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

Introduction

Understanding the max-min theorem and the intermediate-value theorem.

Theorems on Continuous Functions

Various types of discontinuities - meaning and examples.

Types of Discontinuity

Examples of continuous functions, worked problems on continuity.

Continuous Functions

This course provides a focused and rigorous examination of continuity, a fundamental property of functions that underpins the whole of differential and integral calculus. We move from an intuitive understanding of "unbroken" graphs to the formal, limit-based definition of continuity. The course systematically explores the different types of discontinuities and the powerful theorems that apply only to continuous functions.

A command of continuity is essential for understanding why calculus works. This concept ensures that functions are predictable and well-behaved, a necessary condition for modeling real-world phenomena and for the validity of the major theorems of calculus. It is the bridge between the concept of a limit and the concept of a derivative, explaining why a function must be continuous to be differentiable.

By the end of this course, you will be able to use the three-part definition to test for continuity at a point, identify and classify removable, jump, and infinite discontinuities, and apply the Intermediate Value Theorem and the Extreme Value Theorem to analyse function behavior on a closed interval.

This course is designed for first-year undergraduates in science, technology, engineering, and mathematics who have completed a course on limits. It is a critical prerequisite for the study of differentiability and is essential for any student seeking a deep understanding of calculus theory.


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

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

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

Max Engineering Tutorials

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

continuity-of-functions-single-variable-calculus

Single-Variable Calculus

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

Proving Limits Involving Infinity

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

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

Evaluating Finite Limits (1)

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

Evaluating Limits at Infinity

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

Proving Finite Limits

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

Evaluating Finite Limits (2)

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

Evaluating Finite Limits (3)

This course provides a rigorous introduction to the concept of the limit, the theoretical bedrock upon which all of calculus is built. We will move from an intuitive understanding of what it means for a function to 'approach' a value to the algebraic techniques required for precise evaluation. The course covers the limit laws, methods for handling indeterminate forms, and the behavior of functions at infinity.

Understanding limits is non-negotiable for any serious study of calculus. The principles developed here are not merely abstract; they are the tools used to formally define the core concepts of calculus that are used to model the mechanics of change that govern engineering, physics, economics, and computer science.

By the end of this course, you will be able to: evaluate limits graphically, numerically, and algebraically; apply the Squeeze Theorem and L’Hôpital’s Rule; analyse the end-behavior of functions; and identify vertical and horizontal asymptotes.

This course is designed for first-year undergraduates in STEM fields who are beginning their calculus sequence. It is an essential prerequisite for subsequent courses on continuity and differentiability and is also invaluable for any student or professional seeking to rebuild their mathematical foundation from first principles.

limits-of-functions-single-variable-calculus

This chapter establishes the course's core philosophy. We define the modern developer's mindset, outline our practical, tool-centric teaching approach, and address the role of AI in professional workflows. This is the strategic overview for the entire course.

You will grasp the course structure, its practical, tool-focused philosophy, and the professional's approach to using AI in modern development.

To build for the web, you must understand its architecture. This chapter explains the sequence of events that delivers a website to a browser, covering the key components, protocols, and addressing systems that make it work. This knowledge is non-negotiable for any technical professional.

By the end, you will be able to: explain the client-server request-response model, define the roles of HTTP and DNS in communication, and trace the journey of a URL from keyboard to a page rendering on screen.

How the Web Works

Graphical interfaces are for consumption; the command line is for production. This chapter introduces the command-line interface (CLI), the tool developers use to navigate systems and manage files with superior speed and control. Proficiency here is a non-negotiable professional competency.

You will master the essential commands for file system management. This includes orienting yourself within the terminal, navigating directories, and performing all core creation, manipulation, and deletion operations on files and folders.

The Command Line

Your editor is an active development tool, not just a place for typing. This chapter focuses on configuring Visual Studio Code for professional efficiency, transforming it from a default program into a powerful environment that automates tasks and accelerates your workflow.

By the end of the chapter, you will: install and set up the editor, use the integrated terminal for a seamless workflow, automate code formatting for consistency, and leverage AI tools like GitHub Copilot to improve productivity.

Code Editing with VS Code

Version control is the professional standard for managing software projects. This chapter covers Git for tracking local history and GitHub for remote collaboration and backup. Proficiency with these tools is a non-negotiable industry requirement for any developer.

You will master the core version control workflow. This includes creating a local Git repository, saving snapshots of your work using the stage-and-commit process, and synchronizing that history with a remote GitHub repository.

