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.

Introduction

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.

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

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

binomial-theorem-mathematics-undergraduate-foundation

Mathematics

This chapter defines the principle of mathematical induction and its role in formal logic. You will understand the domino effect analogy and why this method is necessary for proving statements across infinite sets of integers. It provides the essential foundation for all subsequent proof techniques in this course.

You will master defining the principle of induction, verifying the base case for n equals one, formulating the inductive hypothesis for n equals k, and executing the inductive step to prove the statement for n equals k plus one.

This chapter covers proofs for the sums of series with terms written as fractions. It is a vital section because it forces you to use advanced algebraic manipulation and common denominators to link the inductive hypothesis to the final goal.

By the end, you will master handling terms with product denominators, using common factors to simplify fractional expressions, and proving telescoping series identities through the inductive step.

Fractional Series

This chapter teaches how to use induction to prove that algebraic expressions are exactly divisible by specific integers. It is a critical skill for number theory and cryptography, moving beyond simple addition to master the manipulation of factors and remainders in a logical sequence.

You will master expressing divisibility as a linear equation, manipulating exponents to isolate common factors, and applying the inductive hypothesis to prove that expressions like n cubed plus 2n are multiples of three.

Divisibility

This chapter focuses on using induction to prove formulas for the sum of arithmetic and geometric sequences. It is the most common application of the method and essential for verifying that a general summation formula holds true for any number of terms.

By the end, you will master proving linear and quadratic series identities, applying the inductive hypothesis to algebraic summations, and simplifying complex fractional series expressions to reach the required right-hand side.

Series

This chapter covers using induction to prove mathematical inequalities where one expression is always greater than another. It is a vital section because it requires logical estimation and transitivity rather than simple equality. This skill is essential for analyzing algorithm efficiency and bounding errors in numerical analysis.

You will master handling base cases where n is greater than one, applying the inductive hypothesis to inequalities, and using algebraic estimation to prove that exponential growth eventually exceeds polynomial growth.

Inequalities

This chapter applies induction to sequences where each term depends on those before it. It is critical for computer science and engineering because it provides the mathematical proof for how recursive algorithms and loops function. You will learn to verify that a general formula correctly predicts the value of any term in a repeating sequence.

You will master defining terms using recurrence relations, proving explicit formulas for recursive sequences, and applying the inductive step to link successive terms.

Recursive Sequences

This chapter covers miscellaneous induction proofs that do not fit into standard categories. It tests your ability to apply logical steps to unfamiliar mathematical structures, ensuring you can use the base case and inductive step on any technical problem.

By the end, you will master proving formulas for the nth power of a matrix, calculating total subsets for a set with n elements, and applying inductive logic to non-standard algebraic identities.

Balancing Redox Reactions

This chapter introduces the fundamental rules of chemical reactions and the law of conservation of mass. It establishes the groundwork for predicting how substances interact and transform, which is necessary for all quantitative work in chemistry and engineering.

By the end of this chapter, you will master defining chemical stoichiometry, identifying reactants and products in an equation, understanding the role of state symbols, and appreciating why equations must be balanced for accurate yield predictions.

This final chapter synthesises all balancing techniques and redox principles covered throughout the course. Mastery here validates your ability to resolve complex reaction stoichiometry, ensuring computational accuracy for theoretical yield predictions and industrial process modelling.

You will master four objectives: integrating conservation of mass across all equation types; resolving multi-step ionic and redox problems; verifying oxidation number assignments; and demonstrating proficiency through comprehensive practice problems.

Conclusion

This chapter provides the conceptual framework for understanding reactions in solution, focusing on ionic and redox chemistry, which is vital for later electrochemistry and analysis. We systematically move from simplifying standard equations to mastering the foundational definitions and rules required to quantify electron transfer reactions.

You will master four key objectives: converting molecular equations to net ionic forms; defining and distinguishing oxidation and reduction; identifying the roles of oxidizing and reducing agents; and accurately assigning oxidation numbers to atoms in any compound or ion.

Redox Reactions

This chapter applies balanced equations to calculate the exact amounts of substances used or made in a reaction. It is the practical core of the course, essential for predicting product yields and managing chemical resources in a laboratory or factory setting.

