A comprehensive treatise of ordinary differential equations covering solutions of first-order and second-order ordinary differential equations, and applications of first-order ordinary differential equations.

Ordinary Differential Equations - Advanced Calculus

Numerical differentiation (finite difference calculus) and its applications.

Finite Differences - Numerical Analysis

Solutions and applications of difference equations.

Difference Equations - Numerical Analysis

Numerical solution of non-linear equations, integration.

Numerical Methods - Advanced Calculus

Comprehensive treatise of advanced calculus covering ordinary differential equations, finite differences, difference equations and numerical integration.

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

Tutorials for undergraduates in engineering and physical sciences.

Max Engineering Tutorials

MAT 241: Ordinary Differential Equations

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.

Kinematics of Particles - Engineering Mechanics (Dynamics)

This course will equip you with the skills and knowledge to tackle various engineering and scientific problems involving kinetics of particles, which is the branch of mechanics that deals with the relation between the forces acting on a particle and its resulting motion. From the basics of mechanics, dynamics, and kinetics to getting comfortable with Newton's second law and momentum methods, you will clearly understand how forces cause motion.

Boost your problem-solving skills and open up new career opportunities in fields like aerospace engineering, mechanical engineering, civil engineering, biomedical engineering, physics, chemistry, and more. Imagine being able to predict how a ball flies through the air, or understanding why a car behaves the way it does in a crash. This course helps you do these and many more!

By the end of this course, you will be able to:
- Explain the fundamentals of mechanics, dynamics, and kinetics
- Use Newton's second law to analyze how forces make bodies speed up or slow down
- Learn about energy, how forces do work, and how these relate to motion of bodies
- Figure out how collisions work and what motion happens before and after collisions

This course is designed for students, engineers, scientists, and anyone who is interested in learning the basics of kinetics of particles. The course assumes that you have a basic background in calculus, physics, vector algebra, and statics.

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.

Kinetics of Particles by Newton's Second Law - Engineering Mechanics (Dynamics)

Analysis of motion of a particle with respect to the forces causing the motion by work-energy and impulse-momentum principles.

Kinetics of Particles by Energy and Momentum Methods - Engineering Mechanics (Dynamics)

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

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

By the end of this course, you will be able to:
- Understand the rotation of rigid bodies about fixed axes and its applications
- Analyze absolute motion to understand how rigid bodies move through space
- Explore relative velocity analysis to solve complex motion problems
- Master the concept of instantaneous centres of rotation for precise motion analysis
- Understand relative acceleration analysis to predict the behavior of moving bodies
- Analyze motion relative to a rotating frame to unravel the dynamics of rotating systems

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

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

2D Kinematics of Rigid Bodies - Engineering Mechanics (Dynamics)

Do you want to learn how to apply the laws of physics to analyze the motion of rigid bodies in two dimensions?
Do you want to understand the concepts and methods of mass moment of inertia, general equations of motion, work, energy, impulse, and momentum?
Do you want to design and optimize mechanical systems and devices that involve the rotation and translation of rigid bodies?
If you answered yes to any of these questions, then this course is for you!

In this course, you will learn the fundamentals of two-dimensional kinetics of rigid bodies, which is the branch of mechanics that deals with the relation between the forces acting on a rigid body and its resulting motion. The course will equip you with the skills and knowledge to solve various engineering and scientific problems involving the motion of rigid bodies in two dimensions. You will be able to apply the concepts and methods of two-dimensional kinetics of rigid bodies to fields such as mechanical engineering, aerospace engineering, civil engineering, robotics, biomechanics, and more. You will also be able to appreciate the power and elegance of the physical laws that govern the motion of rigid bodies.

