Curl and Divergence Operators

These flashcards cover essential vocabulary and concepts related to Curl and Divergence operators in vector calculus.

Curl

An operator that produces a vector field representing the rotation of a vector field.

Divergence

An operator that produces a scalar field representing the spread of a vector field.

Vector Field

A function that assigns a vector to every point in space.

Scalar Field

A function that assigns a scalar value to every point in a space.

Conservative Vector Field

A vector field where the curl is zero, indicating the field can be derived from a scalar potential.

Partial Derivatives

Derivatives of a function with respect to one variable while keeping others constant.

Gradient

A vector field representing the rate and direction of change of a scalar field.

Curl of a Gradient

The curl of the gradient of a scalar function is always zero.

Simply Connected Region

A region without holes, where any loop can be continually shrunk to a point.

Curl F

The notation representing the curl of a vector field F.

Thm. 1

The curl of the gradient of a scalar function f is zero.

Thm. 2

If the curl of a vector field is zero in a simply connected region, the field is conservative.

Continous Partial Derivatives

Partial derivatives exist and are continuous in a region.

Tendency to Rotate

Describes how particles move around in a vector field based on its curl.

Curl & Divergence Theorem

Relates the surface integral of a vector field over a surface to the volume integral of its divergence.

Flux Integral

The surface integral of a vector field across a surface.

Stoke's Theorem

Relates a surface integral over a surface to a line integral around the boundary of that surface.

Conservative Field Test

A field is conservative if its curl is zero.

Area Integral Representation

The formula used to calculate the area over a surface in parametric form.

Parametric Surface

A surface represented by a set of parameterized equations.

Orientation of Surface

The direction in which the surface normal vector points.

Vector Equation of a Surface

An equation that expresses the surface using parameter variables.

Spherical Coordinates

A coordinate system where points are defined by radius and angles.

Normal Vector

A vector that is perpendicular to a surface at a given point.

Surface Area Calculation

The method of finding the area of a parametrically defined surface.

Mass and Center of Mass

The calculation regarding the density and distribution of mass over a surface.