You will master converting between mass and moles using molar ratios, identifying the limiting reagent that stops a reaction, calculating theoretical yields, and determining the amount of excess reactants left over.

Stoichiometric Calculations

This chapter establishes the core skill of balancing chemical equations, which is mandatory for all quantitative chemical analysis; we apply the law of conservation of mass using three distinct, systematic methods to ensure numerical equality between reactants and products.

You will master three key balancing techniques: the inspection method for simple reactions, the algebraic method for complex systems, and the oxidation number change method specifically for redox reactions.

Balancing Chemical Equations

Master the quantitative prediction of chemical change with Stoichiometry of Reactions. This undergraduate foundation course focuses on transforming conceptual chemical reactions into balanced, calculable equations. We begin by applying the conservation of mass principle to balance equations through inspection, the algebraic method, and the oxidation number change method. Next, we simplify reactions in solution by deriving total and net ionic equations, rigorously defining redox reactions, and teaching the precise assignment of oxidation numbers. The final section culminates in balancing complex redox equations using the half-reaction (electron transfer) method in both acidic and basic media.

Accurate reaction stoichiometry is the cornerstone of chemical processing, environmental analysis, and energy generation. Learners will gain the necessary computational skills to predict theoretical yields, determine limiting reagents in industrial synthesis, and analyse electrochemical processes like batteries and corrosion. Correctly balancing and interpreting chemical equations ensures laboratory safety, validates experimental results, and underpins quantitative decision-making in chemical engineering, materials science, and biochemistry.

After completing this course, you will be proficient in balancing any chemical equation based on the conservation of mass; accurately converting molecular equations into total and net ionic forms; defining and identifying oxidation and reduction processes, including their agents; assigning oxidation numbers to elements in compounds; and confidently balancing complicated redox reactions in both acidic and basic solutions. The course concludes with comprehensive practice problems to cement computational accuracy in reaction stoichiometry.

This course is specifically designed for undergraduate students beginning foundational Chemistry or related programmes, including Chemical Engineering, Pharmacy, and pure Sciences, where proficiency in calculating reaction quantities is mandatory. It is also an ideal, focused resource for pre-university students seeking an advanced start, or for any professional needing a rapid, expert-led review of fundamental reaction stoichiometry principles for accreditation or further study.

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 

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

stoichiometry-ii-chemical-equations-and-redox-reactions

Chemistry

This chapter establishes the rigorous classification and fundamental operations of the real number system. You will master the structural hierarchy of numbers and basic algebraic manipulations, providing the essential precision required for solving complex equation systems and inequalities in subsequent chapters.

You will master: classifying real numbers into rational and irrational subsets; applying order of operations to complex expressions; identifying properties of real number subsets; and executing fundamental algebraic simplifications.

This chapter establishes the rigorous mechanical framework for resolving equations of varying degrees and structures. Mastering these transformation techniques is essential for isolating variables in engineering models and scientific formulas with absolute logical consistency.

You will master the systematic isolation of unknowns in linear and quadratic forms; the resolution of rational and radical equations; and the reduction of cubic, biquadratic, and symmetric expressions into solvable factors.

Equations in One Unknown

This chapter governs the manipulation and factorisation of higher-degree algebraic expressions. You will master the structural logic of polynomial functions, establishing the essential skills required for advanced partial fraction decomposition and the analysis of non-linear systems in engineering and physics.

You will master: executing polynomial long division and synthetic division; applying the Remainder and Factor Theorems; factorising cubic and higher-order expressions; and identifying roots of polynomial equations.

Polynomials

This chapter governs the resolution of mixed systems where linear and non-linear dependencies intersect. You will master the algebraic techniques required to solve for variables across multiple constraints, providing the essential logic for intersecting geometric paths and complex equilibrium states in engineering and economics.

You will master: executing substitution and elimination for dual linear systems; identifying points of intersection between straight lines and curves; resolving homogeneous quadratic systems through variable substitution; and simplifying complex rational and product constraints to determine precise coordinate solution sets.

