Multivariable Calculus

Multivariable Calculus explores functions of multiple variables, including partial derivatives, multiple integrals, and vector calculus.

Functions of Several VariablesPartial DerivativesMultiple Integrals
Gradient, Divergence, and CurlLine and Surface IntegralsTheorems of Vector Calculus
Modeling Heat and DiffusionOptimizing Engineering DesignsData Science and Gradient Descent
Visualizing Surfaces and RegionsBreaking Problems into Steps
Functions of Several...Partial DerivativesMultiple IntegralsGradient, Divergence...Line and Surface Int...Theorems of Vector C...Modeling Heat and Di...Optimizing Engineeri...Data Science and Gra...Visualizing Surfaces...Breaking Problems in...
Advanced Topics

Gradient, Divergence, and Curl

Vector Fields in Action

Many physical quantities, like wind or electric fields, have both magnitude and direction—these are vector fields. Calculus tools help us analyze how these fields change.

The Gradient

The gradient of a function points in the direction of steepest increase. If \(f(x, y, z)\) is a temperature field, the gradient shows where it's warming up fastest.

Divergence

Divergence measures how much a vector field spreads out from a point. Positive divergence means "stuff" is flowing out; negative means it's flowing in.

Curl

Curl measures the rotation or swirling of a vector field. It's used to study things like whirlpools or tornadoes.

Physical Connections

  • Gradient helps find the direction of fastest change.
  • Divergence explains sources or sinks in fluids.
  • Curl detects rotation in fields.

Key Formula

\[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)\]

Examples

  • The gradient of elevation gives the direction of steepest slope on a hill.

  • The curl of a wind field reveals the eye of a hurricane.

In a Nutshell

Gradient, divergence, and curl help us understand how vector fields change in space.

Key Terms

Vector Field
A function assigning a vector to every point in space.
Gradient
A vector pointing in the direction of greatest increase of a function.
Divergence
A measure of how much a vector field is expanding or contracting at a point.
Curl
A measure of the rotation of a vector field around a point.
Next
Line and Surface Integrals