Git and GitHub

This concluding chapter consolidates your achievements and directs you to the next stage. We will review the foundational skills you have mastered, place them within a professional context, and provide a clear path forward to ensure your learning momentum continues.

By the end of the chapter, you will: review the core web concepts and the professional toolkit; understand how advanced tools build upon this foundation; and identify the HTML5 course as your immediate next step.

Your Path Forward

Modern web development is not about memorizing code; it is about understanding the system you build for and mastering the tools of the trade. This course focuses on the non-negotiable foundations: how the web functions, how professionals think, and how to operate the tools that matter in real-world environments.

The skills taught here are not theoretical; they are used daily in every technical role. A command of the terminal, a properly configured editor, and version control are what separate efficient professionals from struggling amateurs. This is a practical knowledge required to contribute to projects and pass technical interviews.

You will be able to explain the lifecycle of a web request from a URL to a rendered page. You will configure a professional development environment in VS Code, manage your file system from the command line, and use Git and GitHub to track and share your work.

This course is built for aspiring developers and computer science students seeking to build a professional foundation. It is also essential for product managers, designers, and technical writers who need to understand development workflows to collaborate effectively with technical teams. No prior coding experience is required.

introduction-to-modern-web-development-concepts-and-tools

Web Development

This chapter teaches the fundamental grammar of HTML. We define the core vocabulary of elements and tags, construct a valid page structure, and add basic, semantic content. These are the non-negotiable building blocks for any web page.

Key topics: defining HTML vocabulary (tags, elements); building a valid page structure; adding content with semantic text; and understanding element display behaviour.

The Language of HTML

This chapter establishes a professional workflow for asset management. We will source, create, and organise all required media - logos, headshots, and icons - before building the page. A structured and legally compliant asset base is essential.

Key topics: understanding asset copyright; structuring the project folder; creating a logo and headshot; and choosing a modern icon library.

Preparing Your Project Assets

This final chapter covers the essential "behind-the-scenes" tags in the document <head> that make your page professional, complete, and ready for the world. These details control how your site appears in search engines and social media, and prepare it for styling.

By the end, you will: understand the role of the <head> element; add essential meta tags for SEO and social sharing; set a custom favicon for your site; and understand how to link a CSS file for the next course.

Finalising the Document

This chapter focuses on the correct, professional way to display structured data. We will build a complete "Experience" table for our portfolio, including an interactive, expandable section for details.

By the end, you will: build a complete, semantically correct table; and use the details and summary elements to add interactive, expandable content to your table rows.

Displaying Tabular Data

This chapter focuses on building the high-level skeleton of our portfolio page using semantic HTML. We will translate the main visual sections of our design into the correct, professional HTML5 structural elements.

By the end, you will: structure your page with header and footer landmarks; create a navigation bar with the nav element; and use main and section elements to define the primary content areas of your page.

Building the Page Structure

With the main structure in place, this chapter is about populating our sections with their actual content. We will add the text, images, and icons that make up the hero, skills, and social media sections of our portfolio.

By the end, you will: build a complete hero section with text and an image; understand the critical attributes of the img element; implement SVG icons; and make links functional.

Adding Core Content and Media

This chapter covers the essential HTML for collecting user input. We will build the complete structure for the portfolio's contact form, a non-negotiable component for enabling professional communication. You will learn to handle everything from simple text fields to complex option selectors, ensuring your form is both functional and accessible.

By the end, you will: build a complete and accessible form; use various input types for different data, including text, email, and options; semantically group related inputs using fieldset and legend; and provide a functional submission button.

Building the Contact Form

This chapter focuses on a critical professional skill: building repeatable components. We will build the HTML structure for the "Project Card" from our design, integrating different media types for a dynamic result.

By the end, you will: understand the role of the div element; build a reusable project card structure; and embed static images, local videos, and third-party videos into your project.

Structuring Complex Components

This chapter establishes your professional environment. We will set up the project with version control, generate a design target, and configure a live-preview workflow. This is the standard setup required before writing any code.

Key topics: initialising a project folder and Git repository; generating a target UI Design; and configuring an instant-preview workflow with Live Server.

HTML is not about memorizing tags; it is the professional practice of structuring information. This course teaches HTML5 by guiding you through the construction of a complete, portfolio-ready webpage. We move from a blank file, through a professional asset-preparation phase, to building a multi-section, feature-rich website based on a modern UI design.

