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Calculus 3 Course

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Mathematics

Calculus 3 Course

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Calculus 3 Course

Name: Calculus 3 Course
Price: 204 GBP

Master the full scope of Calculus 3, from vectors and multivariable functions to surface integrals and the major theorems of vector calculus. This course builds the rigorous mathematical foundation required for advanced work in physics, engineering, and applied mathematics. Every core topic is developed systematically, with clear derivations and a wide range of solved problems.

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What you will learn:

You will build a complete understanding of three-dimensional space, vector-valued functions, and multivariable calculus. The course covers partial derivatives, the gradient, optimisation with Lagrange multipliers, and double and triple integrals in multiple coordinate systems. You will study line integrals, surface integrals, and vector fields, then apply Green's theorem, Stokes' theorem, and the Divergence theorem to real problems. Supplementary material includes quadric surfaces, series review, differential equations, numerical methods, and physics applications. By the end, you will have the analytical tools needed for differential equations, real analysis, and advanced engineering coursework.

How you study in practice Calculus 3 Course

How you practise Calculus 3 Course

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Course content

8 Chapters • 41 LessonsDuration between 4 and 360 hours (you decide)

Chapter 1See details

Vectors and Three-Dimensional Space

Lesson 1 • Cross Product and Its Applications
Develops the cross product via the determinant method and explores its geometric meaning. Enables computation of normal vectors, torque, and areas of parallelograms.
Lesson 2 • Coordinates and Points in 3D Space
Introduces the three-dimensional Cartesian coordinate system and distance formula. Provides the spatial framework required for all vector and surface work ahead.
Lesson 3 • Dot Product and Its Applications
Covers the dot product formula, angle between vectors, and orthogonality. Connects directly to projections and normal vectors used in plane equations.
Lesson 4 • Lines and Planes in Space
Derives parametric, symmetric, and standard equations for lines and planes using vectors. Builds the spatial reasoning needed for surfaces and multivariable functions.
Lesson 5 • Vector Algebra and Operations
Defines vectors, scalar multiplication, and vector addition geometrically and algebraically. These operations underpin dot products, cross products, and line equations.

Chapter 2See details

Vector-Valued Functions and Space Curves

Lesson 1 • Arc Length and Curvature
Derives arc length for space curves and defines curvature using the TNB frame. These concepts appear in physics applications and differential geometry.
Lesson 2 • Integration of Vector Functions
Covers definite and indefinite integrals of vector functions component-wise. Connects to position, velocity, and acceleration in motion problems.
Lesson 3 • Introduction to Vector-Valued Functions
Defines vector-valued functions and their domains, limits, and continuity. Establishes the connection between parametric curves and vector functions.
Lesson 4 • Motion in Space: Velocity and Acceleration
Applies vector calculus to analyse projectile and circular motion using tangential and normal acceleration components. Synthesises all prior section skills.
Lesson 5 • Differentiation of Vector Functions
Develops derivative rules for vector functions including product and chain rules. Tangent vectors derived here are foundational for curvature and motion analysis.

Chapter 3See details

Multivariable Functions and Partial Derivatives

Lesson 1 • Tangent Planes and Linear Approximation
Derives tangent plane equations and linearisation for functions of two variables. Connects to differentials and error estimation used in applied problems.
Lesson 2 • Directional Derivatives and the Gradient
Defines directional derivatives and the gradient vector, linking them to steepest ascent. Gradient properties are essential for optimisation and Lagrange multipliers.
Lesson 3 • Functions of Several Variables
Defines multivariable functions, their domains, ranges, and level curves. Visualising level curves and surfaces prepares students for gradient and optimisation work.
Lesson 4 • The Chain Rule for Multivariable Functions
Extends the chain rule to composite multivariable functions using dependency trees. Enables implicit differentiation and related-rate problems in higher dimensions.
Lesson 5 • Limits and Continuity in Several Variables
Extends limit concepts to multivariable settings, including path-dependent limits. Establishes continuity conditions needed for differentiability theorems.
Lesson 6 • Partial Derivatives
Defines and computes first- and higher-order partial derivatives using limit and rule-based methods. Partial derivatives are the building blocks of the gradient and chain rule.

Chapter 4See details

Optimisation of Multivariable Functions

Lesson 1 • Second Derivative Test in Two Variables
Applies the discriminant D to classify critical points as maxima, minima, or saddle points. Extends single-variable second derivative reasoning to surfaces.
Lesson 2 • Local Extrema and Critical Points
Identifies critical points via setting partial derivatives to zero and classifies them. Provides the foundation for the second derivative test and global optimisation.
Lesson 3 • Absolute Extrema on Closed Regions
Finds absolute maxima and minima on bounded closed domains by checking interior and boundary. Mirrors the closed-interval method from single-variable calculus.
Lesson 4 • Lagrange Multipliers
Introduces constrained optimisation using the Lagrange multiplier condition. Solves problems where extrema occur on curves or surfaces defined by constraints.

