CHAPTER 07

Multivariable Calculus

Extend calculus from functions of one variable to functions that depend on several variables.

\[ f(x) \rightarrow f(x,y) \rightarrow f(x,y,z) \]

MULTIVARIABLE CALCULUS

Calculus with several variables

Functions of Several Variables

Understand functions whose outputs depend on two or more independent variables.

Partial Derivatives

Differentiate with respect to one variable while holding the others fixed.

Higher Partial Derivatives

Understand second and higher partial derivatives, including mixed partial derivatives.

Total Derivatives

Describe how a multivariable function changes when several inputs change together.

Multivariable Chain Rule

Track change through functions that depend on other changing variables.

The Gradient

Represent the direction and magnitude of greatest local increase.

Directional Derivatives

Measure how a function changes in a chosen direction.

Tangent Planes and Linear Approximation

Approximate a surface near a point using its local linear behaviour.

Multiple Integrals

Extend integration to regions in two and three dimensions.

Vector Fields

Assign a vector to each point in space and interpret changing directions and magnitudes.

Divergence

Measure the local tendency of a vector field to spread outward or converge inward.

Curl

Measure the local rotational tendency of a vector field.

Line Integrals

Accumulate scalar or vector quantities along a curve.

Surface Integrals

Accumulate quantities over surfaces in three-dimensional space.

The central idea

\[ f(x,y) \quad\Longrightarrow\quad \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \quad\Longrightarrow\quad \nabla f \]

Single-variable calculus studies change along one input direction. Multivariable calculus studies how a quantity changes when several independent variables are involved.