A well-structured portfolio is a non-negotiable tool for any aspiring developer. The quality of your HTML directly impacts your site's performance, search ranking, and accessibility. This course teaches you to write the clean, semantic, and professional markup that separates a hobbyist's page from an impressive portfolio piece that gets you noticed.

You will master the full lifecycle of a modern HTML project. You will prepare all necessary assets using AI and design tools. You will structure a complete page with semantic landmarks, build complex components like project cards and interactive tables, and create a functional contact form. By the end, you will have a complete, well-structured project.

This course is designed for the absolute beginner with no prior coding experience. It is the essential starting point for aspiring developers and is invaluable for designers, content creators, and marketers who need to build or understand the fundamental structure of a professional website.

html-foundations-from-zero-to-your-first-web-page

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

Centroids

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

Position Vectors

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

Division of a Line

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

Vector Projections

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

Vector Components

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

Vector Algebra

This course provides a comprehensive, first-principles guide to vector analysis. We begin with the essential definitions, distinguishing between scalars, vectors, and various vector types. You will then master the core rules of vector algebra—including the triangle and parallelogram laws of addition and scalar multiplication—before applying them to prove geometric properties of polygons. The curriculum progresses logically through key concepts such as position vectors, resolving vectors into Cartesian components, the division of lines, vector projections, and the calculation of centroids.

The true power of vectors lies in their ability to connect abstract algebra to tangible geometry. This course is built around that connection, with a heavy emphasis on practical application. Through dozens of meticulously worked examples, you will move beyond theory and develop a deep, intuitive understanding of how to represent and solve complex geometric problems, building the confidence and skill needed for more advanced study.

This programme is designed for students in late secondary school (such as A-Level or IB) and first-year university students in physics, engineering, mathematics, and computer science. It is the perfect starting point for anyone new to the subject and an invaluable refresher for those who need to solidify their foundational knowledge before tackling more advanced topics like linear algebra or vector calculus.

vector-algebra-and-foundational-geometry-vectors

Vectors

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

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

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

This course is a comprehensive guide to the effects of forces on rigid bodies, a critical topic in engineering mechanics. It moves beyond particles to analyse systems where the point of application of a force is crucial, introducing the concepts of moments and couples. The material covers the calculation of moments in two and three dimensions, the properties of couples, and the methods for reducing complex force systems into simpler, statically equivalent systems.

Understanding moments is fundamental to all structural and mechanical analysis. These principles are used to calculate the bending and twisting stresses in beams, shafts, and machine components, and to ensure the rotational stability of structures and vehicles. A command of this subject is non-negotiable for designing any object intended to withstand real-world loads without failing or rotating unintentionally.

By the end of this course, you will be able to calculate the moment of a force about any point or axis using both scalar and vector methods. You will also be able to determine the moment of a couple, and reduce any complex system of forces and couples acting on a rigid body to its simplest equivalent form, such as a single resultant force and couple moment at a specified point.

This course is for undergraduate students in any engineering discipline who have already completed a course on the statics of particles. It is the direct prerequisite for the study of the equilibrium of rigid bodies, and is a necessary foundation for advanced courses in dynamics, mechanics of materials, and structural analysis.

systems-of-forces-rigid-bodies-engineering-mechanics

Engineering Mechanics

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

Force vectors and how to obtain the resultant of two or more forces - using parallelogram and triangle rules.

Resultant of Forces in a Plane

Resolution of a force into its rectangular and non-rectangular components.

Components of a Force in a Plane

Calculating the resultant of several concurrent forces using their rectangular components.

Resultant by Components in a Plane

Determining the components of forces in three dimensions.

Components of a Force in Space

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

Equilibrium in Space

Equilibrium of particles under the action of planar forces.

Equilibrium in a Plane

Vector addition of force vectors in three dimensions.

Resultant of Forces in Space

This course provides a complete guide to the statics of particles, a foundational topic in engineering mechanics. It covers the principles of force vectors, resultants, and components, first in two dimensions and then extending to three-dimensional space. The core of the course is the analysis of particles in equilibrium, establishing the conditions required for a particle to remain at rest under the influence of multiple concurrent forces.

Statics is the essential science behind structural and mechanical design. The principles covered are the basis for designing every stable structure, from bridges and skyscrapers to aircraft frames and engine components. A command of statics is non-negotiable for any engineer, as it provides the fundamental tools required to analyse forces and ensure the safety and integrity of physical systems.

By the end of this course, you will be able to resolve any force into its components and calculate the resultant of any system of concurrent forces in both 2D and 3D. You will also be able to draw a complete free-body diagram for any particle and apply the equations of equilibrium to solve for unknown forces, tensions, and reactions in a variety of engineering problems.

