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.

Conclusion

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.

Introduction

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.

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

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

This chapter sets the foundation for organic chemistry. It explains why carbon is central to life and modern materials, providing the context needed for advanced study.

You will learn the course structure, the history that disproved vital force theory, and the unique properties of carbon that allow it to form complex molecules.

Carbon forms hollow cages called fullerenes. This chapter links basic chemistry to nanotechnology by exploring these unique molecular structures.

Identify the shape of buckminsterfullerene; understand how closed cages differ from flat graphite; and see why these molecules matter in modern science.

Molecular Allotropes

Carbon nanostructures drive modern tech. This chapter links atomic shape to material strength and electronics, showing why size matters in industry. 

You will master the lattice of graphene, the strength of nanotubes, quantum effects at small scales, and the principles of nanochemistry reactivity.

Nanostructures

This chapter explains how electrons in the carbon atom rearrange to allow for stable bonding. It is the basis for understanding the shape and strength of all organic molecules.

You will master the electronic configuration of the atom, the movement of electrons during bonding, and the formation of sp3, sp2, or sp hybrid orbitals.

Hybridisation

This chapter covers the forces that join carbon atoms and dictate molecular shape. Mastering these principles allows you to predict stability, reactivity, and physical properties. Without this foundation, analysing complex organic reactions becomes impossible.

You will map covalent bond types, calculate formal charge, assess electronegativity and polarity, interpret skeletal formulas, and apply resonance rules to identify stable structures.

Bonding

Same element, different forms. This chapter shows how carbon atom arrangement changes material properties. You will see why structure dictates function in real-world applications.

Master the hard tetrahedral network of diamond; the soft conductive layers of graphite; and the disordered useful structure of amorphous carbon like soot and charcoal.

Basic Allotropes

Carbon is the backbone of life and modern technology, yet most students struggle to visualise how its atoms connect. This course strips away the confusion surrounding organic chemistry by focusing on the fundamental rules that govern carbon behaviour. You will move from the basic history of the subject to the precise electronic arrangements that allow carbon to form millions of compounds. We cover hybridisation, bonding types, and molecular geometry with clarity. You will also explore the various forms of carbon, from diamond and graphite to advanced nanostructures like graphene and carbon nanotubes. This foundation is critical for understanding any advanced topic in chemistry or related sciences.

Understanding these concepts is not just about passing exams; it is about grasping the material world. Knowledge of bonding and polarity helps you predict how drugs interact with the body, how plastics are formed, and why certain materials conduct electricity while others do not. Engineers use these principles to design stronger composites; medical students need them to understand biochemistry; and industrial chemists rely on them to create new products. By mastering the structure of carbon allotropes and nanostructures, you gain insight into the technologies driving the future, from battery storage to nanomedicine. This knowledge turns abstract symbols into tangible tools for problem-solving in real-world scenarios.

By the end of this course, you will be able to determine the hybridisation state of any carbon atom and predict the three-dimensional shape of molecules. You will calculate formal charges, draw accurate resonance structures, and assess molecular polarity using electronegativity values. You will distinguish between the different allotropes of carbon and explain their unique physical properties based on their atomic arrangement. You will also understand the basic principles of nanochemistry and the significance of graphene and carbon nanotubes. These skills provide the necessary toolkit for tackling reaction mechanisms and spectroscopy in higher-level courses.

This course is built for first-year university students in chemistry, engineering, pharmacy, and medicine who need a solid start in organic chemistry. It is also ideal for advanced secondary school students preparing for university entrance exams or those who found their previous chemistry classes confusing. Even if you are not pursuing a science degree, this course offers a clear window into the molecular basis of life and technology. The explanations are direct and free of unnecessary jargon, making complex ideas accessible to anyone with a basic understanding of general chemistry. If you want to stop memorising and start understanding, this is the place to begin.

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

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

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

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

General Chemistry II

introduction-to-organic-chemistry-undergraduate-foundation

Chemistry

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.

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.

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.

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

Vector (cross) product of two vectors and its properties.

Vector Products

Vector triple product of three vectors and its properties.

Vector Triple Products

Scalar triple product of three vectors and its properties.

Scalar Triple Products

Scalar (dot) product of two vectors and its properties.

