Vector Calculus: Gradient, Curl, and Flow (Vocabulary Flashcards)

Vocabulary flashcards covering key concepts from the lecture notes: gradient fields, curl, del operator, flow along curves, parameterization, potential functions, irrotational and conservative fields, net flow, Stokes' theorem, closed curves, and notation.

Vector field

A function that assigns a vector to every point in a region, F(x,y,z) = (P,Q,R).

Gradient field

A vector field that equals the gradient of some scalar function: F = ∇f; it is conservative and has a potential function.

Curl

The curl of F, written ∇ × F, measures local rotation of the field; components are (∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y).

Del (nabla)

The differential operator ∇ = (∂/∂x, ∂/∂y, ∂/∂z) used to compute gradient, divergence, and curl.

Flow along a curve

The line integral of a vector field along a curve C: ∫_C F · dr = ∫ F(r(t)) · r'(t) dt; gives net flow along C.

Parameterization

Expressing a curve as r(t) = (x(t), y(t), z(t)) for t in an interval; used to set up line integrals.

Potential function

A scalar function φ such that F = ∇φ; its existence means F is a gradient (conservative) field.

Irrotational

A vector field with curl F = 0 everywhere in the region; no local rotation.

Conservative field

A vector field with path-independent line integrals; equivalent to a gradient field on suitable domains.

Net flow

The value of the line integral ∫_C F · dr along a curve C, indicating total flow along that path.

Stokes' theorem

Relates a line integral around a closed curve to a surface integral of curl: ∮C F · dr = ∬S (∇ × F) · n dS.

Closed curve

A curve whose endpoints coincide, forming a loop (often described as a wire in notes).

Divergence (del · F)

The scalar measure ∇ · F of how much F “diverges” from a point; relates to sources/sinks of the field.

Curl notation literacy

The idea that curl is ∇ × F (del crossed with F) and that this notation encodes which derivatives are taken and how they’re combined.