This course is for undergraduate students in any engineering discipline, including civil, mechanical, and aerospace. It is a core component of any engineering mechanics curriculum and assumes a prerequisite knowledge of introductory physics and vector mathematics. It is the essential foundation for subsequent courses in dynamics and the mechanics of materials.

statics-of-particles-engineering-mechanics-undergraduate

Understanding the mean-value and Rolle's theorems.

Theorems on Differentiable Functions

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

Higher-Order Derivatives

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

Taylor and Maclaurin series expansion of differentiable functions.

Taylor and Maclaurin Series

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

Derivatives

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

Rules of Differentiation

This course provides a rigorous, theoretical introduction to differential calculus. We will move beyond computation to understand the fundamental nature of the derivative, beginning with its formal definition as the limit of a difference quotient. The curriculum is designed to build a deep conceptual understanding of what it means for a function to be differentiable and the profound consequences of this property.

The primary focus is on formal proofs and theoretical results. We will systematically derive the rules of differentiation, including the product, quotient, and chain rules, directly from first principles. This course emphasizes the "why" behind the mechanics of calculus, establishing the logical framework upon which all applications are built, including the cornerstone theorems of Rolle, the Mean Value Theorem, and Taylor's Theorem.

By the end of this course, you will be able to: explain the derivative from its limit definition; prove the relationship between differentiability and continuity; formally derive all major differentiation rules; and understand the theoretical significance of the Mean Value Theorem and Taylor's Theorem.

This course is designed for first-year undergraduates in mathematics, physics, and engineering who require a deep theoretical foundation for their studies. It is the ideal precursor to subsequent courses on differentiation techniques and applications, providing the essential logical underpinnings for those more practical subjects.


differentiability-and-derivatives-of-functions

This chapter moves from the general definition of a function to a systematic classification of key function families. We will identify and analyse the core algebraic types - including polynomial, rational, and piecewise functions - that form the basis of mathematical modelling across scientific disciplines.

By the end of this chapter, you will be able to: classify functions as polynomial, rational, or algebraic; determine the natural domain for each of these types; interpret and analyse piecewise-defined functions; and test any function for symmetry.

Kinds of Real-Valued Functions

This chapter establishes the foundational concepts of the real number system. A command of these definitions is a prerequisite for defining and analysing the functions that form the core of this course.

By the end of this chapter, you will be able to: differentiate between key number systems from natural to real; define and represent intervals on the real line; and correctly apply the mathematical concept of infinity.

This chapter introduces the function - the central object of study in calculus. We will formally define what constitutes a real-valued function and establish the critical concepts of its valid inputs (domain) and possible outputs (range).

By the end of this chapter, you will be able to: formally define a function; precisely calculate the domain for various real-valued functions; and understand the meaning and importance of a function's range.

Real-Valued Functions

This chapter covers transcendental functions, the non-algebraic tools used to model natural phenomena like population growth, radioactive decay, and wave mechanics. We will systematically analyse the properties and graphs of exponential, logarithmic, trigonometric, and hyperbolic functions.

By the end of this chapter, you will be able to: differentiate between algebraic and transcendental functions; analyse the domains and graphs of exponential, logarithmic, and trigonometric types; and define their corresponding inverse functions.

Transcendental Functions

This course provides a rigorous foundation in the concept of functions, the building block of calculus. We begin with the real number system - rationals, irrationals, intervals, and infinity - before defining real-valued functions, their domains, and ranges. You will systematically study all key function families: polynomial, rational, algebraic, piecewise-defined, and transcendental, including exponential, logarithmic, trigonometric, and hyperbolic types.

A command of functions is essential for all technical and quantitative disciplines. These principles are directly applied in physics to model motion, in economics for supply-demand curves, in engineering for signal processing, and in computer science to analyse algorithm complexity. This course delivers the core mathematical tools required for effective modelling and problem-solving in these fields and beyond.

By the end of this course, you will be able to: calculate the domain and range of any given function; classify functions and identify their fundamental properties; analyse functions for key characteristics like symmetry (odd and even); and accurately construct graphs for a wide range of standard and piecewise-defined functions directly from their equations.

This course is designed for first-year undergraduates in science, technology, engineering, mathematics (STEM), and economics. It is also critical for any student preparing for advanced calculus or for professionals who require a solid mathematical base for roles in finance, data science, or technical research.

functions-and-their-graphs-single-variable-calculus

Transformation of coordinates by rotation.

