This chapter provides a foundational classification of chemical transformations. It introduces the main types of chemical reactions, providing a framework for predicting the products of simple reactions.

Key learning objectives include: identifying and providing examples of the common reaction types, such as synthesis, decomposition, and displacement.

Chemical Reactions

This chapter provides the roadmap for the course. It outlines the foundational concepts of chemistry to be covered, from the scientific method and measurement to the classification of matter and reactions.

Key learning objectives include understanding the overall course structure and appreciating how these distinct topics interconnect to form the basis of chemical science.

Introduction

This chapter covers the fundamental classification of matter as required by the syllabus. It provides the formal definitions for atoms, molecules, elements, compounds, and mixtures, and explores the changes matter undergoes.

Key learning objectives include: defining and differentiating between the classifications of matter; and distinguishing between physical and chemical changes.

Matter

This chapter consolidates your mastery of the non-negotiable concepts required for advanced study in the chemical sciences, acting as a final knowledge checkpoint before you proceed.

Key objectives are to review all core concepts and outline the immediate next steps required in your university-level chemistry curriculum.

Conclusion

Chemistry is the science of matter. This course provides a rigorous introduction to its fundamental principles, moving from the definition of matter and its states to the atomic and molecular structures that govern its behaviour. We will classify matter into elements, compounds, and mixtures, and establish the critical distinction between physical and chemical changes.

A command of chemistry is essential for any quantitative science. The principles covered here are foundational to medicine, engineering, environmental science, and materials development. Understanding how matter is structured and how it transforms is the basis for creating new materials, developing pharmaceuticals, and managing industrial chemical processes safely and efficiently.

Upon completion, you will possess a clear conceptual framework for chemistry. You will be able to classify different types of matter, distinguish between physical and chemical changes, and define elements, compounds, and mixtures. You will also be able to identify the fundamental roles of atoms, molecules, and ions in chemical reactions.

This course is designed for first-year university students beginning their studies in any scientific or engineering discipline. It serves as a mandatory prerequisite for further study in chemistry, chemical engineering, biology, and medicine. It is also suitable for anyone requiring a concise and formal review of foundational chemical concepts.

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

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 

NUC CCMAS

This learning track delivers the complete NUC CCMAS curriculum for General Chemistry I. It is a comprehensive programme designed to build a robust, university-level foundation in modern chemistry. The track systematically covers all essential topics, from atomic theory, chemical bonding, and the states of matter, to the quantitative principles of stoichiometry, equilibrium, thermodynamics, and kinetics.

This programme is for first-year undergraduates in science, technology, engineering, and mathematics (STEM) faculties who are required to take CHM 101. It is also essential for any student or professional globally who needs a rigorous and complete foundation in first-year university chemistry for further study or career development.

This track delivers a full skill set in chemical theory and quantitative problem-solving. Graduates will be able to determine molecular structures, calculate reaction quantities, analyse the energetics and rates of reactions, and solve complex equilibrium problems. This programme provides the non-negotiable prerequisite knowledge for all subsequent chemistry courses and for any degree in the physical sciences, engineering, or medicine.

General Chemistry I

introduction-to-chemistry-undergraduate-foundation

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.

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.

This learning track provides a complete and rigorous treatment of introductory classical mechanics as specified by the NUC Core Curriculum. It is structured to build a comprehensive analytical framework, starting with the mathematical description of motion (kinematics) and progressing through its causes (Newtonian dynamics), the powerful conservation laws, the dynamics of rotating systems, and finally, the principles of universal gravitation. Mastery of this material is the non-negotiable foundation for all subsequent study in physics and engineering.

The principles of classical mechanics are the operational language for analysing the physical world. This track provides the essential toolset for solving problems in every field of engineering, from aerospace to civil, and for understanding phenomena from the trajectory of a projectile to the orbits of planets. By the end of this track, you will be able to analyse motion using vectors and calculus, apply Newton's laws to solve any standard dynamics problem, use conservation laws to analyse complex systems and collisions, analyse rotational motion, and solve problems in celestial mechanics.

This learning track is a mandatory prerequisite for all first-year university students of physics, engineering, and related physical sciences. It provides the foundational knowledge required for all subsequent courses in mechanics, electromagnetism, thermodynamics, and modern physics.