By the end of this course, you will be able to:
- Define and explain the meaning of kinetics, rigid body, and plane motion
- Calculate the mass moment of inertia of a rigid body using integration or parallel axis theorem
- Derive and use the general equations of motion for a rigid body in plane motion
- Analyze the motion of a rigid body undergoing translation, rotation about a fixed axis, or general plane motion
- Apply the principle of work and energy to determine the motion of a rigid body or a system of rigid bodies
- Apply the principle of conservation of energy to determine the motion of a rigid body or a system of rigid bodies under conservative forces
- Apply the principle of impulse and momentum to determine the motion of a rigid body or a system of rigid bodies
- Apply the principle of conservation of momentum to determine the motion of a rigid body or a system of rigid bodies

Tailored for students, engineers, scientists, and anyone passionate about learning the basics of 2D kinetics of rigid bodies, this course assumes a basic background in calculus, physics, vector algebra, and statics. 

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.

2D Kinetics of Rigid Bodies by Newton's Second Law - Engineering Mechanics (Dynamics)

Analysis of motion of a rigid body with respect to the forces causing the motion by work-energy and impulse-momentum principles.

2D Kinetics of Rigid Bodies by Energy and Momentum Methods - Engineering Mechanics (Dynamics)

This course teaches the fundamentals of dynamic systems modelling and simulation. You will learn how to:
- Define and classify static and dynamic systems by their mathematical models
- Use differential equations, transfer functions, and state-space models to describe the behavior of dynamic systems
- Analyze the stability, controllability, and observability of dynamic systems
- Model and simulate translational, rotational, electrical, electromechanical, hydraulic, and thermal systems
- Draw analogies between different types of systems and exploit their similarities
- Use popular software tools such as MATLAB/SIMULINK to implement and test your models

By the end of the course, you will have a solid foundation in dynamic systems modelling and simulation, and you will be able to apply it to your own projects and interests. You will also gain a valuable skill that is in high demand in many industries and disciplines, such as:
- Engineering: design, control, and optimization of dynamic systems such as robots, vehicles, machines, power systems, etc.
- Science: understanding and prediction of natural phenomena such as population dynamics, climate change, epidemics, etc.
- Economics: modelling and analysis of economic systems such as markets, supply chains, production, consumption, etc.
- Policy: evaluation and improvement of policy actions and interventions in complex social systems such as health, education, environment, etc.

Whether you are a student, a professional, or a hobbyist, this course will help you master the art and science of dynamic systems modelling and simulation, and open up new possibilities for your learning and career.

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.

Introduction to Dynamic Systems and Vibration - Engineering Mechanics (Dynamics)

Comprehensive treatise of motion of particles and rigid bodies, with focus on the motion of engineering mechanisms.

Curated for second-year students of engineering and at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

MEE 206: Engineering Mechanics II (Dynamics)

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

Calculus of Scalar and Vector Fields

Do you want to learn how to work with numbers that go beyond the real line? Do you want to understand the concepts of imaginary unit, conjugate, modulus, argument, and polar and exponential forms of complex numbers? Do you want to master the skills of performing algebraic and geometric operations on complex numbers using different methods and tools?
If you answered yes to any of these questions, then this course is for you!

In this course, you will learn how to:
- Define and classify complex numbers and their real and imaginary parts
- Perform addition, subtraction, multiplication, and division of complex numbers using the standard form a + bi
- Find the conjugate, modulus, and argument of a complex number and use them to compare and simplify complex numbers
- Represent complex numbers on the Argand plane and visualize their geometric properties and transformations
- Convert complex numbers from rectangular to polar and exponential forms and vice versa
- Use De-Moivre's theorem and Euler's formula to find the powers and roots of complex numbers in polar and exponential forms
- Use complex numbers to define and manipulate trigonometric and hyperbolic functions and their inverses
- Use complex numbers to define and manipulate logarithmic functions and their properties
- Use complex numbers to graph and solve equations of circles, lines, and other curves on the complex plane

This course is suitable for anyone who wants to learn or review the basics of complex numbers and their applications. It is especially useful for students and professionals in engineering, physics, computer science, cryptography, and other related fields.

By the end of this course, you will have a solid understanding of complex numbers and their operations. You will also be able to apply the knowledge and skills you gain to real-world problems and challenges that involve complex numbers.

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.

