# Calculus 3 : Stokes' Theorem

## Example Questions

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### Example Question #1 : 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

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

### Example Question #2 : 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

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

### Example Question #3 : 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

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

### Example Question #4 : 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

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

### Example Question #5 : 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

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

### Example Question #6 : 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

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

### Example Question #7 : 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

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

### Example Question #8 : 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

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

### Example Question #9 : 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

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

### Example Question #10 : 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

Meaning that

From this we can derive our curl vectors

This allows us to set up our surface integral

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