Multivariable calculus

Definition

Multivariable calculus deals with functions of multiple variables and extends concepts such as limits, derivatives, and integrals from single-variable calculus to higher dimensions. It forms the basis for fields like vector calculus and differential equations.

5 Must Know Facts For Your Next Test

1. Multivariable calculus is essential to understanding changes in systems described by more than one variable.
2. The limit definition extends to multivariable functions using neighborhood concepts around a point.
3. Partial derivatives represent rates of change in multivariable functions with respect to each variable independently.
4. The gradient vector generalizes the derivative for multivariable functions, pointing in the direction of greatest rate of increase.
5. Jacobian matrices are used to transform coordinates and describe how functions change in multivariable contexts.

Review Questions

• What is a partial derivative, and how does it differ from an ordinary derivative?
• Explain how the concept of limits is extended to functions of several variables.
• What is the gradient vector, and what information does it provide about a function?

Related terms

Partial Derivative: The rate at which a function changes with respect to one variable while keeping other variables constant.

Gradient Vector: A vector consisting of all partial derivatives of a function; it indicates the direction of steepest ascent.

Jacobian Matrix: A matrix of all first-order partial derivatives of a vector-valued function; used in coordinate transformations.