Simultaneous Equations

This chapter governs the mechanics of exponential notation and the foundational laws for manipulating powers. You will master these operations to establish the technical proficiency required for solving complex exponential equations and performing the logarithmic transformations essential to scientific and financial data analysis.

You will master: applying the product, quotient, and power laws; evaluating zero, negative, and fractional indices; and simplifying complex exponential expressions through systematic reduction.

Indices

This chapter covers the inverse operation of fraction addition by breaking down complex rational expressions into simpler parts. Mastery of this decomposition is a prerequisite for advanced calculus and engineering system analysis.

You will master identifying proper and improper functions; decomposing expressions with distinct or repeated linear factors; resolving irreducible quadratic denominators; and applying polynomial long division to handle high-degree numerators.

Partial Fractions

This chapter governs the algebra of non-equality and region-based constraints. You will master the rigorous handling of directionality in mathematical relationships, establishing the essential logic required for economic optimisation, engineering tolerance analysis, and defining valid data ranges in computational algorithms.

You will master: executing sign-reversal rules during manipulation; solving linear, quadratic, and rational inequalities; using interval notation; and shading valid regions.

This chapter establishes the inverse relationship between powers and exponents to handle nonlinear data. Mastery of logarithmic transformations is essential for solving complex exponential equations and modelling growth patterns in engineering and financial sectors.

You will master: converting between index and logarithmic forms; applying the product, quotient, and power laws; executing the change of base formula; and solving algebraic equations involving unknown exponents or bases.

Logarithms

This chapter covers surds, which are roots that do not have whole number answers. You must learn to use them to keep your calculations exact, as using decimals in engineering and science leads to errors.

You will master simplifying and performing arithmetic on surds; rationalising denominators using conjugates; finding square roots of compound surds; and solving algebraic equations involving radical terms.

Surds

This chapter governs the synthesis of indices and logarithms to solve equations where variables appear as exponents. You will master the algebraic techniques required to isolate and resolve unknown indices, providing essential analytical tools for modelling exponential growth, decay constants, and complex scaling in engineering and financial data.

You will master: equating bases for direct solution; applying logarithmic transformations; reducing equations to quadratic forms; and solving non-linear simultaneous systems involving mixed bases.

Equations with Unknown Index

Mathematics requires absolute precision. This course provides a rigorous grounding in real number operations, from integers and rational numbers to complex systems and algebraic laws. You will learn to handle polynomials, solve equations ranging from linear to biquadratic, and navigate simultaneous systems. The syllabus covers inequalities, sign tables, partial fractions, indices, logarithms, and the systematic simplification of surds.

These tools are essential for success in engineering, accounting, and the sciences. Clear mathematical thinking allows you to model financial risks, calculate structural loads, and write efficient computer code. Proficiency in these operations ensures accuracy in any career that relies on quantitative data and logical deduction.

You will gain the ability to classify number systems, apply algebraic theorems, and solve complex equations in one or more unknowns. You will acquire the skills to manipulate inequalities, resolve algebraic fractions, and simplify expressions using the laws of indices and logarithms. The course provides the technical competence required to handle surds and find roots of compound expressions effectively.

This training is built for undergraduate foundation students and secondary school leavers entering university. It provides a necessary bridge for anyone needing to strengthen their mathematical base before advanced study. Even those in non-technical roles will find value in the disciplined analytical approach required to master these foundational concepts.

operations-with-real-numbers-mathematics-undergraduate

This chapter provides the roadmap for the course. It outlines the progression from early atomic theories to the modern understanding of atomic structure and its direct relationship to the periodic table.

Key learning objectives include: understanding the overall course structure and appreciating the historical and experimental development of atomic theory.

This chapter traces the origin of atomic theory, starting with the empirical laws that led to Dalton's foundational model. It then covers the pivotal discovery of the electron, the first experimental proof that the atom is divisible, which forced the initial theory to evolve.

Key topics include the laws of chemical combination, Dalton's atomic postulates and their critical shortcomings, and the experiments by Thomson and Millikan that discovered and characterised the electron, leading to the 'plum pudding' model.