Scalar Products

This chapter consolidates the skills from the scalar, vector, and triple product chapters by introducing methods for solving vector equations. We analyse equations containing either unknown vectors or unknown scalars. Mastery of these techniques translates abstract vector theory into a practical problem-solving tool, which is critical for applied mechanics.

By the end of this chapter, you will master: identifying the appropriate solution techniques for different equation types; solving vector equations for the unknown vector; and solving vector equations to find unknown scalar quantities.

Solving Vector Equations

This concluding chapter applies the vector products mastered previously to geometric analysis in three dimensions. We focus specifically on formulating and analysing the vector equations of straight lines. Mastery of this process is the crucial final step in translating abstract vector algebra into a practical tool for spatial geometry.

By the end of this chapter, you will master: deriving the vector equation of a line using direction and a point; deriving the equation using two points; and determining the shortest perpendicular distance from a point to a line.

Vector Equations of Straight Lines

This course moves beyond basic vector algebra to cover the three critical methods of vector multiplication and their geometric applications. We systematically analyse the scalar (dot) product and vector (cross) product, followed by both scalar and vector triple products. The curriculum concludes by consolidating these skills to solve abstract vector equations and formulate the vector equations of straight lines in three dimensions.

These operations are not abstract; they are the language of physical science and engineering. The scalar product is the standard tool for calculating work done by a force and projecting vectors. The vector product is indispensable for defining torque, angular momentum, and magnetic forces. Mastery of these products allows for precise analysis of forces, rotations, and spatial relationships in real-world systems.

Upon completion, you will calculate scalar, vector, and triple products for any given vectors. You will apply these products to solve geometric problems, including testing for perpendicularity and parallelism and finding the area of a parallelogram. Critically, you will master the techniques required to solve complex vector equations and derive the vector equation of a line using various input conditions.

This course is designed for first-year undergraduate students in Engineering, Physics, and Applied Mathematics. It requires prior knowledge of basic vector addition and scalar multiplication. This programme provides the necessary rigorous foundation in vector products, making it a critical prerequisite for subsequent study in classical mechanics, electromagnetism, and linear algebra.

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

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

Vectors

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

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

Linear Combinations and Spans

Linear dependence and independence of vectors in a vector space.

Linear Dependence and Independence

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

Matrix Spaces

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

Sums and Intersections

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

Coordinates

Definition and examples of linear vector subspaces.

Vector Subspaces

Basis and dimension of linear vector spaces.

Basis and Dimension

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

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

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

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

- Perform operations on vectors using linear combinations and scalar multiplication

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

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

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

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

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

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

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

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

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

OAU, Ife

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

linear-vector-spaces-linear-algebra-undergraduate

Linear Algebra

Electric potential maps the energy landscape of charges. We treat it as a scalar sum to avoid vector complexity and examine equipotential surfaces alongside the unique behaviour of charged conductors in equilibrium.

You will apply the superposition principle to find resultant potentials; calculate system assembly energy for charge groups; locate null points where contributions cancel; and determine the constant potential inside conducting spheres.

Electric Potential

Capacitors in circuits follow strict rules for charge and voltage. This chapter explains how series and parallel connections change total storage and potential difference across components.

You will calculate equivalent capacitance for mixed networks; determine charge distribution in parallel branches; analyse voltage division in series paths; and solve problems involving battery-connected or isolated modifications.

Capacitor Networks

This chapter builds the foundation of electric potential energy and voltage. You will learn why electric force is conservative and how to calculate work done on charges. Mastering these concepts allows you to track energy changes in any electric field.

You will define potential difference; convert between joules and electron-volts; apply conservation of energy to find charge speed; and calculate work done by external agents. These skills prepare you for complex electric potential problems later in the course.

Electric potential and field are two sides of the same coin. This chapter links scalar voltage to vector force, showing how one defines the other in space.

You will derive field from potential gradients; calculate voltage drops in uniform fields; extract vectors from multi-variable functions; and interpret graphical data to find field strength.

Field-Potential Relations

This chapter consolidates the principles of electric potential and capacitance. It connects abstract theory to practical circuit design, ensuring you grasp how energy storage works in real hardware.

You will review calculation methods for complex networks; analyse the effect of dielectrics on field strength; and summarise key formulas for exam success. This final overview cements your ability to solve advanced physics problems with confidence.