General Physics I

Mechanics

introduction-to-physics-undergraduate-foundation

Physics

Conditional permutation deals with arrangements limited by specific rules. You will learn to count outcomes when items must stay together or stay apart. This logic is essential for advanced scheduling and security where specific conditions apply.

You will master the tie-together method for items that stay together; apply the gap method for items that must stay apart; solve problems with fixed positions; and calculate arrangements with restricted letter and digit placements.

Conditional Permutation

Permutation with identical objects covers arrangements where some items are not distinguishable. Swapping identical items creates no new pattern, so you must divide to find unique results. This is essential for calculating arrangements in words, passwords, and digital data accurately.

You will master the formula for n items with repetitions; calculate arrangements for words with repeated letters; solve grid-based path problems; and use factorial division to remove duplicate counts.

Permutation with Identical Objects

This chapter introduces the logic of counting required for all arrangement and selection problems. You will master the foundational rules used to calculate total outcomes without manual listing, forming the essential groundwork for the entire course.

You will understand and apply the product rule for sequential events; the sum rule for separate choices; the difference rule for total minus unwanted cases; and the division rule to remove duplicate counts.

Permutation focuses on the arrangement of objects where the specific order is essential. You will learn to calculate total outcomes when selecting and ordering items from a set, a skill required for creating secure PINs and scheduling tasks. This chapter provides the mathematical formulas needed to handle arrangements of all available items or just a specific portion of them.

You will master the use of factorials to count arrangements of all n objects; apply the nPr formula to arrange r items from a larger set; and solve practical problems involving distinct objects in fixed positions.

Permutation

Cyclic permutation covers arrangements in a circle where the starting position is not fixed. You will learn to remove repeated patterns to ensure accurate counting. This section explains the difference between arrangements viewed from one side and those that can be turned over.

You will master the formula for circular arrangements where clockwise and anticlockwise orders differ; calculate unique patterns for items that can be flipped like beads; and solve problems involving symmetry.

Cyclic Permutation

This chapter applies combinatorial logic to solve geometry problems involving points, lines, and shapes. You will learn to calculate the number of possible geometric figures that can be formed from a given set of points, which is essential for spatial reasoning and advanced coordinate geometry.

By the end of this chapter, you will master calculating the number of straight lines from points; determining the number of triangles and quadrilaterals; and finding the total diagonals in a polygon.

Geometric Applications

Derangements cover scenarios where no item in a set returns to its original position. This chapter explains how to count failed matches, such as putting letters into wrong envelopes or seating people in different chairs. Mastering this prevents overcounting and solves problems where specific items must not occupy their natural spots.

You will learn the derangement formula, calculate subfactorials for small sets, solve problems with restricted positions, and apply these rules to real-world logic puzzles.

Derangements

Combination focuses on selection where order does not matter. It is essential for picking committees or groups where picking A then B is the same as B then A. You will learn to identify selection tasks to avoid overcounting and ensure accurate results.

You will master the nCr formula; solve committee problems with group restrictions; calculate lines and triangles in geometry; and handle selections from multiple sets.

Combination

Stop counting one by one. This course provides the mathematical tools to calculate arrangements and selections with speed. You will master the fundamental counting rules including product, addition, subtraction, and division. The curriculum covers linear and cyclic permutations, identical objects, and restricted placements where items must be together or apart. We conclude with combinations, derangements, and the calculation of shapes within geometric frameworks.

Counting is the engine of probability and modern computing. These skills are vital for cryptography, software development, and statistical analysis. You will apply these methods to calculate network paths, secure passwords, and optimise logistics. In any technical career, the ability to quantify possibilities is the difference between guessing and knowing.

You will solve complex arrangement problems using standard formulas and logical shortcuts. You will handle specific constraints like fixed positions or items that cannot be next to each other. You will learn to distinguish between permutations and combinations in practical scenarios. You will also calculate the number of lines, triangles, and diagonals in geometric figures using combinatorial principles.

This foundation is for undergraduate students and secondary school leavers aiming for high scores in competitive exams. It is also essential for professionals in engineering, finance, and data science who require a refresher on discrete mathematics. Anyone seeking to improve their logical deduction and analytical speed will find value here.

