Calculus 3 : Stokes' Theorem

Example Questions

Example Question #33 : Surface Integrals

Let S be a known surface with a boundary curve, C.

Considering the integral , utilize Stokes' Theorem to determine an equivalent integral of the form:

Explanation:

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Example Question #34 : Surface Integrals

Let S be a known surface with a boundary curve, C.

Considering the integral , utilize Stokes' Theorem to determine an equivalent integral of the form:

Explanation:

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Example Question #31 : Surface Integrals

Let S be a known surface with a boundary curve, C.

Considering the integral , utilize Stokes' Theorem to determine an equivalent integral of the form:

Explanation:

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Not as bad as it looked, actually.

Example Question #36 : Surface Integrals

Let S be a known surface with a boundary curve, C.

Considering the integral , utilize Stokes' Theorem to determine an equivalent integral of the form:

Explanation:

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Example Question #21 : Surface Integrals

Let S be a known surface with a boundary curve, C.

Considering the integral , utilize Stokes' Theorem to determine an equivalent integral of the form:

Explanation:

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Example Question #31 : Surface Integrals

Let S be a known surface with a boundary curve, C.

Considering the integral , utilize Stokes' Theorem to determine an equivalent integral of the form:

Explanation:

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Example Question #21 : Surface Integrals

Let S be a known surface with a boundary curve, C.

Considering the integral , utilize Stokes' Theorem to determine an equivalent integral of the form:

Explanation:

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Example Question #40 : Surface Integrals

Let S be a known surface with a boundary curve, C.

Considering the integral , utilize Stokes' Theorem to determine an equivalent integral of the form:

Explanation:

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Example Question #41 : Surface Integrals

Let S be a known surface with a boundary curve, C.

Considering the integral , utilize Stokes' Theorem to determine an equivalent integral of the form:

Explanation:

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function itself integrated over the boundary curve, C, of S.

Note that

From what we're told

Example Question #3861 : Calculus 3

Let S be a known surface with a boundary curve, C.

Considering the integral , utilize Stokes' Theorem to determine an equivalent integral of the form:

Explanation:

In order to utilize Stokes' theorem, note its form

The curl of a vector function F over an oriented surface S is equivalent to the function itself integrated over the boundary curve, C, of S.

Note that

From what we're told