Conservative Vector Fields

From Department of Mathematics at UTSA
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Path Independence

If a vector field F is the gradient of a scalar field G (i.e. if F is conservative), that is,

then by the multivariable chain rule, the derivative of the composition of G and r(t) is

which happens to be the integrand for the line integral of F on r(t). It follows, given a path C , that

In other words, the integral of F over C depends solely on the values of G at the points r(b) and r(a), and is thus independent of the path between them. For this reason, a line integral of a conservative vector field is called path independent.

Stokes' Theorem

Stokes' Theorem is effectively a generalization of Green's theorem to 3 dimensions, and the "curl" is a generalization of the quantity to 3 dimensions. An arbitrary oriented surface can be articulated into a family of infinitesimal surfaces, some parallel to the xy-plane, others parallel to the zx-plane, and the remainder parallel to the yz-plane. Let denote an arbitrary vector field.

Let be a surface that is parallel to the yz-plane with counter-clockwise oriented boundary . Green's theorem gives:

is positive if the normal direction to points in the positive x direction and is negative if otherwise. If the normal direction to points in the negative x direction, then is oriented clockwise instead of counter-clockwise in the yz-plane.

Decomposing a 3D loop into an ensemble of infinitesimal loops that are parallel to the yz, zx, or xy planes.

Repeating this argument for being parallel to the zx-plane and xy-plane respectively gives:

and

Treating as an ensemble of infinitesimal surfaces parallel to the yz-plane, zx-plane, or xy-plane gives:

This is Stokes' theorem, and is the "curl" of which generalizes the "circulation density" to 3 dimensions.

The direction of at is effectively an "axis of rotation" around which the counterclockwise circulation density in a plane whose normal is parallel to is . Out of all planes that pass through , the plane whose normal is parallel to has the largest counterclockwise circulation density at which is .

An arbitrary vector field that is differentiable everywhere is considered to be "irrotational" or "conservative" if everywhere, or equivalently that for all continuous closed curves

Conservative vector fields

A vector field for which at all points is an "conservative" vector field. can also be referred to as being "irrotational" since the gain around any closed curve is always 0.

A key property of a conservative vector field is that the gain of along a continuous curve is purely a function of the curve's end points. If and are two continuous curves which share the same starting point and end point , then . In other words, the gain is purely a function of and . This property can be derived from Stokes' theorem as follows:

Invert the orientation of to get and combine and to get a continuous closed curve , linking the curves together at the endpoints and . Let denote a surface for which is the counterclockwise oriented boundary.

Stokes' theorem states that . The gain around is the gain along minus the gain along : . Therefore:


Resources

Conservative Vector Fields


Finding a Potential Function of a Conservative Vector Field


The Fundamental Theorem of Line Integrals

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