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.

Introduction

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.

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

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

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

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Mathematics

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

Proving Limits Involving Infinity

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

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

Evaluating Finite Limits (1)

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

Evaluating Limits at Infinity

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

Proving Finite Limits

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

Evaluating Finite Limits (2)

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

Evaluating Finite Limits (3)

Limits dictate how mathematical models predict exact behaviour at critical points. This course covers formal and informal definitions of finite and infinite limits. You will study direct substitution, algebraic factorisation, conjugate multiplication, the squeeze theorem, and piecewise function analysis. The material handles indeterminate forms through derivative ratio techniques and rationalisation. Each method includes repeated worked examples to fix the procedure in your mind.

Engineers and analysts use limit evaluation to calculate instantaneous speed, stress thresholds in structures, and signal decay rates. Financial planners apply these rules to compound interest and market trend forecasting. Computer simulation software relies on accurate limit logic to prevent calculation crashes and rounding errors. Mastery of this material gives you reliable tools for real technical modelling and removes guesswork from system analysis.

You will construct formal proofs for finite and infinite limits using strict logical bounds. You will select the correct evaluation method from substitution, algebra, or derivative rules without hesitation. You will resolve indeterminate forms and apply boundary checks to complex piecewise cases. You will verify every answer through graphical inspection and numerical approximation. You will gain complete control over the transition points of any real function.

This course targets university students and postgraduate researchers in engineering, physics, or mathematics. Advanced secondary school students with strong algebra skills will secure a clear advantage before university entry. Professionals preparing for technical exams will rebuild their analytical foundation. The structured progression and repeated practice ensure that any disciplined learner can apply limit theory to academic or industrial problems.

OAU, Ife

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

Advanced calculus forms the backbone of engineering, physics, and data science. This track follows the official NUC CCMAS syllabus for MTH 201 to build your mathematical foundation from scratch. You will master real-valued functions, limits, continuity, and differentiability before moving to partial differentiation and multiple integration. The content moves from single-variable theory to multivariable applications used in real-world modelling. Each module uses strict definitions and repeated worked examples to ensure you can solve problems under exam pressure. This is not just theory; it is the practical toolkit required for technical degrees and professional analysis.

This programme targets undergraduates in engineering, physical sciences, and mathematics. It suits learners who need to pass MTH 201 with high marks or build a strong base for advanced studies. Secondary school leavers with strong algebra skills can use this track to prepare for university-level rigour. Professionals returning to technical fields will refresh their analytical abilities quickly. If you plan to work in structural design, circuit analysis, fluid dynamics, or economic modelling, this track provides the essential mathematical language you must command.

You will analyse domain, range, and behaviour of complex functions without hesitation. You will evaluate limits and prove continuity using formal logical bounds. You will apply differentiation rules, Rolle's theorem, and Taylor series to approximate and optimise systems. You will compute partial derivatives and solve constrained optimisation problems using Lagrange multipliers. You will perform multiple integration over lines, surfaces, and volumes. These skills prepare you for vector calculus, differential equations, and core engineering courses. You will gain the confidence to handle advanced technical coursework and professional modelling tasks with precision.

limits-of-functions-single-variable-calculus

Advanced 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

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

Gradients and Directional Derivatives

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

Applications to Geometry

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

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

Lagrange Multipliers

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

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

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

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

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Multivariable Calculus

Meaning and identification of reflexive relations.

Reflexive Relations

Meaning and properties of the empty relation.

Empty Relation

Introduction to ordered pairs, Cartesian products and relations.

Meaning and identification of symmetric relations.

Symmetric Relations

Meaning and identification of equivalence relations.

Equivalence Relations

Meaning and identification of transitive relations.

Transitive Relations

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

Equivalence Classes

Meaning and examples of congruence relations and residue classes.

Congruence and Residue Classes

Meaning and properties of partially-ordered sets.

Partially-Ordered Sets

Lattices

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

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

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

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

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

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

Sets, Logic and Algebra I

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Introductory Abstract Algebra

Understanding the mean-value and Rolle's theorems.

Theorems on Differentiable Functions

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

Higher-Order Derivatives

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

Taylor and Maclaurin series expansion of differentiable functions.