Complex Numbers

Do you want to learn how to work with abstract spaces and transformations that preserve their structure? Do you want to understand the concepts of vector subspaces, linear combinations, linear dependence, basis, dimension, coordinates, and properties of vector spaces? Do you want to master the skills of defining and manipulating linear maps, their kernels, images, matrix representations, and transition matrices?
If you answered yes to any of these questions, then this course is for you!

Linear Algebra: Linear Vector Spaces and Linear Maps is a comprehensive and engaging course that covers the fundamentals of vector spaces and linear maps and their applications in mathematics and science. You will learn how to:
- Define and classify vector spaces and their subspaces over a given scalar field
- Perform operations on vectors using linear combinations and scalar multiplication
- Determine whether a set of vectors is linearly dependent or independent and find a basis and dimension for a vector space or subspace
- Find the coordinates of a vector with respect to a given basis and change the basis using transition matrices
- Define and classify linear maps between vector spaces and find their domains, codomains, ranges, and null spaces
- Find the kernel and image of a linear map and use them to determine whether a linear map is one-to-one or onto
- Represent a linear map using a matrix and perform matrix operations such as addition, multiplication, and inversion
- Use different methods and tools to solve systems of linear equations, such as Gaussian elimination, row reduction, and inverse matrices

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

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

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

Linear Vector Spaces - Linear Algebra

A comprehensive study of linear transformations (or maps) - definition, kernel, range, matrix representation and related matters.

Linear Maps (Transformations) - Linear Algebra

Do you want to learn how to work with matrices and their properties, operations, and applications? Do you want to understand the concepts of determinants, eigenvalues, eigenvectors, diagonalization, quadratic and canonical forms? Do you want to master the skills of solving systems of linear equations, finding inverses, and computing matrix functions using different methods and tools?
If you answered yes to any of these questions, then this course is for you!

This course covers the fundamentals of matrix theory and its applications in mathematics and science. You will learn how to:
- Define and classify matrices and their special types, such as symmetric, orthogonal, diagonal, and identity matrices
- Perform matrix addition, subtraction, multiplication, and scalar multiplication using the algebraic properties of matrices
- Find the transpose, conjugate, and adjoint of a matrix and use them to simplify matrix operations and expressions
- Perform elementary row and column transformations on matrices and use them to find the row echelon form, reduced row echelon form, rank, and nullity of a matrix
- Find the minors, cofactors, and determinants of matrices and use them to calculate the area, volume, and orientation of geometrical figures
- Find the inverse of a matrix using the adjoint method or the row operations method and use it to solve systems of linear equations
- Find the eigenvalues and eigenvectors of a matrix using the characteristic polynomial and the Cayley-Hamilton theorem and use them to analyze the behavior and stability of dynamical systems
- Diagonalize a matrix using the eigenvalues and eigenvectors and use it to compute matrix functions, such as polynomials, exponentials, sines, and cosines of matrices
- Find the quadratic and canonical forms of an equation using the coefficient matrix and the transformation matrix and use them to classify and graph conics and quadrics
- Use computer software, such as MS-Excel, MATLAB, and Python, to perform matrix operations and computations efficiently and accurately

This course is suitable for anyone who wants to learn or review the basics of matrix theory and its applications. It is especially useful for students and professionals in algebra, calculus, differential equations, linear programming, optimization, cryptography, computer graphics, data science, machine learning, and other related fields.

By the end of this course, you will have a firm grasp of the theory and applications of matrices and determinants. You will also be able to apply the knowledge and skills you gain to solve real-world problems and challenges that involve matrices.

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.

Matrices, Determinants, and Systems of Linear Equations

This course provides a clear and structured exploration of eigenvalues, eigenvectors, and diagonalization, focusing on both theory and real-world applications. 

You’ll learn how to compute eigenvalues and eigenvectors, understand their geometric significance, and apply diagonalization to simplify complex matrix operations. 

Topics include linear transformations, dynamical systems, and applications in physics, engineering, and machine learning. 

