# Calculus 3 : Stokes' Theorem

## Example Questions

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### Example Question #1 : Stokes' Theorem

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

Considering the integral , utilize Stokes' Theorem to determine  for 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

And it can be inferred from this that

A helpful approach can be to look at the right sides of the equations and see what variables are represented compared to what variables a vector companent of F is being derived for. Doing this and integrating, we can infer that

and

### Example Question #2 : Stokes' Theorem

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

Considering the integral , utilize Stokes' Theorem to determine  for 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

And it can be inferred from this that

A helpful approach can be to look at the right sides of the equations and see what variables are represented compared to what variables a vector companent of F is being derived for. Doing this and integrating, we can infer that

and

### Example Question #3 : Stokes' Theorem

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

Considering the integral , utilize Stokes' Theorem to determine  for 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

And it can be inferred from this that

A helpful approach can be to look at the right sides of the equations and see what variables are represented compared to what variables a vector companent of F is being derived for. Doing this and integrating, we can infer that

and

### Example Question #4 : Stokes' Theorem

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

Considering the integral , utilize Stokes' Theorem to determine  for 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

And it can be inferred from this that

A helpful approach can be to look at the right sides of the equations and see what variables are represented compared to what variables a vector companent of F is being derived for. Doing this and integrating, we can infer that

and

### Example Question #5 : Stokes' Theorem

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

Considering the integral , utilize Stokes' Theorem to determine  for 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

And it can be inferred from this that

A helpful approach can be to look at the right sides of the equations and see what variables are represented compared to what variables a vector companent of F is being derived for. Doing this and integrating, we can infer that

and

### Example Question #6 : Stokes' Theorem

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

Considering the integral , utilize Stokes' Theorem to determine  for 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

And it can be inferred from this that

A helpful approach can be to look at the right sides of the equations and see what variables are represented compared to what variables a vector companent of F is being derived for. Doing this and integrating, we can infer that

and

### Example Question #7 : Stokes' Theorem

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

Considering the integral , utilize Stokes' Theorem to determine  for 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

And it can be inferred from this that

A helpful approach can be to look at the right sides of the equations and see what variables are represented compared to what variables a vector companent of F is being derived for. Doing this and integrating, we can infer that

(Note that ; both results are valid)

and

### Example Question #8 : Stokes' Theorem

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

Considering the integral , utilize Stokes' Theorem to determine  for 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

And it can be inferred from this that

A helpful approach can be to look at the right sides of the equations and see what variables are represented compared to what variables a vector companent of F is being derived for. Doing this and integrating, we can infer that

and

### Example Question #9 : Stokes' Theorem

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

Considering the integral , utilize Stokes' Theorem to determine  for 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

And it can be inferred from this that

A helpful approach can be to look at the right sides of the equations and see what variables are represented compared to what variables a vector companent of F is being derived for. Doing this and integrating, we can infer that

and

### Example Question #10 : Stokes' Theorem

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

Considering the integral , utilize Stokes' Theorem to determine  for 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

And it can be inferred from this that

A helpful approach can be to look at the right sides of the equations and see what variables are represented compared to what variables a vector companent of F is being derived for. Doing this and integrating, we can infer that

and

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