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

permutation-and-combination-mathematics-undergraduate

Mathematics

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.

OAU, Ife

Advanced engineering mathematics covering numerical and analytical methods of engineering analysis.

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 II

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

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

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

Mathematical Methods I

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.

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

This comprehensive learning track guides you through the complete world of vector analysis. We begin with the fundamentals of vector algebra and its application to foundational geometry. You will then master scalar, vector, and triple products before using them to construct the vector equations of lines, planes, and conics. The journey culminates in advanced topics, including vector calculus, its applications in classical mechanics, and an introduction to differential geometry.

Vectors are the essential language used to describe our physical world, making their mastery non-negotiable for any serious student of science or engineering. This track is designed to build your intuition for spatial reasoning and equip you with a powerful problem-solving toolkit. You will see direct applications in mechanics, analyzing forces and motion; in geometry, calculating angles and distances; and in calculus, modeling dynamic change over time.

While this track is tailored to the first-year university curriculum for MTH 104 at Obafemi Awolowo University, Ile-Ife, Nigeria, it is an invaluable resource for a wide range of learners. It is ideal for any undergraduate student in mathematics, physics, engineering, or computer science seeking a comprehensive understanding of vector analysis. Furthermore, it serves as an excellent and thorough refresher for professionals who wish to solidify their foundational knowledge of this critical subject.

Elementary Mathematics III

Vectors, Geometry and Dynamics

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

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.

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.

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

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

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

Linear Algebra

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!

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

Calculus

Meaning of mechanics, dynamics, kinematics, kinetics and other terms; general problem solution method; units of measurement.

Analysis of rectilinear motion with variable acceleration.

General Rectilinear Motion

Analysis of relative independent motion of particles, and absolute and relative dependent motion of connected (or constrained) particles.

Special Rectilinear Motion (2)

Analysis of the curvilinear motion of particles using radial and transverse components.

Curvilinear Motion (3)

Analysis of the curvilinear motion of particles using rectangular (Cartesian) coordinates.

Curvilinear Motion (1)

Analysis of the curvilinear motion of particles using components normal and tangential to the trajectory of motion.

Curvilinear Motion (2)

Analysis of rectilinear motion with constant acceleration - uniform motion and uniformly-accelerated motion.

Special Rectilinear Motion (1)

Analysis of erratic rectilinear motion problems using graphs relating the motion parameters.

Graphical Analysis

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

kinematics-of-particles-engineering-mechanics

This chapter establishes the rigorous language of kinematics. It defines the core quantities of motion ??? displacement, velocity, and acceleration ??? with a critical focus on the distinction between their scalar and vector forms. This foundation is the prerequisite for all subsequent analysis.

Key objectives: distinguish between scalar and vector quantities; define average and instantaneous acceleration; and use differentiation to relate displacement, velocity, and acceleration.

This chapter consolidates the core concepts of kinematics. It provides a structured summary of the mathematical tools and physical principles used to describe motion, reinforcing the foundation for the study of dynamics.

The conclusion summarises the key definitions and equations. It also provides a forward look to the next course, where Newton's laws will be used to explain the causes of motion.

This chapter extends kinematic analysis to two and three dimensions. It establishes the general vector framework and calculus-based methods required to describe motion in a plane and in space. This mathematical toolkit is the direct prerequisite for analysing specific cases like projectile motion.

Key objectives: describe position, velocity, and acceleration using standard vector notation; and apply differentiation and integration to these vectors to analyse motion with variable acceleration.

Motion in Two or Three Dimensions

This chapter provides a complete analysis of projectile motion, the primary application of two-dimensional kinematics. The core principle is the complete independence of horizontal motion (at constant velocity) and vertical motion (at constant acceleration).

Key objectives: apply the principle of independent components to projectile problems; derive and use the equations for range, maximum height, and time of flight; and solve projectile trajectory problems.

Projectile Motion

This chapter introduces the vector framework required to relate motion across different inertial frames of reference. Mastering these transformations is critical for solving complex real-world navigation and tracking problems where both the object and the observer are in motion.

You will master four key objectives: applying the subscript notation for relative velocity; calculating resultant velocities in one dimension; resolving perpendicular vector components in two dimensions; and determining optimal headings for aircraft and maritime transit.

