Multivariable calculus

Represent local changes and accumulation over regions and explain partial derivatives and multiple integrals.

Meaning

Multivariable calculus studies functions with several inputs and change and accumulation in space. Partial derivatives, multiple integrals, line integrals, and surface integrals answer different questions within this field.

Example

For holding fixed and differentiating with respect to gives Holding fixed and differentiating with respect to gives 3. Each examines change in one direction.Holding y fixed makes 3y constant in x, so it cancels in the difference. The x-direction difference quotient is 2x+h, tending to 2x. In the y direction x² cancels and the change is 3h, giving quotient 3.A single partial derivative does not describe changes in all directions at once. Change along a path and total differentiability require additional conditions.

Practice for this concept

Scope and references

Topics

Partial derivatives, multiple integrals, vector fields, line and surface integrals

Learning goal

Represent local changes and accumulation over regions and explain partial derivatives and multiple integrals.

Helpful background

Explore the single-variable change and accumulation, coordinates, and vectors needed for the chosen topic.

Partial derivatives and rectangular double integrals

These tasks do not assess integration over general regions, vector fields, line integrals, or surface integrals.

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