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Points and distance in space

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Vectors and space

Step into three dimensions with points, vectors, dot and cross products, lines, planes and moving objects.

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Points and distance in space

Vectors and their length

The dot product

The cross product

Projections and work

Lines in space

Planes in space

Curves and motion

Vectors toolkit

Skill summary

Vectors practice

Practice

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Partial derivatives

Differentiate functions of several variables, find gradients and tangent planes, and optimize in 3D.

Functions of several variables

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Partial derivatives

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Partials with the chain and product rules

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Second partial derivatives

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The multivariable chain rule

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Tangent planes and linear approximation

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The gradient and directional derivatives

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Critical points and the second-derivative test

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Lagrange multipliers

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Partial derivatives toolkit

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Partial derivatives practice

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Find the mistake

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Multiple integrals

Add up volume, mass and averages over flat regions and solid shapes, in any coordinate system.

Double integrals over rectangles

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Switching the order

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Double integrals over general regions

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Double integrals in polar coordinates

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Area, volume, mass and average value

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Triple integrals

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Cylindrical and spherical coordinates

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Change of variables and the Jacobian

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Multiple integrals practice

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Multiple integrals summary

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Vector calculus

Integrate along curves and across surfaces, and meet the big theorems that tie them together.

Vector fields

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Line integrals of functions

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Work and line integrals of vector fields

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Conservative fields and potential functions

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Green's theorem

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Curl and divergence

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Surface integrals and flux

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Stokes' theorem and the Divergence Theorem

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Vector calculus practice

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Vector calculus summary

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Course checkpoint

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