This chapter builds the logic of differentiation from scratch. You will learn why derivatives work before using shortcuts. Mastering these foundations prevents errors in complex calculations later.

You will define the derivative via limits, distinguish continuity from differentiability, and prove the power and sine rules. Practical examples cover quadratics, reciprocals, roots, and trig functions.

Introduction

This chapter consolidates your knowledge of differentiation techniques. It ensures you have the technical foundation needed for the upcoming applications course.

You will review the hierarchy of rules, identify the best technique for any function, and confirm mastery of successive differentiation.

Conclusion

Stop using first principles for every sum. This chapter gives you the standard rules to find gradients fast. You will learn to handle constants, powers and roots without stress. These shortcuts save time and reduce errors in complex calculations. They are the basic tools for all future calculus work.

You will master the power rule for positive and negative indices; convert roots and fractions into easy powers; apply linearity to split long sums; and simplify algebraic terms before differentiating. These skills let you solve polynomial problems quickly and accurately.

Fundamental Rules

Functions multiplied or divided need special rules. The power rule fails here, so you must learn new techniques to find gradients accurately.

Master the product and quotient rules; handle triple products and trigonometric fractions; apply shortcuts for reciprocals; evaluate derivatives at specific points.

Product and Quotient

This chapter provides standard derivatives for trigonometric, exponential, logarithmic, hyperbolic and absolute value functions. These fixed results allow you to solve routine physics and engineering problems without manual proofs.

You will derive key rates from first principles; apply rules to reciprocals and inverses; and handle mixed products. You will memorise standard forms for quick recall during calculations.

Common Derivatives

Functions inside functions require the chain rule. This chapter teaches you to differentiate composite expressions without expanding them, a critical skill for handling complex real-world models in engineering and physics.

Master the general formula and derived shortcuts for powers, roots, exponentials, and logs. Learn to combine this with product rules and solve multi-layer trigonometric and inverse trigonometric problems efficiently.

Chain Rule

Standard rules fail for complex curves. This chapter teaches methods for equations where variables are mixed, defined by parameters, or have variable powers. These techniques handle real-world engineering problems that basic algebra cannot solve.

You will master implicit differentiation for hidden relationships, parametric gradients for motion, and logarithmic differentiation for variable exponents. Learn to simplify tough expressions using log laws before calculating derivatives.

Advanced Techniques

This chapter measures the rate of change of the gradient. It is vital for calculating acceleration and analysing curvature in engineering and physics. You will move beyond simple slopes to understand how functions behave over successive changes.

Master second-order derivatives, successive differentiation, and nth derivative patterns. Learn to handle implicit and parametric forms. Apply these skills to verify solutions for differential equations.

Higher-Order Derivatives

Differentiation measures how quantities change at exact moments. This course builds your complete calculus foundation from first principles to advanced methods. We start with rates of change and formal notation, then cover definitions, continuity, differentiability, and direct proofs. The sequence moves through core rules, composite functions, implicit techniques, parametric systems, and higher-order operations. Each concept links to targeted exercises that test your accuracy before you proceed.

These methods solve real problems in physics, engineering, finance, and software development. You will track speed, predict growth, calculate structural stress, and measure market sensitivity. Software developers apply these calculations to optimise machine learning models. Business analysts use them to adjust variables for target profits. The techniques here convert abstract equations into working tools for daily technical tasks.

You will calculate derivatives for polynomial, trigonometric, exponential, logarithmic, and hyperbolic functions without reference material. You will apply the product rule, quotient rule, and chain rule to complex expressions. You will perform implicit, parametric, and logarithmic differentiation. You will compute second-order and higher-order derivatives for standard forms. You will verify differential equations through systematic steps. Every lesson forces active practice and precise calculation.

The material suits first-year university students, polytechnic learners, and independent learners who require a solid calculus base. Secondary school graduates will gain the exact preparation needed for advanced science and mathematics exams. Technical professionals can use the content to refresh forgotten methods or build reliable calculation habits. The explanations skip heavy theory and focus on working steps. Anyone who handles data, graphs, or mathematical models will apply the methods immediately. You will solve standard calculus problems independently.

