Stokes's Theorem and Vector Fields

Green's Theorem applies to three-dimensional surfaces.

Stokes's Theorem applies to two-dimensional curves.

Green's Theorem is a special case of Stokes's Theorem in two dimensions.

Stokes's Theorem is unrelated to Green's Theorem.

The double integral of the vector field.

The line integral of the gradient of the vector field.

The surface integral of the curl of the vector field.

The surface integral of the vector field itself.

r =

r =

r =

r =

By taking the del cross the vector field.

By taking the cross product of the vector field with itself.

By taking the divergence of the vector field.

By taking the gradient of the vector field.

<1, 1, 1>

<0, 0, 0>

<-1, 0, -x>

0 to 1 - x

0 to 1

x to 1

0 to x

2/3

0

-2/3

1/3

It is only applicable to two-dimensional problems.

It eliminates the need for surface integrals.

It provides exact solutions for all integrals.

It simplifies the calculation of line integrals.

The vector field must be constant.

The boundary must be simple enough to calculate.

The surface must be a sphere.

The curve must be negatively oriented.