Point charges are simple. Real objects like rods and disks have charge spread out. We use integration to find potential for these continuous distributions. This moves you from basic sums to calculus-based physics.

You will integrate for rods, rings, and disks; handle symmetric and off-axis points; solve for non-uniform density. These skills let you calculate voltage for any charged shape.

Continuous Distributions

Insulators reshape electric fields through internal polarisation. This chapter explains how bound charge weakens the net force and boosts capacitance without changing geometry. You will see why dielectrics are vital for safe circuit design and energy storage.

You will calculate the dielectric constant and induced surface charge. We distinguish between isolated and connected scenarios to predict energy shifts. You will also model partial fills and side-by-side materials as series or parallel networks.

Dielectrics

Most capacitors are not flat plates. This chapter covers curved geometries where the electric field changes with distance. You will study cylindrical and spherical designs found in coaxial cables and high-voltage equipment.

You will derive capacitance for coaxial cylinders and concentric spheres. Learn to calculate storage per unit length for transmission lines. Compare scaling effects of length versus radius to optimise design. Solve for charge density and voltage limits in practical systems.

Other Capacitor Geometries

Two opposite charges form a dipole. This chapter shows how to calculate the potential they create at any point in space using superposition and vector maths.

You will derive the exact formula for axial positions; apply the far-field approximation for large distances; and solve problems involving angular dependence.

Capacitance measures how much charge a system holds for a given voltage. This chapter explains the geometric limits of storage in isolated conductors and parallel plates, linking physical dimensions to electrical capacity without relying on complex jargon.

You will calculate capacitance for spheres and plates; derive energy stored in electric fields; analyse how plate separation affects charge and energy under constant voltage or isolation; and solve practical problems involving medical defibrillators and real-world components.

Capacitance

Electric potential governs how charges move and store energy in circuits. This course builds a rigorous foundation in electrostatics by linking field theory to practical capacitor design. You will master the mathematics of potential, understand how continuous charge distributions create fields, and analyse complex capacitor networks with precision.

Engineers use these principles to design circuit boards, power supplies, and medical devices like defibrillators. Students apply this knowledge to solve advanced physics problems and prepare for university examinations. Hobbyists gain the insight needed to build safe electronic gadgets and understand component limits without guesswork.

You will calculate potential from point charges and continuous distributions using integration techniques. The course covers Gauss law applications for cylindrical and spherical capacitors, series and parallel network analysis, and dielectric polarisation effects. You will learn to predict voltage division, charge sharing, and breakdown risks in real-world configurations.

This programme targets undergraduate science and engineering students, secondary school leavers preparing for university physics, and hobbyists building electronic projects. It provides the mathematical rigour required for higher education while maintaining clarity for self-taught learners. Professionals seeking to refresh their core physics knowledge will also find direct value in the structured approach to electrostatics and capacitance.

electric-potential-and-capacitors-physics-undergraduate

This chapter introduces the fundamental language and notation of set theory. Mastering these primary definitions is mandatory for structuring logical arguments and categorising data in all advanced mathematical and computational disciplines.

You will master four objectives: defining sets and elements with precision; using roster and set-builder notation; identifying finite and infinite collections; and applying basic symbols for membership and subsets.

This chapter defines the hierarchical inclusion of sets and the precise conditions for set equality. Mastery of subset logic is critical for constructing rigorous proofs and managing data inheritance in computational architecture.

You will master four objectives: distinguishing between proper and improper subsets; proving set equality through mutual inclusion; calculating the total number of subsets for finite collections; and defining the universal set and empty set within inclusion frameworks.

Subsets

This chapter covers the algebraic manipulation of sets to form new collections through rigorous logical rules. Mastering these operations is foundational for database querying, probability theory, and complex data stratification in engineering and analytical systems.

You will master four objectives: performing unions, intersections, and set differences; calculating absolute and relative complements; applying De Morgan's laws to simplify set expressions; and identifying disjoint sets.

Operations on Sets

This chapter defines the rigorous classification of numerical collections and their hierarchical inclusion. Mastery of these standard sets is mandatory for defining domains in calculus, establishing computational constraints, and ensuring logical consistency in mathematical proofs.

You will master four objectives: categorising natural, integer, rational, and irrational numbers; identifying the real and complex number systems; applying interval notation for continuous subsets; and mapping the nested relationship between numerical sets.