Our mission is to simplify complex concepts by teaching from first principles. We go beyond simple problem-solving to build a deep, foundational understanding of the "why" behind the "how."

Our content is meticulously designed for undergraduates navigating the rigorous curricula of top engineering and physical science programmes in Nigeria and globally.

As a core academic partner on UniDrills, we are committed to upholding the highest standards of clarity, precision, and professional excellence in education.

Max Engineering Tutorials

An academic and graduate engineer dedicated to explaining complex concepts in mathematics, computer science and engineering. I founded UniDrills to bring this vision to life, creating a world-class e-learning marketplace with a global outlook. I apply a first-principles approach to software architecture to build a professional and intuitive learning experience.

NUC CCMAS

Calculus is the mathematical tool for measuring change and finding the best results in any system. This track follows the official NUC CCMAS MTH 102 curriculum to teach you how functions behave, how to calculate exact rates of change, and how to sum up tiny movements to find total areas or volumes. You will move from basic limits to complex integration techniques used to solve practical problems in the physical world.

This track is for first-year university students in engineering, science, and economics departments across Nigeria. It also serves secondary school leavers preparing for technical degrees or university entrance exams. Anyone needing to build a strong foundation in mathematical logic for data analysis or professional licensing will find these lessons essential.

After finishing this programme, you will calculate derivatives using first principles and shortcuts, find turning points for optimisation, and solve definite integrals for land measurement or tank design. You will use the trapezium and Simpson's rules to handle experimental data accurately. These skills ensure you pass your university exams and excel in technical careers like civil engineering, physics research, or financial forecasting.

Elementary Mathematics II

Calculus

Management and administration require precise numerical logic for decision-making. This track follows the official NUC CCMAS syllabus for AMS 102, covering real numbers, set theory, and complex systems. You will master algebraic operations, permutations, and combinations alongside the foundations of trigonometry and calculus. It provides the mathematical tools to model financial trends, handle business data, and solve optimisation problems in corporate settings.

This programme is built for first-year university students in Accounting, Business Administration, Banking, Finance, and Public Administration. It also serves secondary school leavers preparing for management degrees and professionals needing a refresher on business mathematics. It is suitable for anyone moving into roles that demand accurate quantitative analysis and logical deduction.

You will gain the ability to simplify complex business formulas, calculate probabilities for risk assessment, and find the exact points where profits are highest. You will be able to handle sequences for interest calculations and use differentiation to manage production costs. Completing this track ensures success in university examinations and builds the analytical strength required for careers in financial analysis, auditing, and strategic management.

Basic Mathematics

fundamentals-and-techniques-of-differentiation-calculus

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

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

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

Conditional Permutation

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

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

Permutation with Identical Objects

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

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

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

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

Permutation

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

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

Cyclic Permutation

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

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

Geometric Applications

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

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

Derangements

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

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

Combination

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

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

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

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

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science.

This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions.

Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Elementary Mathematics I

Algebra and Trigonometry

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

permutation-and-combination-mathematics-undergraduate

Mathematics

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

Analysis of rectilinear motion with variable acceleration.

General Rectilinear Motion

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

Special Rectilinear Motion (2)

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

Curvilinear Motion (3)

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

Curvilinear Motion (1)

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

Curvilinear Motion (2)

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

Special Rectilinear Motion (1)

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

Graphical Analysis

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

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

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

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

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

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

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

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

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

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

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

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

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

OAU, Ife

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

kinematics-of-particles-engineering-mechanics

Engineering Mechanics

This chapter establishes the foundation of organic chemistry by explaining how functional groups define molecular behaviour. You will see why specific atom arrangements classify substances and determine their chemical properties.

You will identify major functional groups; distinguish between similar structures like aldehydes and ketones; and map every class to its defining bond arrangement. This knowledge allows you to predict reactivity and apply correct IUPAC categories.

Alkanes form the parent chains for all organic nomenclature. This chapter explains how saturated hydrocarbons serve as the base structure and how removing hydrogen creates reactive alkyl substituents.

You will learn to identify the longest carbon chain; assign correct locants to branches; and apply IUPAC rules to complex structures with groups like isopropyl. This ensures you can name any branched alkane without ambiguity.