The course is designed for students, engineers, and data scientists seeking a strong foundation in matrix methods. By the end, you'll confidently apply these concepts in problem-solving and computational modeling.

Eigenvalues, Eigenvectors and Diagonalization of Matrices

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.

MTH 202: Mathematical Methods II

Do you want to learn how to work with quantities that have both magnitude and direction? Do you want to understand the concepts of scalars, vectors, tensors, components, projections, products, and equations of vectors? Do you want to master the skills of performing algebraic, geometric, differential, and integral operations on vectors using different methods and tools?
If you answered yes to any of these questions, then this course is for you!

This course covers the fundamentals of vectors and their applications in mathematics and mechanics. You will learn how to:
- Define and classify scalars, vectors, and tensors and their properties
- Represent vectors by directed line segments, unit vectors, direction cosines, and coordinates
- Perform vector addition, subtraction, and multiplication by a scalar using the triangle and parallelogram laws
- Apply vector algebra to various geometrical problems involving mid-points, parallelism, and collinearity
- Find the position vector of a point and use it to locate the point in space
- Resolve vectors into components in two and three dimensions and use them to simplify vector operations
- Divide a line in a given ratio internally or externally using vectors
- Project a vector on another vector or a plane and use it to find the angle or distance between them
- Find the centroid of a set of points or a polygon using vectors
- Find the scalar and vector products of two vectors and use them to calculate the area, volume, and orthogonality of geometrical figures
- Find the scalar and vector triple products of three vectors and use them to determine the coplanarity and linear dependence of vectors
- Solve vector equations with unknown vectors or scalars using various techniques
- Find the vector equation of a line or a plane and use it to describe the direction, intersection, angle, and distance of lines and planes
- Find the parametric and non-parametric equations of circles, parabolas, ellipses, and hyperbolas using vectors
- Differentiate and integrate vector-valued functions and use the rules of vector differentiation and integration
- Apply vector differentiation and integration to find the derivatives, arc length, and curvature of parametric curves
- Find the tangential, normal, and binormal vectors and the osculating, normal, and rectifying planes of a parametric curve using the Frenet-Serret formulas
- Apply vectors to various problems in mechanics, such as forces, equilibrium, work, energy, momentum, displacement, velocity, acceleration, and motion in different coordinate systems and frames of reference

This course is suitable for anyone who wants to learn or review the basics of vectors and their applications. It is especially useful for students and professionals in engineering, physics, geometry, calculus, and other related fields.

By the end of this course, you will have a firm understanding of vectors and their operations. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve vectors.

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.

Vector Algebra and Geometry - Introduction to Vectors

Vector products and vector equations (standard and parametric) equations of geometries.

Vector Products and Vector Equations of Geometries - Introduction to Vectors

Differentiation and integration of vectors and their applications to mechanics and differential geometry.

Vector Differentiation, Integration and Their Applications - Introduction to Vectors

Introduction to vectors, covering vector algebra, geometry, products, vector equations of geometries, vector differentiation, integration and their applications to mechanics and differential geometry.

Curated for first-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.

MTH 104: Vectors

Analysis of systems of concurrent forces in two and three dimensions; equilibrium of particles in two and three dimensions.

Statics of Particles - Engineering Mechanics (Statics)

Analysis of systems of forces on rigid bodies in two and three dimensions - moments of forces about a point or axis, couples and equivalent systems of forces.

Systems of Forces on Rigid Bodies - Engineering Mechanics (Statics)

Analysis of equilibrium of rigid bodies in two and three dimensions, and an introduction to statically-indeterminate reactions.

Equilibrium of Rigid Bodies - Engineering Mechanics (Statics)

Analysis of trusses, frames and machines.

Analysis of Structures - Engineering Mechanics (Statics)

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

Dry Friction - Engineering Mechanics (Statics)

Analysis of centroids and moments of inertia.

Centroids and Centres of Gravity - Engineering Mechanics (Statics)

Analysis of the second moments of areas and its applications.