Taylor and Maclaurin Series

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

Derivatives

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

Rules of Differentiation

Differentiation measures how a function changes at any given point. This course moves from the basic slope of a line to advanced derivative theory. You will define differentiability at a point and on an interval. You will link differentiability to continuity to spot where functions break. The material covers standard rules for sums, products, quotients, and composites. You will study Rolle's theorem and the mean-value theorem to understand function behaviour between points. The course extends to higher-order derivatives using Leibniz's formula. You will finish with Taylor and Maclaurin series to approximate complex functions with polynomials.

Engineers use derivatives to calculate velocity, acceleration, and force in moving systems. Electrical engineers apply these rules to analyse current change in circuits. Economists use differentiation to find marginal cost and maximise profit. Computer scientists deploy Taylor series for fast approximation in graphics and machine learning algorithms. Mastery of these tools allows you to model real-world dynamics with precision. You will gain the ability to predict system responses and optimise designs without relying on trial and error.

You will determine if a function is differentiable at any point or interval. You will apply product, quotient, and chain rules to differentiate complex expressions. You will use Rolle's theorem and the mean-value theorem to prove existence of critical points. You will compute higher-order derivatives and apply Leibniz's formula for product terms. You will construct Taylor and Maclaurin polynomials to approximate functions near a centre. You will estimate error bounds for these approximations. You will solve applied problems that merge multiple differentiation techniques.

This course targets undergraduates and graduate students in mathematics, engineering, and physics. It suits students preparing for advanced calculus examinations or research projects. Secondary school leavers with strong algebra skills can use this material to bridge the gap to university-level analysis. Professionals returning to technical work will refresh their understanding of rate of change. The clear structure and repeated practice ensure that any disciplined learner can master differentiation for academic or industrial application.

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

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

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

Real Analysis I

differentiability-and-derivatives-of-functions

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

Centroids

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

Position Vectors

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

Division of a Line

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

Vector Projections

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

Vector Components

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

Vector Algebra

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

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

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

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

Vector analysis is the mathematical backbone of classical mechanics, electromagnetism, and fluid dynamics. This learning track delivers the complete NUC CCMAS MTH 210 curriculum, rigorously progressing from fundamental vector algebra to the advanced differential and integral calculus of scalar and vector fields used in complex engineering and scientific modelling.

This programme is targeted at undergraduates in engineering, physics, mathematics, and computer science. It provides the essential mathematical toolkit for students entering disciplines that rely on applied mathematics and spatial analysis, and serves as a rigorous refresher for professionals needing to solidify their command of vector principles.

You will master the full spectrum of vector operations including dot, cross, and triple products, and apply them to solve geometric problems and vector equations. You will acquire the skills to analyze the differential geometry of curves using the Frenet-Serret framework and apply the powerful gradient, divergence, curl, and Laplacian operators in various coordinate systems. Completion establishes the critical mathematical foundation demanded for advanced studies in continuum mechanics, electrodynamics, and theoretical physics.

Vector Analysis

vector-algebra-and-foundational-geometry-vectors

Vectors

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.

University

Statics decides if a structure stands or fails. This track teaches force calculation, support reactions, and equilibrium analysis for stable engineering systems. You will move from particle resolution to full structural modelling. The curriculum delivers the exact mathematical foundation required for real design work.

This programme is for first and second-year engineering undergraduates. It suits mechanical, civil, aerospace, and structural students, plus polytechnic learners. You only need basic physics and vector algebra. The sequence provides the precise calculation methods your degree demands.

You will finish able to resolve two and three dimensional forces, compute moments and couples, and solve rigid body equilibrium problems. You will calculate truss loads, evaluate friction in screws and belts, locate centres of gravity, and verify systems using virtual work. These results secure high grades in advanced mechanics modules and prepare you for structural design, machine engineering, and civil site roles.

Engineering Mechanics I

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

statics-of-particles-engineering-mechanics-undergraduate

Engineering Mechanics

This chapter provides the roadmap for the course. It introduces the concept of a dynamic, reversible chemical reaction and outlines the progression from the principles of equilibrium to their application in acid-base chemistry.

Key learning objectives include: understanding the overall course structure and appreciating the importance of equilibrium in controlling the outcomes of chemical reactions.

This chapter establishes the foundational mechanics of reversible reactions and the dynamic state of equilibrium. Mastering these principles is vital for quantifying reaction yields and predicting the behaviour of chemical systems in industrial and biological environments.

You will master five objectives: defining the dynamic equilibrium state; interpreting rate and concentration graphs; deriving Kc expressions via the law of mass action; differentiating phase-dependent equilibria; and calculating Kp for gaseous systems.