Early Atomic Models

This chapter covers the development of the nuclear atom, from Rutherford's definitive gold foil experiment to Chadwick's discovery of the neutron. It establishes the modern planetary model of the atom and immediately addresses its critical failure, demonstrating why classical physics is insufficient.

Key topics include the experimental evidence for the nucleus, the discovery of the neutron, the shortcomings of Rutherford's classical model, and the fundamental properties of protons, neutrons, and electrons.

The Nuclear Model

This concluding chapter summarises the key concepts of atomic theory. It reinforces the understanding of atomic structure and its connection to the periodic properties of the elements.

This summary prepares the student for the next course, 'Chemical Bonding and Molecular Geometry', where electronic structure is used to predict how atoms form molecules.

This chapter introduces the modern quantum mechanical model, replacing Bohr's flawed orbits with probabilistic orbitals. Mastery of this model is non-negotiable, as it provides the definitive framework for describing electron behaviour and predicting all chemical properties.

Key topics include: the four quantum numbers, defining orbital shapes (s, p, d, f) and their spatial orientation, and applying the Aufbau principle, Pauli exclusion, and Hund's rule to write correct electronic configurations.

The Quantum Model

This chapter details Niels Bohr's model, the first attempt to solve the failures of the classical nuclear atom using quantum theory. Understanding this transitional model is critical to grasping the full conceptual leap to modern quantum mechanics.

Topics include: Bohr's postulates, explaining hydrogen's emission spectrum, energy level calculations, and the model's ultimate shortcomings.

Bohr's Model

This chapter details wave-particle duality, the principle that all matter has wave properties. This concept explains the failure of Bohr's fixed orbits and establishes the necessary theoretical foundation for the modern quantum mechanical model of the atom.

Key topics include: De Broglie???s wave hypothesis, applying the de Broglie relation (?? = h/mv), and calculating the wavelength of moving particles based on their mass and velocity.

Wave-Particle Duality

This chapter connects electronic configuration to the predictable, repeating properties of the elements. Mastery of these trends is the practical application of atomic theory and is essential for predicting chemical behaviour, bonding, and reactivity.

Key topics: trends in atomic radii, ionisation energy, electron affinity, and electronegativity; and the classification of elements as metals, non-metals, and metalloids.

Periodicity

This course provides a complete guide to the modern theory of the atom. It traces the historical development of atomic models, from Dalton's foundational theory to the modern understanding of electronic structure. The material covers subatomic particles, the contributions of Thomson and Rutherford, the Bohr model, wave-particle duality, and culminates in a full treatment of electronic configuration and its direct relationship to the periodic trends of the elements.

A command of atomic theory is the absolute foundation of modern chemistry. The electronic structure of the atom dictates all chemical properties and bonding behaviour. This knowledge is essential for understanding the periodic table, predicting chemical reactions, and is the prerequisite for the study of spectroscopy, materials science, and quantum mechanics.

By the end of this course, you will be able to describe the historical evolution of atomic theory. You will also be able to explain the atom's structure, including the roles of protons, neutrons, and electrons. You will determine the electronic configuration of any element and use periodic trends to predict and justify atomic properties like size, ionisation potential, and electronegativity.

This course is for students who have completed an introductory chemistry course. It is a mandatory prerequisite for any student pursuing a degree in chemistry, chemical engineering, materials science, or physics.

atomic-structure-and-periodicity-of-elements-chemistry

General analysis of problems involving dry friction - for particles.

Problems Involving Dry Friction (1)

Analysis of frictional forces on journal bearings.

Journal Bearings

Analysis of frictional forces (rolling resistance) on wheels.

Wheels

Analysis of dry friction on square-threaded screws.

Screws

Wedges

Fundamental principles of dry friction - meaning of friction, coefficients of friction, angles of friction, etc.

Laws of Dry Friction

Welcome to the course and definition of terms - friction, dry friction, fluid friction, lubrication, etc.

Analysis of frictional forces on flat belts.

Belts

Analysis of frictional forces on thrust bearings - collar bearings, pivot (end) bearings and disks.

Thrust Bearings

General analysis of problems involving dry friction - for rigid bodies.

