Vector Fields and Line Integrals

These flashcards cover key definitions and concepts related to vector fields, line integrals, and integral theorems in multivariable calculus.

Vector Field

A function that assigns a vector to every point in a space, for example, R² or R³.

R²

A two-dimensional space represented by ordered pairs (x,y).

R³

A three-dimensional space represented by ordered triples (x,y,z).

Gradient Field

A vector field that is the gradient of a scalar function, indicating the direction of greatest increase of that function.

Arc Length Formula

The formula to find the length of a curve in R², given by s(t) = ∫ from a to b √((dx/dt)² + (dy/dt)²) dt.

Line Integral

An integral that calculates the effect of a vector field along a curve.

Conservative Field

A vector field where the line integral between two points is independent of the path taken.

Curl

A measure of the rotation of a vector field in three-dimensional space.

Divergence

A scalar operation on a vector field that measures the rate of change of density at a point.

Fundamental Theorem for Line Integrals

If f is continuous on [a, b], then ∫ f(x) dx from a to b equals F(b) - F(a), where F is any antiderivative of f.

Green's Theorem

A theorem relating the double integral over a region to a line integral around its boundary in R².

Parameterization

The process of expressing a curve using a variable that represents the position on the curve as a function of time.

Line Segment

The straight path connecting two points in space.

Parametrized Curve

A curve defined by functions for x and y in terms of one or more parameters.

Work (Physics)

The line integral of a force vector field along a curve, representing the energy transferred by the force.

Potential Function

A scalar function f such that a vector field
\vec{F}
is its gradient, i.e.,
\vec{F} = \nabla f
. This function exists for conservative fields.

Path Independence

A property of a vector field where the line integral between two points is the same regardless of the path taken, characteristic of conservative fields.

Circulation

The line integral of a vector field along a closed curve, measuring the tendency of the field to rotate objects along that curve.

Scalar Function

A function that assigns a single scalar value (a number) to every point in a space, for example, R

² or R

³.

Gradient

A vector operator that indicates the direction and rate of the greatest increase of a scalar field, represented as \nabla f = \left< \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right> in R³.

Flux

The measure of a vector field passing through a given surface, often calculated using a surface integral. It quantifies the flow of a vector quantity across an oriented surface.

Stokes' Theorem

A theorem that relates the integral of the curl of a vector field over a surface to the line integral of the vector field around the boundary curve of that surface. It is a generalization of Green's Theorem to three dimensions.

Divergence Theorem (Gauss's Theorem)

A theorem that relates the flux of a vector field out of a closed surface to the triple integral of the divergence of the field over the volume enclosed by the surface.

Simply Connected Region

A region in which every closed loop can be continuously shrunk to a point within the region. This property is crucial for a vector field to be conservative in that region (assuming certain continuity conditions).