Alkanes and Alkyls

Geometric isomerism occurs when restricted rotation locks atoms in fixed positions. This chapter explains how double bonds create distinct spatial arrangements that alter chemical behaviour. You will learn why shape matters for drug function and material properties.

You will distinguish stereoisomer types; assign cis and trans labels; apply E and Z priority rules; resolve ties using atomic number lists; and handle multiple bonds with phantom atoms.

Geometric Isomerism

Molecules with multiple functional groups create naming conflicts. This chapter resolves which group claims the parent suffix and which becomes a prefix using strict IUPAC rules.

You will master the priority hierarchy for competing groups; assign correct locants to complex structures; and name compounds containing acids, alcohols, ketones, and amines accurately.

Polyfunctional Compounds

Rings and aromatic systems form the backbone of many drugs and fuels. You will learn to name these stable structures using strict IUPAC rules. This chapter covers everything from simple saturated rings to complex fused benzene derivatives.

You will name cyclic compounds with multiple substituents; distinguish ortho, meta and Para positions on benzene rings; resolve priority conflicts between functional groups; and apply correct nomenclature to poly-aromatic systems.

Cyclics and Aromatics

Some molecules are mirror images yet cannot be superimposed. This chapter explains why identical formulas behave differently in the human body. You will learn to identify chirality and understand how spatial arrangement affects biological function.

You will assign R-S configurations; distinguish enantiomers from diastereomers; calculate stereoisomer counts; identify meso compounds; and interpret optical activity using Fischer projections.

Optical Isomerism

Same formula does not mean same substance. This chapter shows how atoms link in different orders to create distinct chemicals with unique properties. You will see why structure dictates behaviour in organic compounds.

You will define isomerism and separate structural types from spatial ones. You will identify chain, position, and functional group variations. You will also explain how tautomers shift hydrogen to change their class.

Structural Isomerism

Organic chemistry relies on precise naming to avoid confusion in laboratories and industries across the world. This course teaches you the standard rules for naming carbon-based molecules and understanding their three-dimensional shapes. You will move from simple chains to complex rings, learning how to identify functional groups and assign priorities correctly. The curriculum covers structural variations and spatial arrangements that define molecular behaviour in real chemical systems.

Accurate nomenclature is essential for reading scientific literature, writing lab reports, and communicating with peers in university or industry. You will apply these skills to interpret drug structures, analyse polymer compositions, and solve examination questions with confidence. Understanding stereochemistry helps you predict how molecules interact with light and biological receptors, which is critical for pharmacy and materials science. This knowledge forms the backbone of advanced organic synthesis and analytical techniques used in professional practice.

You will master the IUPAC rules for alkanes, polyfunctional compounds, cyclics, and aromatic systems. You will learn to distinguish chain, position, and functional group isomers, as well as tautomeric forms. The course trains you to assign E/Z configurations using priority rules and phantom atoms for geometric isomerism. You will also determine R/S configurations using Fischer projections and calculate stereoisomer counts for chiral centres, including meso compounds and diastereomers.

This course is primarily for first-year university students in chemistry, pharmacy, and engineering who need a solid foundation in organic structure. It suits Nigerian secondary school leavers preparing for advanced science exams like JAMB or WAEC who want to get ahead of university coursework. Non-science majors seeking basic chemical literacy will also benefit from the clear explanations of molecular architecture. Professionals refreshing their knowledge of systematic naming conventions will find the structured approach efficient and practical for immediate application.

organic-nomenclature-and-stereochemistry-undergraduate

This chapter sets the foundation for organic chemistry. It explains how electrons drive bonding and molecular behaviour. You will grasp the core principles needed to predict reaction outcomes and understand stability.

You will learn the course structure, define key electronic terms, and understand the role of valence electrons in bond formation.

This chapter covers quantitative analysis methods for determining elemental composition in organic compounds. Accurate percentage values are essential for verifying purity and establishing empirical formulas in laboratory work.

You will master combustion analysis for carbon and hydrogen, Kjeldahl and Dumas methods for nitrogen, Carius technique for halogens and sulphur, and stoichiometric calculations from experimental data to determine exact mass percentages of each element.

