Vector Fields, Line and Surface Integrals with Green, Stokes and Divergence Theorems: every key term you need (+ practice quiz)
25 flashcard terms for Multivariable Calculus Topic 8, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
An assignment of a vector to every point of a plane or spatial region, used to model velocity of a fluid, force of gravity or an electric influence.
Gradient field
A vector field that is the gradient of some scalar function. Such fields are exactly the conservative ones on suitable domains.
Potential function
A scalar function whose gradient reproduces a given vector field. It is determined only up to an additive constant on a connected region.
Conservative vector field
A field whose line integral depends only on the endpoints of the path. Equivalent conditions are the existence of a potential and vanishing circulation on every closed loop.
Line integral of a scalar function
Accumulates a scalar along a curve weighted by arc length, giving quantities such as the mass of a wire with variable density.
Line integral of a vector field
Accumulates the tangential component of the field along an oriented curve, computing work done by a force along a path.
Orientation of a curve
The choice of direction of travel. Reversing it flips the sign of a vector line integral while leaving a scalar arc length integral unchanged.
Fundamental theorem for line integrals
For a conservative field the line integral equals the potential at the terminal point minus the potential at the initial point, making the path irrelevant.
Path independence
The property that any two paths sharing endpoints give the same line integral, equivalent to conservativeness on a connected open region.
Cross partials test
On a simply connected domain a field is conservative exactly when the appropriate mixed partial derivatives of its components agree everywhere.
Simply connected region
A connected region in which every loop can be contracted to a point without leaving it. Puncturing a plane region destroys this and invalidates the cross partials test.
Circulation
The line integral of a vector field around a closed oriented curve, measuring the net tendency of the field to push material around that loop.
Flux across a plane curve
The line integral of the normal component of a plane field along a curve, measuring net outward flow rate across it.
Divergence
The scalar field measuring the net outflow per unit volume at a point, computed as the sum of the partials of each component with respect to its own variable.
Curl
The vector field measuring local rotation of a field, computed formally as the cross product of the derivative operator with the field. It is zero for every gradient field.
Irrotational field
A field whose curl vanishes identically. On a simply connected domain irrotational is the same as conservative, but on a punctured domain it is strictly weaker.
Incompressible field
A field of zero divergence everywhere, meaning no point acts as a source or a sink, as in an idealised steady fluid flow.
Green theorem
Relates the circulation of a plane field around a positively oriented simple closed curve to a double integral of a partial derivative difference over the enclosed region.
Positive orientation of a plane boundary
Traversal that keeps the enclosed region on the left, counterclockwise for an outer boundary and clockwise for a hole. Wrong orientation flips the sign of the result.
Parametrised surface
A surface described by a vector function of two parameters, whose partial derivative vectors span the tangent plane where their cross product is nonzero.
Surface area element
The magnitude of the cross product of the two parameter partial derivatives, times the changes in both parameters, giving the area of a small patch.
Surface integral of a scalar function
Integrates a scalar over a surface using the area element, producing quantities such as the mass of a thin curved sheet.
Flux integral
The surface integral of the normal component of a vector field across an oriented surface, measuring the rate at which the field crosses it.
Stokes theorem
Equates the flux of the curl through an oriented surface with the circulation of the field around its boundary curve, provided the boundary orientation matches the surface normal by the right-hand rule.
Divergence theorem
Equates the outward flux of a field through a closed surface with the triple integral of the divergence over the enclosed solid, requiring a closed surface and continuously differentiable components.