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Multivariable Calculus

Calculus in more than one dimension: surfaces instead of curves, gradients instead of slopes, volumes instead of areas. Everything here is drawn in 3D — a surface you can rotate is worth a page of symbols.

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Introductory Functions of several variables Surfaces, level curves, domains — reading z = f(x, y). x^2 - y^2 Core Partial derivatives and the gradient Differentiating in one direction at a time; the gradient as the vector of slopes. gradient of x^2*y - 3x + y^3 Core Optimisation in several variables Critical points, the second-derivative test, and Lagrange multipliers. gradient of x^2 + y^2 - 2x - 4y Core Double and triple integrals Volume under a surface by iterated integration; changing the order and the coordinates. integrate x^2 dx from 0 to 1 Core Vector fields, line integrals and the big theorems Gradient fields, curl and divergence, Green, Stokes and Gauss. gradient of x^2 + y^2

Chapters from Boelkins et al., Active Calculus Multivariable

Every section of the book, condensed into a lesson with its own practice problems.

Chapters from OpenStax Calculus Volume 3

Every section of the book, condensed into a lesson with its own practice problems.

1. Parametric Equations and Polar Coordinates

Parametric Equations Calculus of Parametric Curves Polar Coordinates Area and Arc Length in Polar Coordinates Conic Sections

2. Vectors in Space

Vectors in the Plane Vectors in Three Dimensions The Dot Product The Cross Product Equations of Lines and Planes in Space Quadric Surfaces Cylindrical and Spherical Coordinates

3. Vector-Valued Functions

Vector-Valued Functions and Space Curves Calculus of Vector-Valued Functions Arc Length and Curvature Motion in Space

4. Differentiation of Functions of Several Variables

Limits and Continuity Tangent Planes and Linear Approximations The Chain Rule Directional Derivatives and the Gradient Maxima/Minima Problems Lagrange Multipliers

5. Multiple Integration

Double Integrals over Rectangular Regions Double Integrals over General Regions Double Integrals in Polar Coordinates Triple Integrals Triple Integrals in Cylindrical and Spherical Coordinates Calculating Centers of Mass and Moments of Inertia Change of Variables in Multiple Integrals

6. Vector Calculus

Line Integrals Conservative Vector Fields Green’s Theorem Divergence and Curl Surface Integrals Stokes’ Theorem The Divergence Theorem

Symbols used here

\frac{\partial f}{\partial x}

partial derivative
Derivative with respect to x, holding the other variables fixed.

\nabla f

gradient (nabla, del)
Vector of partial derivatives; points uphill.

\int f(x)\,dx,\ \int_a^b

integral
Antiderivative (indefinite) or signed area from a to b (definite).

\iint,\ \oint

double / contour integral
Integral over a region of the plane; integral around a closed curve.

\mathbf{v},\ \vec{v}

vector
A quantity with magnitude and direction; a column of numbers.

\mathbf{u} \cdot \mathbf{v},\ \|\mathbf{v}\|

dot product, norm
Σ u_i v_i; the length of v, √(v·v).

Questions people ask

What is a partial derivative?

The ordinary derivative with respect to one variable while every other variable is frozen — the slope of the surface in one coordinate direction.

What does the gradient point at?

Uphill: the direction of steepest increase, with length equal to that steepest slope. It is perpendicular to the level curves.

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∇ Multivariable Calculus