### All Calculus 3 Resources

## Example Questions

### Example Question #27 : Line Integrals

Find the divergence of the force field

**Possible Answers:**

**Correct answer:**

The correct formula for divergence is .

Remember that the result of a divergence calculation must be a scalar, not a vector.

.

### Example Question #28 : Line Integrals

Evaluate the divergence of the force field

**Possible Answers:**

None of the other answers

**Correct answer:**

The correct formula for divergence is .

Remember that the result of a divergence calculation must be a scalar, not a vector.

.

### Example Question #29 : Line Integrals

Find the divergence of the force field .

**Possible Answers:**

None of the other answers

**Correct answer:**

The correct formula for divergence is .

Remember that the result of a divergence calculation must be a scalar, not a vector.

.

### Example Question #30 : Line Integrals

Find the divergence of the force field .

**Possible Answers:**

None of the other answers

**Correct answer:**

The correct formula for divergence is .

Remember that the result of a divergence calculation must be a scalar, not a vector.

.

### Example Question #21 : Divergence

Find the divergence of the vector

**Possible Answers:**

**Correct answer:**

To find the divergence of a vector , we apply the following definition: . Applying the definition to the vector from the problem statement, we get

### Example Question #22 : Divergence

Find , where F is given by

**Possible Answers:**

**Correct answer:**

The divergence of a vector is given by

, where

Taking the partial respective partial derivatives of the x, y, and z components of our curve, we get

The rules used to find the derivatives are as follows:

,

### Example Question #23 : Divergence

Calculate the divergence of the following vector:

**Possible Answers:**

**Correct answer:**

For a given vector

The divergence is calculated by:

For our vector

The answer is, therefore:

### Example Question #24 : Divergence

Given that **F** is a vector function and *f *is a scalar function, which of the following operations results in a vector?

**Possible Answers:**

**Correct answer:**

For all the given answers:

- The curl of a vector is also a vector, so the divergence of the term in parenthesis is scalar.

- The divergence of a vector is a scalar. The divergence of a scalar is undefined, so this expression is undefined.

- The gradient of a scalar is a vector, so the divergence of the term in parenthesis is a scalar.

- The divergence of a scalar doesn't exist, so this expression is undefined.

The remaining answer is:

- The curl of a vector is also a vector, so the curl of the term in parenthesis is a vector as well.

### Example Question #25 : Divergence

Given that **F** is a vector function and *f *is a scalar function, which of the following expressions is valid?

**Possible Answers:**

**Correct answer:**

For each of the given answers:

- The curl of a scalar is undefined, so the term in parenthesis is invalid.

- The divergence of a scalar is also undefined, so the term in parenthesis is invalid.

- The gradient of a scalar is undefined as well, so the term in parenthesis is invalid.

- The divergence of a vector is a scalar. The divergence of a the term in parenthesis, which is a scalar, is undefined, so the expression is invalid.

The remaining answer must be correct:

- The gradient of a scalar is a vector, so the divergence of the term in parenthesis is a scalar. The expression is valid.

### Example Question #26 : Divergence

Compute the divergence of the following vector function:

**Possible Answers:**

**Correct answer:**

For a vector function ,

the divergence is defined by:

For our function:

Thus, the divergence of our function is:

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