Meaning of the range of a linear map, with worked examples.

Range

Meaning of the kernel of a linear map, with worked examples.

Kernel

Definition, examples and general properties of linear maps.

Linear Maps

Matrix representation of a linear map, with worked examples.

Matrix Representation

Meaning and properties of the transition matrix between two bases of a vector space.

Transition Matrix

Welcome and course outline. Review of elementary matrix and linear vector space concepts.

Introduction

Definition of a map (also mapping or transformation) between two sets, notations and some general properties of maps.

Maps

Further operations on linear maps involving their definition and properties.

Operations on Linear Maps

Analysis of the vector space of all linear maps between two given vector spaces. Addition, scalar multiplication, composition and inverse of linear maps.

Algebra of Linear Maps

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

Tutorials for undergraduates in engineering and physical sciences.

Max Engineering Tutorials

Linear Maps (Transformations) - Linear Algebra

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

Divergence of a Vector Field

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

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

Gradient of a Scalar Field

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

Curl of a Vector Field

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

Scalar and Vector Fields

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

Scalar and Vector Products

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

Orthogonal Curvilinear Coordinates (1)

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

Use of Sign Functions

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

Orthogonal Curvilinear Coordinates (3)

Divergence and curl operators in general orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (4)

The Laplacian operator in general orthogonal curvilinear coordinates.

Orthogonal Curvilinear Coordinates (5)

Some special curvilinear coordinates - cylindrical and spherical coordinate systems.

Orthogonal Curvilinear Coordinates (6)

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

Orthogonal Curvilinear Coordinates (2)

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

Calculus of Scalar and Vector Fields

Welcome to the course, review of fundamental concepts - equilibrium of rigid bodies, internal forces, external forces, differences between trusses, frames and machines.

Analysis of frames under various loading conditions.

Frames

Analysis of machines under various loading conditions.

Machines

Meaning, classification, load transmission and modelling of planar trusses.

Planar Trusses (1)

Analysis of planar trusses by the method of sections.

Planar Trusses (3)

Analysis of planar trusses by the method of joints.

Planar Trusses (2)

Modelling and analysis of space trusses - by methods of joints and sections.

Space Trusses

Analysis of trusses, frames and machines.

Analysis of Structures - Engineering Mechanics (Statics)

Meaning, operations and properties of eigenvalues and eigenvectors of matrices.

Eigenvalues and Eigenvectors

Diagonalization of matrices; evaluating polynomials and transcendentals of matrices.

Diagonalization of Matrices

Quadratic and canonical forms; transformations using symmetric matrices and orthogonal diagonalizing matrices.

Quadratic and Canonical Forms

Welcome and course outline. Review of elementary matrix concepts.

Diagonalization of symmetric matrices - orthogonal diagonalizing matrix, the Gram-Schmidt orthogonalization procedure, and related concepts.

Symmetric Matrices

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

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

Applications of first-order ordinary differential equations.

Applications of First-Order Ordinary Differential Equations

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

Solutions of Second-Order Ordinary Differential Equations (1)

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

Solutions of First-Order Ordinary Differential Equations

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

Solutions of Second-Order Ordinary Differential Equations (2)

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

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

Analysis of rectilinear motion with variable acceleration.

General Rectilinear Motion

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

Curvilinear Motion (1)

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

Special Rectilinear Motion (2)

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

 Curvilinear Motion (2)

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

Curvilinear Motion (3)

Analysis of rectilinear motion with constant acceleration - uniform motion and uniformly-accelerated motion; graphical analysis of erratic motion problems.

Special Rectilinear Motion (1)

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)

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

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

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

Vector Projections

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

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

Natural numbers, integers, rational numbers, real numbers, and an introduction to complex numbers and their descriptions.

Exponential (Euler's) representation of complex numbers; powers of complex numbers in exponential form.

Exponential Form

Operations on complex numbers; conjugates of complex numbers and their properties; equality of complex numbers.

Algebra of Complex Numbers

Polar representation of complex numbers; multiplication and division of complex numbers in polar form; powers of complex numbers in polar form (De-Moivre's theorem).

Polar Form

Manipulating sines and cosines with complex numbers.

Trigonometric Functions

Manipulating hyperbolic functions with complex numbers.

Hyperbolic Functions

Manipulating logarithms with complex numbers.

Logarithmic Functions

Equality and roots of complex numbers in polar form.

Roots

Geometric representation of complex numbers on the Argand plane; modulus of a complex number; general and principal arguments of a complex number.

Complex Numbers on the Argand Plane

Equations in two-dimensional coordinate geometry using complex numbers.

Graphing on the Complex Plane

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

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

Rotation About a Fixed Axis (2)

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

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

 Relative Velocity Analysis

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

Rotation About a Fixed Axis (1)

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

Relative Acceleration Analysis

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

Instantaneous Centre of Rotation

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

Absolute Motion Analysis

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

Motion Relative to a Rotating Frame

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

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

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

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

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

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

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

Definition and examples of linear vector subspaces.

Vector Subspaces

Basis and dimension of linear vector spaces.

Basis and Dimension

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

Linear Combinations and Spans

Linear dependence and independence of vectors in a vector space.

Linear Dependence and Independence

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

Matrix Spaces

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

Sums and Intersections

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

Coordinates

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

Meaning, operations and properties of matrix inverses.

Matrix Inverses

Meaning of matrices, types of matrices, and other descriptive terminologies. 

Addition of matrices, multiplication of matrices by scalars, multiplication of matrices by matrices, etc.

Algebra of Matrices

Transposition of matrices and properties of matrix transposes.

Transposition of Matrices

Meaning, operations and properties of determinants.

Determinants

Elementary row and column operations on matrices; row echelon and reduced row echelon  forms; rank and nullity of matrices.

Elementary Transformations

Manipulating matrices with Microsoft Excel (Google Sheets) software.

Computer-Aided Handling

Application of matrices, determinants and inverses in solving systems of linear equations, consistency of systems of linear equations and applicable theorems.

Systems of Linear Equations

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

Numerical solutions of equations in one variable - Newton-Raphson's iterative method.

Equations in One Variable (2)

Introduction, existence of solutions, and bisection method of numerical solutions of equations in one variable.

Equations in One Variable (1)

Introduction to numerical integration of single-variable functions. Trapezoidal rule.

Integration (1)

Simpson's 1/3 rule of numerical integration of single-variable functions.

Integration (2)

Numerical solution of non-linear equations, integration.

Numerical Methods - Advanced Calculus

Continuity of functions of several variables.

Continuity

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

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

Limits

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.

Courses

Customize your learning experience

Linear Maps (Transformations) - Linear Algebra

Calculus of Scalar and Vector Fields

Analysis of Structures - Engineering Mechanics (Statics)

Eigenvalues, Eigenvectors and Diagonalization of Matrices

Ordinary Differential Equations - Advanced Calculus

Kinematics of Particles - Engineering Mechanics (Dynamics)

Vector Algebra and Geometry - Introduction to Vectors

Complex Numbers

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

Linear Vector Spaces - Linear Algebra

Matrices, Determinants, and Systems of Linear Equations

Numerical Methods - Advanced Calculus

Partial Differentiation and Its Applications - Advanced Calculus

3D Coordinates and Quadric Surfaces - Analytic Geometry

Equilibrium of Rigid Bodies - Engineering Mechanics (Statics)

Statics of Particles - Engineering Mechanics (Statics)

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

Dry Friction - Engineering Mechanics (Statics)

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