Problems Involving Dry Friction (2)

Analysis of equilibrium of rigid bodies subjected to dry (Coulumb) friction, and its applications to screws, wedges, belts and bearings.

OAU, Ife

Master the science of structural stability. This programme provides a complete education in engineering statics, covering the analysis of forces, moments, and equilibrium in rigid bodies. It delivers the non-negotiable principles required to design safe and reliable structures.

This learning track is for first or second-year undergraduate students in Civil, Mechanical, Aerospace, and Structural Engineering. It is the foundational mechanics course upon which all subsequent design and analysis subjects are built.

Analyse the forces within any static structure, from simple trusses to complex machines. You will master free-body diagrams, equilibrium equations, and structural analysis methods. This programme equips you with the core analytical skills for a career in structural, mechanical, or civil design.

Engineering Mechanics

Statics

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

dry-friction-engineering-mechanics-undergraduate

Introduction to engineering mechanics, fundamental concepts, systems of units and general analysis procedure.

Force vectors and how to obtain the resultant of two or more forces - using parallelogram and triangle rules.

Resultant of Forces in a Plane

Resolution of a force into its rectangular and non-rectangular components.

Components of a Force in a Plane

Calculating the resultant of several concurrent forces using their rectangular components.

Resultant by Components in a Plane

Determining the components of forces in three dimensions.

Components of a Force in Space

Equilibrium of particles under the action of three-dimensional forces.

Equilibrium in Space

Equilibrium of particles under the action of planar forces.

Equilibrium in a Plane

Vector addition of force vectors in three dimensions.

Resultant of Forces in Space

This course provides a complete guide to the statics of particles, a foundational topic in engineering mechanics. It covers the principles of force vectors, resultants, and components, first in two dimensions and then extending to three-dimensional space. The core of the course is the analysis of particles in equilibrium, establishing the conditions required for a particle to remain at rest under the influence of multiple concurrent forces.

Statics is the essential science behind structural and mechanical design. The principles covered are the basis for designing every stable structure, from bridges and skyscrapers to aircraft frames and engine components. A command of statics is non-negotiable for any engineer, as it provides the fundamental tools required to analyse forces and ensure the safety and integrity of physical systems.

By the end of this course, you will be able to resolve any force into its components and calculate the resultant of any system of concurrent forces in both 2D and 3D. You will also be able to draw a complete free-body diagram for any particle and apply the equations of equilibrium to solve for unknown forces, tensions, and reactions in a variety of engineering problems.

This course is for undergraduate students in any engineering discipline, including civil, mechanical, and aerospace. It is a core component of any engineering mechanics curriculum and assumes a prerequisite knowledge of introductory physics and vector mathematics. It is the essential foundation for subsequent courses in dynamics and the mechanics of materials.

statics-of-particles-engineering-mechanics-undergraduate

Heat tracking is the basis of thermodynamics. This chapter uses enthalpy and the first law to explain how reactions exchange energy with their surroundings. These tools simplify calculations by focusing only on initial and final states.

You will learn to calculate internal energy; distinguish exothermic from endothermic reactions; apply standard heats of formation; and use Hess's law to sum intermediate reaction steps.

Enthalpy

The first law tracks energy but cannot predict if a reaction will occur. This chapter introduces entropy to explain why processes move in specific directions. It is essential for determining reaction spontaneity.

You will learn to state the second law of thermodynamics; predict disorder changes from physical states; calculate entropy for phase transitions; and evaluate total entropy using system and surroundings data.

Entropy

Substances absorb or release heat when changing state even at a constant temperature. This chapter explains how to quantify these energy shifts during fusion, vaporisation, and sublimation. Mastering these calculations is essential for understanding physical transitions and industrial thermal management.

You will learn to define specific enthalpies of phase changes; classify transitions as endothermic or exothermic; calculate molar enthalpy from mass and heat data; and apply Hess's law to relate sublimation to fusion and vaporisation.

Enthalpy of Phase Changes

Gibbs free energy is the final way to tell if a reaction will happen on its own. It joins heat and disorder into one value to show spontaneity. This chapter provides the ultimate rules for chemical stability and reaction direction.