Quantitative Analysis

Pure chemicals are the foundation of valid science. This chapter teaches you to isolate target compounds from complex mixtures using physical properties like boiling point, solubility, and polarity.

Master purity checks, distillation methods, solvent extraction calculations, and chromatographic analysis to ensure your laboratory samples are clean and ready for use.

Purification Methods

This chapter covers the final steps in identifying a compound by calculating its empirical and molecular formulas. This process turns raw analytical data into a usable chemical representation.
You will learn to derive the simplest whole-number ratio of atoms and use molar mass to find the true molecular formula of a substance.

Formula Determination

Electron shifts dictate reactivity. This chapter maps how density moves through sigma and pi systems, defining the electronic landscape of organic molecules.

Master inductive and mesomeric effects, steric hindrance, and their impact on acidity, basicity, and resonance stability to predict chemical behaviour.

Electronic Effects

This chapter teaches you to identify elements inside organic compounds. You will learn standard tests that reveal the chemical makeup of unknown samples, a vital skill for accurate laboratory analysis and quality control in industry.

You will detect carbon and hydrogen using copper oxide; prepare sodium fusion extracts via the Lassaigne test; and identify nitrogen, sulfur, and halogens through specific colour changes with nitroprusside, silver nitrate, and iron(III) chloride.

Qualitative Analysis

Organic chemistry starts with how electrons move and hold atoms together. This course explains electronic theory, covering how bonds form and why electrons sit in specific places within a molecule. You will learn to isolate pure substances from mixtures using standard laboratory methods. We also cover how to identify what is inside a compound and calculate its exact chemical formula through qualitative and quantitative analysis.

Knowing how to purify chemicals is vital for making medicines, refining oil, and ensuring food safety. If you cannot separate a target substance from waste, you cannot manufacture a useful product. These techniques are the same ones used by forensic scientists to solve crimes and by engineers to develop new materials. Mastering these basics allows you to work accurately in any professional science lab or industrial setting.

By the end of this course, you will understand inductive effects, resonance, and how electron distribution dictates chemical reactions. You will be able to perform solvent extraction, chromatography, and distillation to get pure samples. You will also know how to test for specific elements like nitrogen or halogens and use mass data to determine empirical and molecular formulas. These skills ensure your laboratory results are reliable and scientifically sound.

This course is for first-year university students and science enthusiasts who want to move beyond textbook theory into practical lab work. It provides a solid base for anyone entering pharmacy, chemical engineering, or medical sciences. Even if you are just starting out, these lessons simplify complex ideas into clear steps, making it easier to grasp how modern science identifies the invisible components of the world around us.

electronic-theory-and-purification-techniques-chemistry

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

Applications of first-order ordinary differential equations.

Applications of First-Order Ordinary Differential Equations

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

Solutions of Second-Order Ordinary Differential Equations (1)

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

Solutions of Second-Order Ordinary Differential Equations (2)

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

Solutions of First-Order Ordinary Differential Equations

This course provides a complete guide to solving ordinary differential equations (ODEs). It covers the classification of differential equations and details the solution methods for first-order equations, including separable, homogeneous, exact, and linear types. The course then moves to second-order linear equations with constant coefficients and covers the methods of undetermined coefficients and variation of parameters.

Differential equations are the mathematical language used to model dynamic systems in science and engineering. They are used to describe the motion of objects, the flow of electric circuits, population growth, radioactive decay, and chemical reaction rates. A command of this subject is a non-negotiable requirement for any serious study in physics, engineering, or applied mathematics.

By the end of this course, you will be able to classify any ordinary differential equation by its order, degree, and linearity. You will be able to solve a wide variety of first-order ODEs and constant-coefficient second-order ODEs. You will also be able to model and solve real-world problems, such as orthogonal trajectories, exponential growth and decay, and simple electric circuits.

This course is for undergraduate students in mathematics, physics, engineering, and chemistry. It is a standard module in any mathematical methods curriculum and assumes a full prerequisite knowledge of single and multivariable calculus. It is an essential foundation for the study of partial differential equations, control theory, and advanced physics.

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

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

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

Mathematical Methods I

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

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

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

Elementary Differential Equations

ordinary-differential-equations-mathematical-methods

Mathematical Methods

Definition, evaluation and properties of the divergence of a vector field, and an introduction to the Laplacian of scalar and vector fields.

