Calculus 3 : Divergence

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #1 : Divergence

Compute  for 

Possible Answers:



Correct answer:

Explanation:

In order to find the divergence, we need to remember the formula.

Divergence Formula:

, where , and  correspond to the components of a given vector field .

 

Now lets apply this to our situation.

 

Example Question #2 : Divergence

Compute  for 

Possible Answers:

Correct answer:

Explanation:

In order to find the divergence, we need to remember the formula.

Divergence Formula:

, where , and  correspond to the components of a given vector field .

 

Now lets apply this to our situation.

 

Example Question #3 : Divergence

Compute  for 

Possible Answers:

Correct answer:

Explanation:

In order to find the divergence, we need to remember the formula.

Divergence Formula:

, where , and  correspond to the components of a given vector field .

 

Now lets apply this to our situation.

 

Example Question #4 : Divergence

Find , where 

Possible Answers:

Correct answer:

Explanation:

In order to find the divergence, we need to remember the formula.

Divergence Formula:

, where , and  correspond to the components of a given vector field .

 

Now lets apply this to our situation.

 

Example Question #5 : Divergence

Compute , where 

Possible Answers:

Correct answer:

Explanation:

All we need to do is calculate the partial derivatives and add them together.

Example Question #6 : Divergence

Given the vector field

find the divergence of the vector field:

.

Possible Answers:

Correct answer:

Explanation:

Given a vector field 

we find its divergence by taking the dot product with the gradient operator:

We know that , so we have 

Example Question #7 : Divergence

Suppose that . Calculate the divergence.

Possible Answers:

Correct answer:

Explanation:

We know,

Use this to obtain the correct answer

Example Question #8 : Divergence

Given that

calculate 

Possible Answers:

Correct answer:

Explanation:

using this formula we have

Example Question #9 : Divergence

Find , where F is given by the following curve:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector is given by

where 

So, we take the partial derivative of each component of our vector with respect to x, y, and z respectively and add them together:

The derivatives were found using the following rules:

Example Question #10 : Divergence

Find  where F is given by

Possible Answers:

Correct answer:

Explanation:

The divergence of a curve is given by

where 

Taking the dot product of the gradient and the curve, we end up summing the respective partial derivatives (for example, the x coordinate's partial derivative with respect to x is found).

The partial derivatives are:

The following rules were used to find the derivatives:

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