You will master how to define Gibbs free energy; use its sign to predict if a reaction is spontaneous; explain how temperature changes affect if a reaction occurs; and calculate free energy changes using standard data.

Gibbs Free Energy

Chemical reactions involve breaking and making bonds, leading to energy shifts known as enthalpy changes. This chapter explains how to measure and calculate these heats of reaction under standard conditions. Understanding these energy balances is vital for predicting reaction behaviour and industrial process efficiency.

You will learn to define standard enthalpy changes; distinguish between spontaneous and non-spontaneous processes; calculate reaction heats using bond dissociation energies; and write accurate thermochemical equations from experimental data.

Enthalpy of Reactions

Thermodynamics begins with defining boundaries and energy movement. This chapter teaches you about systems, heat, and work. You need these to track energy in chemical reactions accurately.

You will learn to identify system types; explain different forms of work; define specific and molar heat capacities; and calculate heat using mass and temperature changes.

Energy drives every chemical change in our universe. This course explains how heat, work, and internal energy dictate the behaviour of matter from simple phase changes to complex chemical reactions. You will master the laws of thermodynamics, enthalpy calculations, entropy changes, and the ultimate predictor of spontaneity, Gibbs free energy.

These principles are the backbone of modern industry. Chemical engineers use them to design efficient industrial reactors and cooling systems. Materials scientists apply them to develop better batteries and energy sources, while biochemists use these same laws to understand the energy flow within living cells. Mastering these concepts is essential for anyone aiming to solve energy-related problems in the real world.

By the end of this course, you will be able to distinguish between open, closed, and isolated systems and calculate heat changes using specific heat capacities. You will apply Hess’s law and bond energies to find reaction enthalpies and use the second law of thermodynamics to predict disorder through entropy. Finally, you will calculate Gibbs free energy to determine exactly if and when a reaction will occur naturally under various temperatures.

This course is built for undergraduate students in chemistry, chemical engineering, and materials science who need a solid foundation in physical chemistry. It also benefits secondary school leavers preparing for university-level science and professionals looking to refresh their technical knowledge of energy systems. Even those outside these fields will gain a logical framework for understanding how energy and disorder shape the physical world.

chemical-thermodynamics-chemistry-undergraduate-foundation

This chapter provides the roadmap for the course. It introduces volumetric analysis, or titration, as a major practical application of all preceding stoichiometric principles.

Key learning objectives include: understanding the overall course structure and appreciating the role of titration as a fundamental technique in analytical chemistry.

This final chapter synthesises all volumetric principles and stoichiometric calculations covered throughout the course. Mastery here is essential to validate your analytical precision and ensure readiness for advanced laboratory modules or professional certification exams.

You will master three objectives: integrating concentration and dilution laws; resolving multi-step solution stoichiometry problems; and verifying computational accuracy through comprehensive practice questions.

This chapter establishes the quantitative basis for volumetric analysis by defining how substance amounts are expressed in liquid media. Precision here is mandatory, as these concentration units are the primary variables in all subsequent titration calculations and laboratory protocols.

You will master four key objectives: defining and calculating molarity; determining mass concentration; performing precise dilution calculations; and interconverting between various concentration units to prepare accurate standard solutions.

Composition of Solutions

This chapter provides the mathematical and practical framework for reducing solution concentration without altering the total amount of solute. Precise dilution is a mandatory laboratory skill, as preparing working standards from concentrated stocks is standard practice in all volumetric analysis.

You will master four objectives: applying the dilution equation; calculating stock volumes for target molarities; performing multi-step serial dilutions; and solving complex mixture concentration problems.

Dilution of Solutions

This chapter bridges pure concentration calculations and chemical reactivity by examining how dissolved substances interact. Understanding stoichiometric ratios within a liquid phase is critical for predicting product yields and determining chemical equivalence points in analytical procedures.

You will master four objectives: applying mole ratios to solution-phase reactions; calculating precipitate mass; determining limiting reagents in aqueous systems; and computing theoretical yields from known molar concentrations.