Divergence of a Vector Field

Definition, evaluation and properties of the gradient of a scalar function.

Gradient of a Scalar Field

Definition, evaluation and properties of the curl (spin or rotor) of a vector field.

Curl of a Vector Field

Definitions, examples and some visualizations of scalar and vector fields.

Scalar and Vector Fields

Scalar (dot) and vector (cross) products of vectors using sign functions.

Scalar and Vector Products

Definitions and illustrations of coordinates, coordinate lines and coordinate surfaces for the Cartesian coordinate system, general curvilinear coordinates, and general orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (1)

Definitions and operations of gradient of a scalar field and divergence and curl of a vector field using sign functions and conventions such as the Einstein summation, Kronecker delta and Levi-Civita notations.

Use of Sign Functions

The gradient operator and contravariant bases vectors in general orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (3)

Divergence and curl operators in general orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (4)

The Laplacian operator in general orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (5)

Some special curvilinear coordinates - cylindrical and spherical coordinate systems.

Orthogonal Curvilinear Coordinates (6)

Meaning of right-handed coordinate systems, vector products of covariant base vectors in right-handed systems, length and volume elements in general orthogonal curvilinear coordinate systems.

Orthogonal Curvilinear Coordinates (2)

Meaning of scalars, vectors and tensors; Einstein's summation convention, Kronecker delta and Levi-Civita notations; orthogonal and orthonormal vector bases.

Do you want to learn how to manipulate and analyze scalar and vector fields using advanced calculus techniques? Do you want to understand the concepts and methods of scalar and vector products, gradient, divergence, curl, and Laplacian operators? Do you want to apply the principles of vector analysis to solve problems in physics, engineering, and other disciplines?

If you answered yes to any of these questions, then this course is for you!

In this course, you will learn the fundamentals of calculus of scalar and vector fields, which is the branch of mathematics that deals with functions and operators that map points in space to scalars or vectors. You will learn how to use different notations and conventions to express scalar and vector fields, such as Einstein's summation, Kronecker delta, and Levi-Civita symbols. You will learn how to perform scalar and vector products of vectors, which are useful for measuring angles, areas, and volumes. You will learn how to compute and interpret the gradient of a scalar field, which measures the rate and direction of change of a scalar function. You will learn how to compute and interpret the divergence and curl of a vector field, which measure the sources and vortices of a vector field. You will also learn how to use the Laplacian operator, which combines the divergence and the gradient, and is useful for studying heat conduction, electrostatics, and fluid dynamics.

This course will equip you with the skills and knowledge to solve various problems involving scalar and vector fields. You will be able to apply the concepts and methods of calculus of scalar and vector fields to fields such as physics, engineering, chemistry, biology, and more. You will also be able to appreciate the power of vector analysis theorems, such as Gauss', Green's, and Stokes' theorems, which relate the integrals of scalar and vector fields over different domains.

By the end of this course, you will be able to:

- Define and explain the meaning of scalars, vectors, and tensors

- Identify and use different notations and conventions for scalar and vector fields

- Perform scalar and vector products of vectors and calculate angles, areas, and volumes

- Compute and interpret the gradient of a scalar field and its properties

- Compute and interpret the divergence and curl of a vector field and their properties

- Compute and interpret the Laplacian of a scalar or a vector field and its applications

- Use different coordinate systems and bases to express scalar and vector fields and operators

- Compute and interpret the gradient, divergence, curl, and Laplacian in orthogonal curvilinear coordinates

- Recognize and use some special curvilinear coordinates, such as cylindrical and spherical coordinates

- Apply the vector analysis theorems, such as Gauss', Green's, and Stokes' theorems, to relate the integrals of scalar and vector fields over different domains

Created for students, engineers, scientists, and anyone passionate about learning the calculus of scalar and vector fields, this course assumes a basic background in calculus, linear algebra, and vector algebra. With engaging video lessons, interactive quizzes, practice problems, and supportive peer interactions, you'll embark on a journey of discovery and mastery like never before.

Seize the opportunity to advance your mathematical skills and unlock new problem-solving possibilities with an enrolment right away!

