Contents of Calculus III

An Introduction to Vectors

Vectors in 3-D Coordinate System

Using the Dot Product

The Cross Product

Lines and Planes in 3-D

Cylinders and Surfaces in 3-D

Using Cylindrical and Spherical Coordinates

An Introduction to Vector Functions

Derivatives and Integrals of Vector Functions

Arc Length / Parameterization, TNB (Frenet-Serret) Intro

TNB Frames, Curvature, Torsion, Encapsulation

Velocity and Acceleration of Vector Functions

Intro to Multivariable Functions

Limits and Continuity of Multivariable Functions

Partial Derivatives

Finding Differentials

The Chain Rule for Multivariable Functions

Directional Derivatives and Gradients

Tangent Planes and Normal Lines

Finding Extrema (Max and Min)

Constrained Optimization with Lagrange Multipliers

Introduction to Double Integrals

Solving Double / Iterated Integrals

Double Integrals over Polar Regions

Center of Mass & Moments for Lamina

Triple Integrals & Center of Mass / Volume

Triple Integrals in Cylindrical / Spherical Coordinates

Changing Variables with the Jacobian

Introduction to Vector Fields

Divergence and Curl of Vector Fields

Line Integrals over Non-Conservative Fields

Line Integrals over Conservative Fields

Green’s Theorem

Surface and Flux Integrals

Divergence Theorem & Stokes’ Theorem

Calculus III

Multivariable calculus, 3D geometry, vector calculus, and advanced integration methods—this page takes you through the key concepts of functions of several variables, multiple integrals, and vector fields. Use the table of contents on the left to jump to any topic.

What you’ll learn

Work with vectors in 3D space: dot/cross product, planes, surfaces, and coordinate systems.
Analyze vector functions: derivatives, integrals, curvature, torsion, and motion.
Differentiate multivariable functions: partial derivatives, gradients, tangent planes, and extrema.
Perform double, triple, and polar/cylindrical/spherical integrations with applications to mass, volume, and change of variables.
Understand vector fields, compute divergence & curl, and apply Green’s, Divergence, and Stokes’ theorems.