Reactions in Solution

Precision in volumetric analysis is the difference between experimental success and catastrophic failure in the laboratory. This course provides a rigorous mathematical framework for quantifying dissolved substances, beginning with fundamental concentration units like molarity and mass concentration before progressing to the principles of dilution and standard solution preparation. You will master the stoichiometric requirements of complex chemical interactions, including precipitation, acid-base neutralisation, and redox reactions, through a series of intensive worked examples covering advanced techniques such as back-titration and standardisation.

Quantitative titration skills are the industrial standard for quality control and analytical research. Proficiency in solution stoichiometry allows for the exact determination of drug purity in pharmaceuticals, contaminant levels in environmental monitoring, and reactant concentrations in industrial chemical synthesis. Mastering these computational methods ensures the accuracy required for professional competency in any clinical or academic laboratory setting.

By the conclusion of this course, you will be able to calculate precise concentrations from raw experimental data across multiple reaction types. You will acquire the technical ability to prepare primary standard solutions, apply dilution laws to reach target molarities, resolve limiting reagent problems in aqueous phases, and execute complex multi-step stoichiometric calculations for redox and back-titration scenarios. These skills are essential for converting laboratory observations into reportable analytical results.

This course is a mandatory requirement for undergraduate students in Chemistry, Pharmacy, Biochemistry, and Chemical Engineering. It also serves as a critical computational review for laboratory technicians and research assistants preparing for technical certification or professional exams. Even for those in peripheral scientific fields, the rigorous logic and mathematical precision developed here provide a necessary foundation for any discipline requiring core analytical data processing.

stoichiometry-iii-volumetric-analysis-chemistry

This chapter covers arithmetic sequences where terms increase or decrease by a constant common difference. Mastering these linear patterns is essential for understanding uniform growth, simple interest, and steady accumulation in science and finance.

You will master calculating general terms and sequence lengths; evaluating finite series sums; inserting arithmetic means; and using symmetric notation to solve algebraic progression problems involving ratios and unknown variables.

Arithmetic Sequences and Series

This chapter covers geometric sequences where terms change by a constant ratio. Mastering these exponential patterns is essential for modelling compound interest, asset depreciation, and physical rebound heights in science and finance.

You will master calculating general terms and finite sums; inserting geometric means; and applying symmetric notation or logarithms to solve for unknown variables in real-world growth and decay scenarios.

Finite Geometric Sequences and Series

This chapter establishes the fundamental definitions and notation for ordered numerical lists and their summations. Technical precision here is vital for distinguishing between sequences and series before advancing to complex progressions.

You will master defining formal sequence notation; distinguishing between finite and infinite structures; identifying individual terms via subscripts; and utilising sigma notation for compact series representation.

This chapter covers the algebraic mechanics of ordered numerical lists defined by functional rules. Mastery of these structures is essential for translating discrete patterns into precise mathematical models using both direct and step-by-step methods.

You will master generating terms from explicit formulae; defining sequences through recurrence relations; deriving general rules from numerical patterns; and verifying term accuracy through algebraic substitution.

Sequences

This chapter covers the algebraic mechanics of discrete summation and the rigorous application of sigma notation. Technical proficiency here is vital for condensing and expanding series, providing the foundational logic for calculating total accumulations in financial and physical models.

You will master executing summations using sigma notation; applying linear properties to simplify complex sums; extracting individual terms from partial sums; and classifying series as convergent, divergent, or oscillatory.

This chapter covers infinite geometric series that sum to a finite limit. Mastering convergence is vital for limits, calculus, and professional financial modelling.

You will define conditions for series convergence; calculate the sum to infinity; expand recurring decimals into geometric series; and convert these decimals into exact fractions.

Infinite Geometric Series

This chapter covers complex sequences and special series that fall outside standard linear or exponential models. You will learn to synthesise arithmetic and geometric properties to resolve hybrid patterns and use standard identities for power summations.

You will master identifying harmonic progressions through reciprocals; calculating nth terms for arithmetic-geometric sequences; and applying formulae for the sums of natural numbers and their squares.

Other Sequences and Series

Master the algebraic rules governing ordered lists of numbers and their total sums. This course establishes a solid foundation in the mechanics of sequences and series. We investigate precise definitions for arithmetic and geometric progressions; use formal notation for general terms; and handle finite and infinite series using sigma notation. The curriculum focuses on the structural properties required for university-level mathematical analysis.