Sets of Numbers

This chapter covers the fundamental laws governing set operations and the formal proof of set identities. Mastering this symbolic logic is essential for simplifying complex boolean expressions and ensuring the validity of analytical arguments in computer science and mathematical engineering.

You will master four objectives: applying idempotent, associative, and commutative laws; implementing distributive and absorption laws; proving identities using De Morgan's laws; and simplifying expressions using identity and complement laws.

Algebra of Sets

This chapter introduces the visual representation of set relationships and the spatial logic of inclusions. Mastery of Venn diagrams is essential for mapping data overlaps, solving combinatorial problems, and verifying set identities through geometric interpretation.

You will master four objectives: constructing diagrams for two and three sets; shaded representation of unions and intersections; interpreting regions of relative and absolute complements; and solving practical survey problems using the principle of inclusion-exclusion.

Venn Diagrams

This chapter defines the construction of ordered pairs and the product of distinct sets. Understanding these mappings is mandatory for defining relations, functions, and the multidimensional coordinate systems used in database schema design and linear algebra.

By the end of this chapter, you will master: constructing ordered pairs and n-tuples; calculating the cardinality of product sets; applying the non-commutative property; and executing distributive operations over unions and intersections.

Cartesian Products

This chapter explains how elements from different sets interact through specific rules. These principles are the building blocks for computer programming and advanced calculus. You will learn to link data points accurately and transform inputs into valid outputs.

You will master identifying relations and functions; performing algebraic and composite operations; classifying injective, surjective, and bijective mappings; and calculating inverse functions.

Relations, Mappings and Functions

Mathematics begins with sets. This course covers everything from basic definitions and membership notations to complex set algebra and De Morgan's laws. You will master cardinality, power sets, and the classification of number systems including rational, irrational, and complex numbers. The curriculum moves from simple operations like union and intersection into element-wise proofs, Cartesian products, and the mechanics of relations and functions.

Set theory is the language of modern data and logic. These concepts are essential for computer programming, database management, and statistical analysis. Understanding functions and mappings allows you to model real-world dependencies in engineering, economics, and the sciences. Mastering these foundations provides the exact logical framework needed to solve complex problems in technology and research.

Upon completion, you will be able to simplify set expressions and solve grouping problems using Venn diagrams and the inclusion-exclusion principle. You will know how to perform element-wise proofs and calculate set cardinalities. You will also gain the ability to evaluate composite functions and prove whether a mapping is one-to-one, onto, or bijective.

This course is designed for undergraduate students and secondary school leavers entering STEM disciplines. It provides a necessary logical foundation for anyone moving into calculus, data science, or advanced mathematics. The clear, direct instruction ensures that any student can develop the systematic thinking required for professional technical roles.

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

Build the mathematical foundation required to excel in social science degrees and data-driven careers. This track follows the official NUC syllabus for DSS 103 to teach sets, number operations, counting methods, logical proofs, binomial expansions, complex numbers, and trigonometry. Each topic connects directly to the quantitative skills you need for statistics, research methods, and economic analysis.

This programme is for first-year university students studying economics, sociology, political science, psychology, geography, or other social science disciplines. It is also for secondary school leavers preparing for university entry and professionals who must strengthen their maths for postgraduate study or analytical roles in government, NGOs, or private sector research.

After completing this track, you will solve equations, handle algebraic proofs, manipulate trigonometric formulas, and apply counting principles to real-world data problems. These skills prepare you for advanced coursework in econometrics, statistical analysis, social research, and policy evaluation. You will have the confidence to tackle quantitative modules and compete for roles in banking, public service, consulting, and development work.

Mathematics for Social Scientists I

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

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Mathematics

This chapter covers basic binomial expansion and Pascal's triangle. Identifying these patterns early allows you to skip long multiplication and build algebraic speed. Accuracy in these fundamentals is required for all advanced scientific computations.

You will learn to find the total number of terms; use Pascal???s triangle for coefficients; observe the symmetry of powers; and expand simple binomials to small positive indices.

This chapter covers the formal binomial formula, combination notation, and rigorous proof by mathematical induction. These tools allow you to find specific terms and coefficients for large powers without full expansion. Mastering these techniques is essential for accurate decimal approximations and university-level algebraic proofs.

