### All Calculus 3 Resources

## Example Questions

### 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:

**Possible Answers:**

**Correct answer:**

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 **F **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:

**Possible Answers:**

**Correct answer:**

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 **F **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:

**Possible Answers:**

**Correct answer:**

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 **F **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**.

**Possible Answers:**

**Correct answer:**

In order to utilize Stokes' theorem, note its form

**F** over an oriented surface **S** is equivalent to the function **F **itself integrated over the boundary curve, **C**, of **S**.

Note that

From what we're told

And it can be inferred from this that

**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**.

**Possible Answers:**

**Correct answer:**

In order to utilize Stokes' theorem, note its form

**F** over an oriented surface **S** is equivalent to the function **F **itself integrated over the boundary curve, **C**, of **S**.

Note that

From what we're told

And it can be inferred from this that

**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**.

**Possible Answers:**

**Correct answer:**

In order to utilize Stokes' theorem, note its form

**F** over an oriented surface **S** is equivalent to the function **F **itself integrated over the boundary curve, **C**, of **S**.

Note that

From what we're told

And it can be inferred from this that

**F** is being derived for. Doing this and integrating, we can infer that

and

### Example Question #1 : Surface Integrals

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

**Possible Answers:**

**Correct answer:**

In order to utilize Stokes' theorem, note its form

**F** over an oriented surface **S** is equivalent to the function **F **itself integrated over the boundary curve, **C**, of **S**.

Note that

From what we're told

And it can be inferred from this that

**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**.

**Possible Answers:**

**Correct answer:**

In order to utilize Stokes' theorem, note its form

**F** over an oriented surface **S** is equivalent to the function **F **itself integrated over the boundary curve, **C**, of **S**.

Note that

From what we're told

And it can be inferred from this that

**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**.

**Possible Answers:**

**Correct answer:**

In order to utilize Stokes' theorem, note its form

**F** over an oriented surface **S** is equivalent to the function **F **itself integrated over the boundary curve, **C**, of **S**.

Note that

From what we're told

And it can be inferred from this that

**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**.

**Possible Answers:**

**Correct answer:**

In order to utilize Stokes' theorem, note its form

**F** over an oriented surface **S** is equivalent to the function **F **itself integrated over the boundary curve, **C**, of **S**.

Note that

From what we're told

And it can be inferred from this that

**F** is being derived for. Doing this and integrating, we can infer that

and

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