Knowing how to handle numerical patterns is vital for many modern careers. Financial analysts use geometric series to calculate compound interest, loan repayments, and annuities. Computer scientists use sequence structures to organise data sets and analyse computer loops. This knowledge is necessary for measuring growth, decay, and accumulation in business and science with absolute accuracy.

You will learn to identify and define sequences using general terms and recurrence formulas. You will calculate the nth term and the sum of terms for arithmetic, geometric, and harmonic progressions. You will master sigma notation; determine when infinite series converge; convert recurring decimals into fractions; and use formulas to sum natural numbers, squares, and cubes.

This course is for students entering university foundation programmes in mathematics, finance, or computer science. It serves as an intensive refresher for professionals returning to academia or shifting into roles that demand precise numerical work. Even those outside these fields will benefit from the sharp logical thinking and problem-solving speed developed through pattern analysis.

sequences-and-series-mathematics-undergraduate-foundation-2

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

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

Kinds of Real-Valued Functions

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

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

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

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

Real-Valued Functions

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

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

Transcendental Functions

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

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

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

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

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

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

UI, Ibadan

This learning track provides the complete theoretical machinery of single-variable calculus and analysis. We build the subject from first principles, establishing the rigorous logical framework required for advanced quantitative disciplines. This is the 'why' behind the mathematics that powers science and engineering.

This track is built for second-year engineering and physical science students, particularly those at the University Of Ibadan. It is also structured for any student requiring the same rigorous theoretical foundation for advanced quantitative study.

On completion, you will command the complete theoretical basis of single-variable calculus. You will construct formal proofs, rigorously analyse function behaviour, and determine the convergence of infinite series. This programme provides the non-negotiable prerequisite knowledge for advanced study in differential equations, complex analysis, and theoretical physics.

Real Analysis

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

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

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

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Single-Variable Calculus

Continuity of functions of several variables.

Continuity

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

Limits

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

Homogeneous Functions

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

Composite Functions

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

Partial Derivatives

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

Extreme Values

Taylor's theorem for functions of two variables.

Taylor's Theorem

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

Implicit Differentiation

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

Theorems on Jacobians

This course provides a complete guide to the calculus of several variables. It builds from the foundational concepts of multivariable functions, limits, and continuity to the core techniques of differentiation, including partial derivatives, the chain rule, and implicit differentiation. The material culminates in advanced topics such as Taylor's theorem for several variables and the use of Jacobians.

Multivariable calculus is the language of modern science, engineering, and economics. Its principles are used to model complex surfaces, analyse thermodynamic systems, create 3D computer graphics, and solve critical optimisation problems in business and finance. This is the mathematical toolkit for working with systems that have multiple interacting variables.

By the end of this course, you will be able to calculate partial derivatives, apply the multivariable chain rule, and find directional derivatives using the gradient vector. You will also be able to solve both unconstrained and constrained optimisation problems by finding extreme values and using the method of Lagrange multipliers, and apply these derivatives to find tangent planes to surfaces.

This course is for students who have completed a full single-variable calculus sequence. It is the standard curriculum for a multivariable calculus (Calculus III) module and is a direct prerequisite for the study of vector calculus, differential equations, and advanced courses in physics, engineering, and economics.

Courses

Binomial Theorem - Mathematics (Undergraduate Foundation)

Mathematical Induction - Mathematics (Undergraduate Foundation)

Stoichiometry of Reactions - Chemistry (Undergraduate Foundation)

Operations with Real Numbers - Mathematics (Undergraduate Foundation)

Atomic Structure and Periodicity of Elements - Chemistry (Undergraduate Foundation)

Dry Friction - Engineering Mechanics (Undergraduate Advanced)

Statics of Particles - Engineering Mechanics (Undergraduate Advanced)

Chemical Thermodynamics - Chemistry (Undergraduate Foundation)

Stoichiometry of Solutions - Chemistry (Undergraduate Foundation)

Sequences and Series - Mathematics (Undergraduate Foundation)

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

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