Chemical Equilibrium

This chapter bridges chemical equilibria with the practical solubility of salts in water. It explains why some substances dissolve while others form precipitates, providing the tools needed to predict reaction outcomes in industrial and environmental chemistry.

You will master the solubility product constant, the common ion effect, and the use of the reaction quotient to determine if a precipitate will form. You will also learn to calculate molar solubility and understand how pH influences the dissolving process.

Solubility and Solubility Products

This chapter applies equilibrium principles to acid-base chemistry, a critical area for understanding chemical reactivity in industrial and biological systems. You will master the quantitative tools needed to predict and control the behaviour of aqueous solutions.

By the end, you will define acids and bases using Arrhenius, Bronsted-Lowry, and Lewis theories; calculate pH and pOH for strong and weak electrolytes; predict salt hydrolysis outcomes; and design buffer systems to resist pH changes.

Acids, Bases, and Salts

This chapter covers the qualitative analysis of equilibrium shifts in response to external stresses. Mastering Le Chateliers principle is mandatory for predicting and controlling the direction of reversible reactions in industrial and biological systems.

You will master the mechanical prediction of directional shifts caused by concentration changes; the resolution of gaseous equilibria under varying pressure or volume; the effect of temperature on exothermic and endothermic reactions; and the role of catalysts in kinetic acceleration.

Le Chatelier's Principle

This chapter examines the mathematical dependence of the equilibrium constant on the stoichiometry of the chemical equation. Understanding these correlations is essential for correctly modifying Kc values when reversing reactions, scaling coefficients, or combining multiple elementary steps in industrial process design.

You will master four objectives: calculating the reciprocal constant for reversed reactions; determining new Kc values for scaled stoichiometric coefficients; applying the rule of multiple equilibria for multi-step reactions; and establishing consistent units for the equilibrium constant.

Stoichiometric Relations

This chapter explains how solids dissolve in water until they reach a balance. It is vital for understanding how scale forms in pipes, how bones grow, and how to remove pollutants from liquid waste. You will learn to predict if a solid will settle out of a liquid and how to control this process in the lab or industry.

By the end, you will master writing solubility product expressions, calculating molar solubility from constants, and predicting precipitate formation using the ion product.

Solubility Equilibria and Precipitation

This course provides a complete guide to chemical equilibria, the state where the rates of forward and reverse reactions are equal. It covers the law of mass action, the definition and calculation of the equilibrium constant, and the factors that can cause a shift in the equilibrium position. The course then applies these principles to the study of aqueous equilibria, including the properties of acids, bases, and salts, and the calculation of pH.

The principles of equilibrium govern the outcomes of all reversible reactions, from industrial synthesis to biological processes. A command of this topic is essential for chemists and chemical engineers to maximise the yield of a desired product, for environmental scientists to understand natural water systems, and for biochemists to analyse metabolic pathways.

By the end of this course, you will be able to write the expression for the equilibrium constant for any reversible reaction, use Le Chatelier's principle to predict how a system at equilibrium will respond to changes in concentration, pressure, or temperature, and perform calculations involving the pH of acidic and basic solutions.

This course is for students who have a solid foundation in stoichiometry. It is a mandatory course for all students of chemistry and chemical engineering and is a direct prerequisite for the study of analytical chemistry, biochemistry, and environmental science.

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

chemical-equilibria-and-acid-base-chemistry-undergraduate

Chemistry

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

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

Conclusion

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

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

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

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

The Mole Concept

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

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

Chemical Formulae

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

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

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

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

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

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

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

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

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

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

Kinetic Theory

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

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

Gas Laws

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

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

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

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

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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

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.

Courses

Permutation and Combination - Mathematics (Undergraduate Foundation)

Limits of Real-Valued Functions - Advanced Calculus (Undergraduate Advanced)

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

Relations - Introductory Abstract Algebra (Undergraduate Advanced)

Differentiability of Real-Valued Functions - Advanced Calculus (Undergraduate Advanced)

Vector Algebra and Geometry - Vectors (Undergraduate Foundation)

Statics of Particles - Engineering Mechanics (Undergraduate Advanced)

Chemical Equilibria and Acid-Base Chemistry (Undergraduate Foundation)

Stoichiometry of Composition - Chemistry (Undergraduate Foundation)

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

Chemical Thermodynamics - Chemistry (Undergraduate Foundation)

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