Moments of Inertia - Engineering Mechanics (Statics)

Analysis of equilibrium of rigid bodies by the method of virtual work.

Method of Virtual Work - Engineering Mechanics (Statics)

Comprehensive treatise of static equilibrium of particles and rigid bodies, with focus on the stability of machines and structures.

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

MEE 205: Engineering Mechanics I (Statics)

This course comprehensively treats limits, continuity and differentiability of real-valued single-variable functions. You will learn how to:
- Define and classify real-valued functions and their properties, such as domain, range, and graphs
- Identify and compare different kinds of real-valued functions, such as composites, piecewise functions, polynomials, rational functions, algebraic functions, transcendental functions, odd and even functions
- Find the limit of a function as the independent variable approaches a certain value, and use the formal (epsilon-delta) and informal definitions of limits
- Find the limit of a function using various methods and techniques, such as direct substitution, theorems, graphing, factorization, conjugates, L’Hôpital’s rule, use of known special limits, the squeeze theorem, two-sided limits, etc.
- Find the limit of a function at infinity or an infinite limit, and use the formal and informal definitions of limits at infinity and infinite limits
- Find the continuity of a function at a point or on an interval, and use the graphical, formal, and informal definitions of continuity
- Identify and understand different types of discontinuities, such as removable, jump, and infinite discontinuities
- Apply the max-min theorem and the intermediate-value theorem to continuous functions and their graphs
- Find the differentiability of a function at a point or on an interval, and use the graphical and formal definitions of differentiability
- Find the derivative of a function as the slope of the tangent line or the rate of change of the function, and use the first-principle method and the rules of differentiation to calculate derivatives
- Apply the mean-value theorem and the Rolle's theorem to differentiable functions and their graphs
- Find the higher-order derivatives of a function by applying the differentiation rules repeatedly, and use the notation and terminology for higher derivatives
- Apply the Leibniz's formula to find the derivatives of products of functions
- Find the Taylor and Maclaurin series expansions of differentiable functions and use them to approximate functions and their values

This course is suitable for anyone who wants to learn or review the advanced topics of calculus and its applications. It is especially useful for students and professionals in analysis, differential equations, optimization, physics, engineering, and other related fields. 

By the end of the course, you will have a firm grasp of advanced single-variable differential calculus concepts and their applications. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve analyzing the behavior, trends, and optimization of functions and their graphs.

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.

Limits, Continuity and Differentiability of Real-Valued Functions - Advanced Calculus

Do you want to learn how to work with infinite sums of numbers and functions and their properties, operations, and applications? Do you want to understand the concepts of progressions, sequences, series, and power series, and how they relate to the approximation, convergence, and divergence of functions? Do you want to master the skills of finding and applying various tests and criteria of convergence and divergence to different types of series, such as arithmetic, geometric, harmonic, p-series, and alternating series?
If you answered yes to any of these questions, then this course is for you!

This course reviews the fundamental concepts of finite progressions and provides a thorough treatise of the convergence of infinite real sequences and series. You will learn how to:
- Define and classify progressions and their properties, such as common difference, common ratio, and sum to infinity
- Define and classify sequences and their properties, such as terms, general term, and boundedness
- Find the convergence or divergence of a sequence using the formal definition and various theorems and examples
- Define and classify series and their properties, such as partial sums, absolute and conditional convergence, and uniform convergence
- Find the convergence or divergence of a series of real numbers using various methods and techniques, such as direct comparison, integral test, p-series test, ratio test, Raabe's test, and alternating series test
- Define and classify sequences and series of functions and their properties, such as pointwise and uniform convergence, and term-by-term differentiation and integration
- Find the convergence or divergence of a sequence or series of functions using various methods and techniques, such as Weierstrass M-test, Abel's test, and Dirichlet's test
- Define and classify power series and their properties, such as radius and interval of convergence, and term-by-term differentiation and integration
- Find the convergence or divergence of a power series using various methods and techniques, such as ratio test, root test, and Cauchy-Hadamard theorem

This course is suitable for anyone who wants to learn or review the advanced topics of calculus and its applications. It is especially useful for students and professionals in analysis, differential equations, optimization, physics, engineering, and other related fields. By the end of this course, you will have a solid foundation of the theory and practice of calculus and its operations. You will also be able to apply the knowledge and skills you learned to real-world problems and challenges that involve infinite sums of numbers and functions and their approximation, convergence, and divergence.