Relative Motion

This chapter examines uniform circular motion, a special case of two-dimensional motion where an object's speed is constant but its velocity is not. We will introduce the concept of an acceleration that serves only to change the direction of the velocity vector.

Key objectives: define the descriptive quantities of circular motion (period, speed); explain the nature and direction of centripetal acceleration; and apply its formula to solve problems.

Uniform Circular Motion

This chapter applies kinematic principles to the simplest case: motion along a straight line. We will cover motion with variable acceleration using calculus, and then focus on the critical case of constant acceleration and its set of standard equations. Mastery of 1D motion is the required foundation for all further analysis.

Key objectives: use calculus to relate displacement, velocity, and acceleration; apply the kinematic equations to solve constant acceleration problems; and analyse vertical motion under gravity.

Motion in One Dimension

Motion is the most fundamental concept in physics. This course provides the mathematical framework to describe it with quantitative precision. We will systematically analyse motion in one, two, and three dimensions, progressing from motion along a straight line to the vector analysis of projectiles and general spatial motion. This requires a full command of both vector algebra and introductory calculus.

The principles of kinematics are the bedrock of physical analysis, engineering design, and biomechanics. This knowledge is required to analyse everything from vehicle performance and projectile trajectories to the movement of organisms and robotic motion. A command of this material is essential for any work involving the dynamics of moving systems, providing the tools to predict and control motion in the real world.

Upon completion, you will be able to analyse motion in multiple dimensions. You will solve one-dimensional problems using the standard kinematic equations for constant acceleration. You will analyse two-dimensional projectile motion by resolving vectors into independent components. Critically, you will use vector calculus to describe the general case of motion with variable acceleration.

This course is designed for first-year university students of the physical, biological, and medical sciences, as well as engineering and computer science. A firm command of vector algebra and introductory calculus (differentiation and integration) is a mandatory prerequisite. This material is the necessary foundation for the subsequent study of Newtonian dynamics and biomechanics.

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This final chapter is a mandatory comprehensive assessment designed to confirm full mastery of the course material. Successful completion ensures you can confidently apply both the theoretical principles of the kinetic theory and the practical calculations derived from all gas laws. This is the required preparation for 'Stoichiometry I'.

Key objectives include: application of the Ideal Gas Equation and combined gas laws; explaining gas behaviour using kinetic postulates; and validating understanding of the states of matter.

This chapter establishes the course roadmap and introduces the Kinetic Theory of Matter. The theory provides the essential microscopic framework - the motion of particles - required to understand and explain the macroscopic properties and behaviour of gases studied in the following chapters.

Upon completion, you will be able to: outline the core postulates of the Kinetic Theory of Matter and appreciate the direct link between particle motion and measurable physical properties like pressure and temperature.

This chapter systematically applies the Kinetic Theory of Matter to explain the states of matter. We establish how the balance between particle kinetic energy (temperature) and intermolecular forces dictates whether a substance exists as a solid, liquid, or gas, governing phase changes.

Upon completion, you will be able to: differentiate the three states of matter based on particle movement, explain the microscopic basis for phase transitions (boiling, melting), and use the kinetic theory to describe the influence of temperature on state.

Kinetic Theory

This chapter establishes the core empirical laws that describe gas behaviour, starting with the kinetic explanation for pressure. Command of these relationships is essential for accurately calculating unknown state variables and understanding the practical limits of gas systems in engineering and physical science applications.

Key objectives include: applying Boyle's, Charles's, Avogadro's, and Gay-Lussac's Laws; deriving and calculating gas density; and mastering the Ideal Gas Equation, PV = nRT, for complex state variable analysis.

Gas Laws

This course provides a rigorous treatment of the Kinetic Theory and the Gas Laws, explaining the physical principles that control the behaviour of solids, liquids, and gases. We move from the foundational kinetic postulates to the precise mathematical relationships required to model matter's response to energy, pressure, and temperature.

These principles dictate real-world engineering, not just textbook theory. You will apply this knowledge to model atmospheric dynamics, design efficient chemical reactors, and optimise engine performance. A command of these laws allows you to quantify substance behaviour under varying conditions, solving the calculation pain points often faced in process engineering and physical science.