Chapter 5See details

Multiple Integrals

Lesson 1 • Double Integrals over General Regions
Extends double integrals to type I and type II regions with variable limits. Changing integration order often simplifies otherwise intractable integrals.
Lesson 2 • Triple Integrals in Spherical Coordinates
Introduces spherical coordinates and the Jacobian rho squared sin phi for triple integrals. Ideal for spheres, cones, and other radially symmetric solids.
Lesson 3 • Double Integrals in Polar Coordinates
Converts double integrals to polar form using the Jacobian factor r. Polar coordinates simplify integrals over circular and symmetric regions.
Lesson 4 • Triple Integrals in Cylindrical Coordinates
Transforms triple integrals using cylindrical coordinates for solids with axial symmetry. The Jacobian r simplifies integration over cylinders and cones.
Lesson 5 • Triple Integrals in Rectangular Coordinates
Sets up and evaluates triple integrals over box and general solid regions. Computes volume, mass, and moments of solid objects.
Lesson 6 • Double Integrals over Rectangles
Defines double integrals as limits of Riemann sums and evaluates them as iterated integrals. Fubini's theorem justifies switching the order of integration over rectangles.

Chapter 6See details

Change of Variables and Applications of Multiple Integrals

Lesson 1 • Change of Variables and the Jacobian
Derives the general change-of-variables formula using the Jacobian determinant. Unifies polar, cylindrical, and spherical substitutions under one framework.
Lesson 2 • Surface Area via Double Integrals
Computes surface area of graphs z = f(x, y) using the double integral formula. Extends arc length ideas from single-variable calculus to surfaces.
Lesson 3 • Probability Applications of Double Integrals
Applies double integrals to joint probability density functions and expected values. Demonstrates the breadth of multiple integration beyond geometry.
Lesson 4 • Moments, Centres of Mass, and Centroids
Uses double and triple integrals to find moments and centres of mass for laminas and solids. Connects integration to physical equilibrium and engineering design.

Chapter 7See details

Vector Fields and Line Integrals

Lesson 1 • Line Integrals of Vector Fields
Defines and computes line integrals of vector fields representing work done by a force. Connects to circulation and flux interpretations in physics.
Lesson 2 • Line Integrals of Scalar Functions
Evaluates line integrals of scalar functions with respect to arc length. Computes mass and centre of mass of wire-shaped objects.
Lesson 3 • Vector Fields in Two and Three Dimensions
Defines vector fields, sketches them, and identifies gradient fields. Recognising conservative fields reduces line integral computation significantly.
Lesson 4 • Fundamental Theorem for Line Integrals
States and applies the fundamental theorem to conservative fields, showing path independence. Finding potential functions reduces line integrals to boundary evaluations.
Lesson 5 • Green's Theorem
Relates a line integral around a closed curve to a double integral over the enclosed region. Enables efficient computation of circulation and flux for planar fields.

Chapter 8See details

Surface Integrals and the Theorems of Stokes and Gauss

Lesson 1 • Surface Integrals of Vector Fields and Flux
Defines flux as the surface integral of a vector field dotted with the unit normal. Orientation of surfaces determines the sign of flux integrals.
Lesson 2 • Surface Integrals of Scalar Functions
Evaluates surface integrals of scalar functions over parametric and graph surfaces. Computes mass and average value of a surface distribution.
Lesson 3 • The Divergence Theorem
Relates the flux through a closed surface to the triple integral of divergence over the enclosed volume. Completes the unification of vector calculus theorems.
Lesson 4 • Parametric Surfaces and Their Areas
Represents surfaces parametrically and computes surface area using the cross product of partial derivatives. Generalises the earlier surface area formula to arbitrary surfaces.
Lesson 5 • Stokes' Theorem
Relates the surface integral of curl F to the line integral around the boundary curve. Generalises Green's theorem to surfaces in three dimensions.
Lesson 6 • Curl and Divergence of Vector Fields
Defines curl and divergence using the del operator and interprets them physically. These operators appear directly in Stokes' and the Divergence theorem.

Certification

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This course is for you:

College students: currently enrolled in a third-term calculus course needing support.
Engineering undergraduates: preparing for fluid mechanics, electromagnetics, or structural analysis coursework.
Physics majors: needing rigorous vector calculus before tackling classical or quantum mechanics.
Data scientists: seeking the mathematical backbone behind gradient-based machine learning algorithms.
Self-taught programmers: filling gaps in mathematical training to move into technical research roles.
STEM graduates: refreshing multivariable calculus skills before entering postgraduate programmes.

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