You will master the nCr formula for coefficients; calculate the general term of an expansion; identify terms independent of x; prove the theorem using mathematical induction; and solve small decimal approximations.

The Binomial Formula

This chapter extends binomial expansion to expressions with negative and fractional indices. These expansions result in infinite series rather than finite terms, providing a critical tool for approximating roots and reciprocal functions. Engineers and scientists use these series to simplify complex equations where absolute precision is required for very small values.

You will master the general expansion formula for any rational index; determine the range of values for which an expansion is valid; calculate approximations for square and cube roots; and find the first few terms of infinite binomial series.

Negative and Fractional Powers

This chapter focuses on isolating specific parts of a binomial expansion without writing out the full polynomial. It is a critical skill for solving complex engineering and probability problems where only a single value or coefficient is required.

By the end, you will master calculating the general term formula; finding the coefficient of a specific power of x; and identifying the term independent of x. You will also learn to determine middle terms and specific term positions.

Individual Terms

Stop manually multiplying brackets. This course teaches the systematic expansion of binomial expressions using Pascal's triangle and the formal Binomial Theorem. We cover the fundamental proof for positive integers before moving to individual term identification and the general expansion for negative or fractional powers. You will master the mechanics of series expansion from simple squares to complex infinite series.

The Binomial Theorem is a core tool in probability, statistics, and financial engineering. It allows for the approximation of complex functions and the calculation of compounding interest or risk factors. Engineers and data scientists use these principles to manage error margins and optimise algorithms. Understanding these patterns is essential for any career involving quantitative analysis or technical forecasting.

You will expand binomials to any power and identify specific terms within a series without full expansion. You will demonstrate the theorem's proof and apply Pascal's triangle for rapid computation. You will also calculate expansions for non-integer indices and determine the range of validity for these series. These skills enable the simplification of advanced algebraic expressions.

This curriculum is for undergraduate students and secondary school leavers entering science or technology tracks. It provides the necessary foundation for engineering, physics, and economics degrees. Professionals needing a mathematical refresher for data analysis will find the direct approach efficient. Any learner requiring precise algebraic tools for higher education will gain immediate value.

binomial-theorem-mathematics-undergraduate-foundation

Kinematics of curvilinear motion of a point on a rigid body undergoing rotation about a fixed axis, and its applications in gears, pulleys, etc.

Rotation About a Fixed Axis (2)

Kinematics of motion of a rigid body undergoing general plane motion using the velocity of a point relative to another point on the same rigid body.

Relative Velocity Analysis

Kinematics of angular motion of a rigid body (or a line on a rigid body) undergoing rotation about a fixed axis.

Rotation About a Fixed Axis (1)

Acceleration analysis of the motion of a rigid body undergoing general plane motion using the acceleration of a point relative to another point on the same rigid body.

Relative Acceleration Analysis

Analysis of motion of a rigid body undergoing general plane motion by locating an instantaneous centre of zero velocity.

Instantaneous Centre of Rotation

Meaning of rigid bodies; different kinds of motion undergone by rigid bodies.

Kinematics of general plane motion of rigid bodies by relating the angular motion of a line to the motion of a point on the same rigid body or different rigid bodies with dependent motion.

Absolute Motion Analysis

Kinematics of motion of a rigid body undergoing general plane motion using parameters measured relative to a reference frame in rotation.

Motion Relative to a Rotating Frame

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

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

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

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

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

- Explore relative velocity analysis to solve complex motion problems

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

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

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

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

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

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

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

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

Engineering Mechanics II

Dynamics

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

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

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

Applied Mechanics

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Engineering Mechanics

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)

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!

Courses

Static Charges and Fields - Physics (Undergraduate Foundation)

Introduction to Organic Chemistry (Undergraduate Foundation)

Functions, Limits and Continuity - Calculus (Undergraduate Foundation)

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

Linear Vector Spaces - Linear Algebra (Undergraduate Advanced)

Electric Potential and Capacitors - Physics (Undergraduate Foundation)

Set Theory - Mathematics (Undergraduate Foundation)

Binomial Theorem - Mathematics (Undergraduate Foundation)

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

Calculus of Scalar and Vector Fields (Undergraduate Advanced)

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

Kinematics of Particles - Engineering Mechanics (Undergraduate Advanced)