Calculus 3 : Divergence

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #27 : Line Integrals

Find the divergence of the force field 

Possible Answers:

Correct answer:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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:

Explanation:

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 is a scalar function, which of the following operations results in a vector?

 

 
Possible Answers:

Correct answer:

Explanation:

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:

Explanation:

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:

Explanation:

For a vector function ,

the divergence is defined by:

For our function:

Thus, the divergence of our function is:

 

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