line internally and externally; collinearity of points.

Division of a Line

Projection of a vector on another vector; projection of a vector on a plane.

Vector Projections

Meaning of vector components; resolution of vectors into components in two and three dimensions; unit vectors, direction cosines and angle between two vectors in the three-dimensional Cartesian coordinate system.

Vector Components

Vector addition - triangle and parallelogram laws; multiplication of a vector by a scalar; relations on mid-points of sides of a triangle; vector algebra on quadrilaterals and other polygons; parallel vectors; laws of vector algebra.

Vector Algebra

Vectors are the primary tool for describing quantities with both magnitude and direction. This course provides a complete foundation in their algebraic and geometric properties. We move systematically from the basic definition and classification of vectors to the core operations of vector algebra: addition and scalar multiplication. The curriculum then progresses to essential geometric applications, including position vectors, Cartesian components, direction cosines, the division of lines, vector projections, and centroids.

A command of vector algebra is not optional; it is essential for any technical or scientific discipline. This knowledge is the bedrock of classical mechanics, electromagnetism, and fluid dynamics. Engineers use these principles to analyse forces, computer scientists use them to build 3D graphics engines, and data scientists apply them in advanced linear algebra. This course provides the indispensable mathematical toolkit required for these fields.

Upon completion, you will be able to perform all fundamental vector operations with precision. You will resolve vectors into Cartesian components and use direction cosines. You will solve geometric problems involving position vectors, the internal and external division of lines, and collinearity. Furthermore, you will master the calculation of vector projections and the determination of centroids in geometric systems.

This course is designed for first-year undergraduate students in Engineering, Physics, Mathematics, and Computer Science. It serves as a critical foundation for anyone beginning studies that rely on applied mathematics. It is also a rigorous and efficient refresher for professionals or advanced students who need to solidify their understanding of foundational vector principles before tackling more complex material.

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

vector-algebra-and-foundational-geometry-vectors

Vectors

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!

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.

Courses

Functions, Limits and Continuity - Calculus (Undergraduate Foundation)

Limits of Functions - Single-Variable Calculus (Undergraduate Foundation)

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

Relations - Introductory Abstract Algebra (Undergraduate Advanced)

Differentiability of Functions - Single-Variable Calculus (Undergraduate Foundation)

Functions and Their Graphs - Single-Variable Calculus (Undergraduate Foundation)

Static Charges and Fields - Physics (Undergraduate Foundation)

Ordinary Differential Equations - Mathematical Methods (Undergraduate Advanced)

Vector Algebra and Geometry - Vectors (Undergraduate Foundation)

Calculus of Scalar and Vector Fields (Undergraduate Advanced)

An Introduction to Modern Web Development Concepts and Tools (Professional Practice)

Kinematics of Particles - Engineering Mechanics (Undergraduate Advanced)