By the end of this course, you will confidently apply Boyle's, Charles's, Avogadro's, and the Combined Gas Laws to determine unknown state variables. You will derive and utilize the Ideal Gas Equation, PV = nRT, and explain gas properties using kinetic theory postulates. Additionally, you will analyse the structure of solids and calculate gas density. 

This is a mandatory foundation for Chemistry and Chemical Engineering undergraduates and a prerequisite for thermodynamics. It is intended for learners with a grasp of chemical bonding who require a structured, student-centred alternative to disjointed free resources.

kinetic-theory-of-matter-and-gas-laws-chemistry

This concluding chapter summarises the core concepts of chemical measurement. It reinforces the understanding of the mole and chemical formulae.

This summary prepares the student for the next course - Stoichiometry of Reactions - where these accounting tools will be applied to chemical equations.

This chapter establishes the mass scale for chemical measurement. We define relative atomic mass using the carbon-12 standard and determine the weighted average mass of elements based on natural isotopic variance.

You will master the course objectives, the carbon-12 reference definition, and the calculation of weighted atomic mass from isotopic abundance data.

This chapter introduces the single most important quantitative concept in chemistry: the mole. We formally define this SI unit and establish molar mass as the essential conversion factor for all chemical accounting.

You will master the formal definition of the mole, the calculation of molar mass for any substance, and the ability to execute fundamental mass-to-mole conversions.

The Mole Concept

This chapter focuses on converting laboratory mass data into a definitive chemical formula. We cover all analytical methods required to determine a compound's simplest (empirical) and actual (molecular) whole-number atomic ratios.

You will master defining empirical and molecular formulae, performing calculations from percentage and combustion analysis data, deriving the final molecular formula, and solving hydrate analysis problems.

Chemical Formulae

Grasp the fundamental language of chemistry through Stoichiometry of Composition. This concise undergraduate-foundation course systematically establishes core quantitative concepts essential for all subsequent chemistry studies. We will precisely define and calculate relative atomic mass, master the critical mole concept to transition between the microscopic and macroscopic worlds, determine molar masses, and then apply these principles to calculate percent composition and accurately derive both empirical and molecular formulae from experimental data. The content is structured for immediate application, moving from foundational definitions to complex problem-solving.

Quantitative chemistry is not abstract theory; it is the practical basis for chemical synthesis and analysis across science and industry. Learners will develop the ability to accurately predict reactant and product ratios in chemical reactions, interpret laboratory data for compound identification, and correctly formulate materials in fields such as chemical engineering, pharmaceuticals, and materials science. Mastery of these stoichiometric calculations is non-negotiable for success in laboratory work and industrial scale-up.

Upon completion, you will be able to calculate and interpret relative atomic and molecular masses; confidently apply the mole concept to interconvert mass, moles, and number of particles; calculate the molar mass of any compound; precisely determine the percent composition of a substance; and successfully solve multi-step problems to deduce a compound's empirical and molecular formulae. The course culminates in extensive practice to secure computational fluency.

This course is essential for undergraduate students beginning any science or engineering programme that requires a solid foundation in Chemistry, particularly those preparing for university-level coursework. It also serves as an intensive, structured refresher for advanced students, technical professionals needing to reinforce core computational skills, and anyone using UniDrills to self-study for competitive professional examinations.

Courses

Introduction to Chemistry (Undergraduate Foundation)

Introduction to Physics (Undergraduate Foundation)

Permutation and Combination - Mathematics (Undergraduate Foundation)

Ordinary Differential Equations - Mathematical Methods (Undergraduate Advanced)

Vector Algebra and Geometry - Vectors (Undergraduate Foundation)

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

Calculus of Scalar and Vector Fields (Undergraduate Advanced)

Kinematics of Particles - Engineering Mechanics (Undergraduate Advanced)

Kinematics: Motion in One and Two Dimensions - Physics (Undergraduate Foundation)

Kinetic Theory of Matter and Gas Laws - Chemistry (Undergraduate Foundation)

Stoichiometry of Composition - Chemistry (Undergraduate Foundation)

Set Theory - Mathematics (Undergraduate Foundation)