Vector Calculus Course

Lesson 1 • Review of Vectors and Coordinate Systems

Establishes vector algebra and 3D coordinate geometry as the language of vector calculus. Connects dot and cross products to later gradient and curl computations.

Lesson 2 • Parametric Curves and Vector Functions

Defines vector-valued functions and parametric representations of curves. Lays groundwork for line integrals and arc-length calculations in later chapters.

Lesson 3 • Functions of Several Variables

Introduces scalar-valued functions of two and three variables and their domains. Provides the functional foundation needed before differentiation is introduced.

Lesson 1 • Partial Derivatives: Definition and Computation

Defines partial derivatives geometrically and algebraically for functions of two or more variables. Anchors the chapter by establishing the core differentiation operation.

Lesson 2 • Directional Derivatives

Extends differentiation to arbitrary directions using the gradient dot product formula. Builds intuition for flux and rate-of-change concepts used in integration chapters.

Lesson 3 • The Gradient Vector

Introduces the gradient as the vector of partial derivatives and its geometric meaning. Connects to directional derivatives and later to conservative fields.

Lesson 4 • Tangent Planes and Linear Approximation

Derives the tangent plane equation and the total differential for error estimation. Provides the local linearisation framework underlying the chain rule and optimisation.

Lesson 1 • Critical Points and the Second Derivative Test

Identifies critical points via zero-gradient conditions and classifies them using the Hessian determinant. Establishes the core unconstrained optimisation workflow.

Lesson 2 • Absolute Extrema on Closed Regions

Extends optimisation to compact domains by combining interior critical points with boundary analysis. Prepares students for constrained problems by reinforcing boundary reasoning.

Lesson 3 • Lagrange Multipliers: Single Constraint

Introduces the method of Lagrange multipliers for optimising subject to one equality constraint. Connects gradient geometry to constraint surfaces for intuitive understanding.

Lesson 1 • Triple Integrals and Coordinate Systems

Introduces triple integrals over 3D regions in Cartesian, cylindrical, and spherical coordinates. Connects coordinate choice to symmetry for computational efficiency.

Lesson 2 • Double Integrals over Rectangular Regions

Defines the double integral via Riemann sums and evaluates it using Fubini's theorem. Establishes the iterated-integral technique that all subsequent integration builds upon.

Lesson 3 • Double Integrals in Polar Coordinates

Transforms double integrals to polar form using the Jacobian factor r. Enables efficient evaluation over circular and radially symmetric regions.

Lesson 4 • Double Integrals over General Regions

Extends double integration to type I and type II non-rectangular regions. Develops the skill of choosing and reversing integration order for efficiency.

Lesson 1 • Introduction to Vector Fields

Defines vector fields as assignments of vectors to points and visualises them via field plots. Establishes the physical context of force, velocity, and flux fields.

Lesson 2 • Green's Theorem in the Plane

States and applies Green's theorem to convert line integrals to double integrals over enclosed regions. Bridges 2D line integrals to surface and volume theorems ahead.

Lesson 3 • Line Integrals of Scalar Functions

Defines and evaluates line integrals of scalar functions with respect to arc length. Connects to mass-of-wire and average-value problems along curves.

Lesson 4 • Conservative Fields and Potential Functions

Identifies conservative vector fields via path independence and the curl-free condition. Develops the technique of finding potential functions for exact fields.

Lesson 1 • Curl: Computation and Meaning

Computes the curl of a vector field and interprets it as local rotation or circulation density. Connects to Stokes' theorem and irrotational field conditions.

Lesson 2 • The Del Operator and Its Forms

Introduces the nabla operator and its three primary forms: gradient, divergence, and curl. Provides a unified symbolic framework for all differential operations in vector calculus.

Lesson 3 • Divergence: Computation and Meaning

Computes divergence of vector fields and interprets it as local source or sink strength. Connects to the divergence theorem and fluid continuity equations.

Lesson 4 • The Laplacian and Harmonic Functions

Defines the Laplacian as divergence of the gradient and identifies harmonic functions. Prepares students for applications in potential theory and partial differential equations.

Lesson 1 • Flux Integrals of Vector Fields

Defines flux as the surface integral of a vector field dotted with the outward unit normal. Provides the physical foundation for the divergence theorem.

Lesson 2 • Parameterising Surfaces in 3D

Defines surface parameterisations using two parameters and computes the tangent-vector cross product. Establishes the surface element dS needed for all surface integrals.

Lesson 3 • The Divergence Theorem

Converts flux integrals over closed surfaces to volume integrals of divergence. Completes the trio of fundamental theorems and unifies the course's integration theory.

Lesson 4 • Surface Integrals of Scalar Functions

Evaluates integrals of scalar functions over parameterised surfaces for mass and average value. Connects to the double-integral framework established in the multiple-integrals chapter.

Lesson 1 • Applications in Fluid Mechanics

Applies divergence, curl, and integral theorems to model incompressible and irrotational fluid flow. Demonstrates how vector calculus directly drives physical modelling.

Lesson 2 • Connecting the Three Fundamental Theorems

Unifies the fundamental theorem of line integrals, Green's, Stokes', and divergence theorems under one framework. Reveals the boundary-operator duality common to all four results.

Lesson 3 • Synthesis Problems and Proof Strategies

Presents multi-step problems requiring selection and combination of theorems from the entire course. Builds strategic problem-solving fluency as the capstone skill.

Lesson 4 • Generalised Stokes' Theorem Overview

Introduces differential forms and the exterior derivative to state the generalised Stokes' theorem. Provides conceptual depth without requiring full differential geometry prerequisites.