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.

Convergence of Infinite Sequences and Series - Advanced Calculus

This course introduces multi-variable real-valued functions and thoroughly addresses their limits, continuity, partial and total derivatives, applications and related concepts. You will learn how to:
- Define and classify real-valued functions of several variables and their properties, such as domain, range, graphs, level curves and surfaces
- Find the limit of a function of several variables as the independent variables approach certain values, and use the formal and informal definitions of limits
- Find the continuity of a function of several variables at a point or on a set, and use the graphical, formal, and informal definitions of continuity
- Find the partial derivative of a function of several variables with respect to one of the independent variables, and use the formal definition and various rules of partial differentiation
- Find the higher-order partial derivatives of a function of several variables by applying the partial differentiation rules repeatedly, and use the notation and terminology for higher partial derivatives
- Find the partial derivative of a function of several variables that is composed of other functions, and use the chain rule to differentiate composite functions
- Find the partial derivative of a function of several variables that is homogeneous, and use the Euler's theorem to simplify the calculation
- Find the total differential and derivative of a function of several variables, and use them to approximate the change in the function value
- Find the partial derivative of a function of several variables that is defined implicitly by an equation
- Find the Jacobian determinant of a function of several variables, and use it to obtain derivatives of implicit functions
- Find the Taylor series expansion of a function of several variables, and use it to approximate the function value and its derivatives
- Find the equation of the tangent plane and the normal line to a surface defined by a function of two variables, and use them to analyze the local behaviour of the surface
- Find the extreme values of a function of several variables, and use the partial derivatives and the second derivative test to determine the nature of the extrema
- Find the extreme values of a function of several variables subject to some constraint, and use the partial derivatives and the method of Lagrange multipliers to solve the constrained optimization problem

This course is suitable for anyone who wants to learn or review the advanced topics of calculus and its applications. It is especially useful for students and professionals in analysis, differential equations, optimization, physics, engineering, and other related fields. By the end of this course, you will have a solid understanding of derivatives of functions of several variables. You will also be able to apply the knowledge and skills you gain to real-world problems and challenges that involve functions of several variables and their derivatives.

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.

Partial Differentiation and Its Applications - Advanced Calculus

Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations.

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.

MTH 201: Mathematical Methods I

Learn to analyze the three-dimensional Euclidean space using coordinates and equations with pin-point precision.
Designed for students, researchers, and professionals in fields such as engineering, computer graphics, physics, astronomy, medicine, biology, architecture, and building technology, this course provides a thorough understanding of coordinate systems, equations, and quadric surfaces.

On completing this course, you will have mastered:
- One-dimensional coordinate system and locating points
- Two-dimensional coordinate system, lines, and curves
- Conic sections and their equations
- Three-dimensional coordinate system and quadric surfaces
- Computer-aided visualization using Geogebra
- Coordinate transformation and rotation
- Equations in polar, cylindrical, and spherical coordinates

Throughout the course, you'll engage with high-quality video lessons, completely-solved examples, free-hand sketching techniques, and computer-aided visualizations. Multiple-choice quizzes will keep you focused and help reinforce your learning.

Prerequisites:
This course is designed for first-year undergraduate students in engineering and sciences. Prior knowledge of basic trigonometric ratios, determinants, and elementary row operations on matrices is helpful but not required, as the course is designed to be accessible to learners with varying backgrounds.

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.

3D Coordinates and Quadric Surfaces - Analytic Geometry

Multiple integration; line, surface and volume integrals.

Multiple Integration and Its Applications

Complex numbers, variables, functions and series in the complex plane.

Complex Variables

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.

Learning Tracks