Coordinates Transformation

Equations of circles, ellipses, parabolas and hyperbolas.

Second-Degree Plane Curves

Three-dimensional rectangular coordinates system, graphs.

Three-Dimensional Rectangular Coordinates

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

Two-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

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

Quadric Surfaces

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.

3-d-coordinates-and-quadric-surfaces-analytic-geometry

Analytic Geometry

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

Applications of first-order ordinary differential equations.

Applications of First-Order Ordinary Differential Equations

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

Solutions of Second-Order Ordinary Differential Equations (1)

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

Solutions of Second-Order Ordinary Differential Equations (2)

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

Solutions of First-Order Ordinary Differential Equations

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

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

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

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

ordinary-differential-equations-mathematical-methods

Mathematical Methods

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

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

Analysis of trusses, frames and machines.

analysis-of-structures-engineering-mechanics

Vector (cross) product of two vectors and its properties.

Vector Products

Vector triple product of three vectors and its properties.

Vector Triple Products

Scalar triple product of three vectors and its properties.

Scalar Triple Products

Scalar (dot) product of two vectors and its properties.

Scalar Products

This course delves into the essential topic of vector multiplication. We systematically explore the four main types of vector products, beginning with the scalar (dot) and vector (cross) products of two vectors. You will then advance to products involving three vectors: the scalar triple product and the vector triple product, uncovering their unique properties and applications.

Vector products are the key to unlocking the geometric power of vector analysis. They provide the tools to calculate angles, projections, areas, and volumes with elegant algebraic formulas. This course focuses on revealing these deep connections, showing how abstract multiplication rules translate into powerful methods for solving real-world problems in physics and engineering.

This is an intermediate course designed for students who have a solid grasp of foundational vector algebra and components. It is the perfect next step for students in A-Level or university STEM programmes looking to deepen their understanding of vector analysis. A strong foundation in the topics from our introductory course is highly recommended for success.

scalar-vector-and-triple-products-of-vectors-undergraduate

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

Divergence of a Vector Field

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

Gradient of a Scalar Field

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

Curl of a Vector Field

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

Scalar and Vector Fields

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

Scalar and Vector Products

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!

Courses

Customize your learning experience

Continuity of Functions - Single-Variable Calculus (Undergraduate Foundation)

Limits of Functions - Single-Variable Calculus (Undergraduate Foundation)

An Introduction to Modern Web Development Concepts and Tools (Professional Practice)

HTML Foundations: From Zero To Your First Web Page - Web Development (Professional Practice)

Vector Algebra and Foundational Geometry - Vectors (Undergraduate Foundation)

Systems of Forces on Rigid Bodies - Engineering Mechanics (Undergraduate Advanced)

Statics of Particles - Engineering Mechanics (Undergraduate Advanced)

Differentiability of Functions - Single-Variable Calculus (Undergraduate Foundation)

Functions and Their Graphs - Single-Variable Calculus (Undergraduate Foundation)

3D Coordinates and Quadric Surfaces - Analytic Geometry (Undergraduate Advanced)

Ordinary Differential Equations - Mathematical Methods (Undergraduate Advanced)

Analysis of Structures - Engineering Mechanics (Undergraduate Advanced)

Scalar, Vector and Triple Products of Vectors (Undergraduate Foundation)

Calculus of Scalar and Vector Fields (Undergraduate Advanced)

2D Kinematics of Rigid Bodies - Engineering Mechanics (Undergraduate Advanced)

Kinematics of Particles - Engineering Mechanics (Undergraduate Advanced)

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

Kinetics of Systems of Particles - Engineering Mechanics (Undergraduate Advanced)

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

Matrices, Determinants, and Systems of Linear Equations - Linear Algebra (Undergraduate Advanced)

Linear Maps (Transformations) - Linear Algebra (Undergraduate Advanced)

Eigenvalues, Eigenvectors and Diagonalization of Matrices - Linear Algebra (Undergraduate Advanced)

Linear Vector Spaces - Linear Algebra (Undergraduate Advanced)

Complex Numbers - Mathematical Methods (Undergraduate Advanced)

Introduction to Numerical Methods (Undergraduate Foundation)

Relations - Introductory Abstract Algebra (Undergraduate Advanced)

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

Equilibrium of Rigid Bodies - Engineering Mechanics (Undergraduate Advanced)

Dry Friction - Engineering Mechanics (Undergraduate Advanced)