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Multivariable Calculus

Partial derivatives, gradients, multiple integrals, vector fields.

Lessons

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

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.

More lessons

Precalculus of Multivariable Functions

Why Multivariable?Three Dimensional Space Vectors Lines in Space Planes in Space Common Graphs in Three Dimensions Polar, Cylindrical, and Spherical Coordinates

Parametric Equations and Polar Coordinates

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

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

Vector-Valued Functions of One Variable

Classic Calculus Approach Vector-Valued Functions of One Variable Calculus of Vector-Valued Functions of One Variable Arc Length The TNB Frame Curvature Splitting the Acceleration Vector

Vector-Valued Functions

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

Derivatives of Multivariable Functions

Calculus of Several Variables Limits First-Order Partial Derivatives Second-Order Partial Derivatives Linearization: Tangent Planes and Differentials The Multivariable Chain Rule Higher Dimensions Optimization Constrained Optimization: Lagrange Multipliers

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

Multiple Integrals

Integrating Functions of Several Variables Double Riemann Sums and Double Integrals over Rectangles Iterated Integrals Applications of Double Integrals

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

Derivatives in many variables

Multivariable calculus Partial derivative Gradient Directional derivative Total derivative Chain rule Jacobian matrix and determinant Hessian matrix Implicit function theorem Inverse function theorem Taylor series Lagrange multiplier Critical point (mathematics)

Integrals in many variables

Multiple integral Iterated integral Change of variables Polar coordinate system Spherical coordinate system Line integral Surface integral Jacobian

Some of these lessons are adapted from open texts: Boelkins et al., Active Calculus Multivariable, OpenStax Calculus Volume 3, Wikipedia. Each lesson page names its source and licence.

Symbols used here

Tap any symbol for the full definition, a picture, and what every letter in it means.

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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