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

Differentiability of Functions (Higher-Order Derivatives)

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

Differentiability of Functions (Definition)

Continuity (at an interior point, at an endpoint, on an interval) of real-valued functions - graphical illustration, formal and informal definitions.

Continuity of Functions (Definition)

Understanding the mean-value and Rolle's theorems.

Differentiability of Functions (Theorems)

Understanding removable discontinuities, the max-min theorem and the intermediate-value theorem.

Continuity of Functions (Types, Theorems)

Real-valued functions - meaning, domain, range and graphs.

Real-Valued Functions

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

Limits of Functions (Definition)

Formal and informal definitions of limits at infinity and infinite limits; different methods of evaluating limits of real-valued functions at infinity and infinite limits.

Limits of Functions (Infinity)

Taylor and Maclaurin series expansion of differentiable functions.

Taylor and Maclaurin Series

What are real numbers, and how do they differ from natural numbers, integers, rational numbers, irrational numbers? Meaning of infinity; defining finite and infinite intervals.

Real Numbers

A look at different kinds of real-valued functions - composites, piecewise functions, polynomials, rational functions, algebraic functions, transcendental functions, odd and even functions.

Kinds of Real-Valued Functions

Different methods of evaluating limits of real-valued functions - direct substitution, theorems, graphing, factorization, conjugates, L’Hôpital’s rule, use of known special limits, squeeze theorem, limits of piecewise-defined functions, etc.

Limits of Functions (Evaluation)

Differentiability on an interval; relation of differentiability and continuity; review of how to find the derivatives of real-valued functions; rules of derivatives of sums, products and quotients of functions and their proofs.

Differentiability of Functions (Derivatives)

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.

Tutorials for undergraduates in engineering and physical sciences.

Max Engineering Tutorials

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

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

Introduction

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)

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

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

Points, distances, lines, angles, gradients, etc., on the two-dimensional coordinates system.

Two-Dimensional Rectangular Coordinates

Transformation of coordinates by rotation.

Coordinates Transformation

Graphs of second-degree equations in three variables - ellipsoids, cones, cylinders, hyperboloids, paraboloids, etc.

Quadric Surfaces

Equations of circles, ellipses, parabolas and hyperbolas.

Second-Degree Plane Curves

Three-dimensional rectangular coordinates system, graphs.

Three-Dimensional Rectangular Coordinates

Direct classification of quadric surfaces.

Classifying Quadric Surfaces I

Classification of quadric surfaces by variable substitution.

Classifying Quadric Surfaces II

The polar, cylindrical and spherical coordinates systems.

Other Coordinates Systems

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.

Analytic Geometry: Master 3D Coordinates and Quadric Surfaces

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

Vector addition of force vectors in three dimensions.

Resultant of Forces in Space

Equilibrium of particles under the action of planar forces.

Equilibrium in a Plane

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

Equilibrium in Space

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

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)

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)

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.

Master Complex Numbers - Theory and Applications

Welcome to the course and review of fundamental concepts - meaning of rigid bodies, principle of transmissibility of forces, vector products, etc.

Meaning and calculation of moment of a force about a point in three dimensions.

Moment about a Point in Three Dimensions

Meaning and calculation of the moment of a force about an arbitrary axis.

Moment about an Axis

Meaning and calculation of the moment of a force about a point in two dimensions.

Moment about a Point in Two Dimensions

Simplifying general systems of forces and couples on rigid bodies.

General Systems of Forces

Meaning and analysis of couples, moments of couples, and couple vectors.

Couples

Meaning, scalar and vector forms of the moment of a force about a point and its relation to the moment about an axis; Varignon's theorem.

Moment of a Force

Analysis of simple force-couple systems - resolution of a single force into a force and a couple, and reduction of a force-couple system to a single equivalent force.

Force-Couple Systems

Simplifying systems of forces on rigid bodies involving distributed loads.

Distributed Loads

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.

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