Multivariable Calculus

Multivariable calculus, also known as calculus of several variables, extends the concepts of calculus to functions of multiple variables.

Functions of Several Variables

Functions of several variables take multiple inputs and produce an output. For example, f(x, y) = x^2 + y^2 is a function of two variables.

Partial Derivatives

Just as the derivative of a function of one variable gives the rate of change of the function at a point, the partial derivatives of a function of several variables give the rate of change of the function in each of the coordinate directions.

For example, the partial derivative of f(x, y) with respect to x is the rate of change of f as x changes while y is held constant.

Multiple Integrals

Multiple integrals extend the concept of integration to functions of several variables. A double integral, for example, computes the volume under a surface in three dimensions.

Vector Calculus

Vector calculus is a branch of multivariable calculus that deals with vector fields, which assign a vector to every point in space.

Key concepts in vector calculus include:

  • Gradient: The gradient of a scalar field is a vector field that points in the direction of greatest rate of increase of the scalar field.

  • Divergence: The divergence of a vector field is a measure of how much the vector field spreads out from a point.

  • Curl: The curl of a vector field is a measure of its rotation.

  • Line Integrals, Surface Integrals, and Volume Integrals: These are integrals taken over a line, surface, or volume, respectively.

  • Stokes' Theorem and Divergence Theorem: These are fundamental theorems that relate line, surface, and volume integrals.

Applications

Multivariable calculus has applications in physics, engineering, computer graphics, economics, and many other fields. It's used to describe the motion of planets, the flow of fluids, the curvature of surfaces, and much more.