Advanced engineering mathematics covering solid analytical geometry, integrals, scalar and vector fields, matrices and determinants and complex variables.

Curated for third-year students of engineering at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Engineering Analysis I

Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors.

Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Mathematical Methods II

Master the 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

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

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This chapter explains how electrons move in organic reactions. You will learn to identify reagents and map bond changes using standard notation to predict products accurately.

You will distinguish electrophiles from nucleophiles; differentiate polar and radical pathways; apply curved arrow notation correctly; and track electron flow to determine reaction outcomes without guessing.

Electronic Movements

This chapter establishes the foundation of organic reaction mechanisms. You will learn how bonds break and why specific intermediates form during chemical changes.

You will distinguish homolytic from heterolytic fission; identify carbocations and carbanions; explain stability trends for methyl, primary, secondary, and tertiary species.

This chapter explains the energy changes that control whether reactions occur and how fast they proceed. You will learn to read energy diagrams and identify the barriers molecules must overcome during bond breaking and formation.

You will define activation energy and Gibbs free energy; distinguish exergonic from endergonic pathways; locate transition states and intermediates on profiles; and interpret multi-step reaction diagrams accurately.

Reaction Energetics

This chapter covers the mathematics of reaction speed. You will learn to write rate equations and distinguish molecularity from reaction order to predict how concentration changes affect industrial output.

You will derive rate laws from concentration data; define reaction order precisely; identify unimolecular and bimolecular steps; and separate molecularity from kinetic order without confusion.

Chemical Kinetics

This chapter classifies organic reactions by their bond changes. You will learn to identify the four main types and predict products by inspecting reactant structures before applying specific mechanisms.

You will distinguish addition from substitution; identify elimination and rearrangement patterns; classify reactions by structural signatures; and predict major products without memorising every equation.

Reaction Types

Organic reactions follow specific pathways called mechanisms rather than happening by chance. This course covers how molecules break and form bonds through step-by-step processes and measured reaction rates. You will track electron movement, identify intermediates, and understand the kinetics governing reaction speed to predict system behaviour using fundamental principles instead of memorisation.

Understanding these mechanisms is vital for pharmacy, oil and gas, and food processing industries. Knowing how a reaction works allows you to control it to produce better medicines or stronger materials. These skills apply directly to laboratory research, chemical manufacturing, and quality control. Mastering reaction kinetics enables the optimisation of industrial processes, which saves time and reduces waste in commercial production.

You will gain the ability to draw reaction mechanisms using curly arrows to show electron flow and calculate reaction rates. The course teaches how temperature and catalysts change reaction speeds and helps you predict final products. You will explain why specific pathways are preferred over others, master the relationship between energy changes and reaction timing, distinguish bond fission types, and classify reactions into addition, substitution, elimination, or rearrangement categories.

This course is for undergraduate chemistry and engineering students needing a solid foundation in organic chemistry. It also suits science teachers or laboratory assistants wanting to refresh their knowledge of reaction pathways. Even those new to the subject will benefit from clear explanations of bond-breaking and bond-making. Any student preparing for advanced exams will find these logical steps useful for solving complex chemical problems without guessing.

organic-reaction-mechanisms-and-kinetics-chemistry

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.

Electric Dipoles

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.

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.

Courses

Fundamentals and Techniques of Differentiation - Calculus (Undergraduate Foundation)

Introduction to Organic Chemistry (Undergraduate Foundation)

Permutation and Combination - Mathematics (Undergraduate Foundation)

Kinematics of Particles - Engineering Mechanics (Undergraduate Advanced)

Organic Nomenclature and Stereochemistry (Undergraduate Foundation)

Electronic Theory and Purification Techniques - Chemistry (Undergraduate Foundation)

Ordinary Differential Equations - Mathematical Methods (Undergraduate Advanced)

Calculus of Scalar and Vector Fields (Undergraduate Advanced)

Organic Reaction Mechanisms and Kinetics - Chemistry (Undergraduate Foundation)

Electric Potential and Capacitors - Physics (Undergraduate Foundation)

Introduction to Physics (Undergraduate Foundation)

Static Charges and Fields - Physics (Undergraduate Foundation)