Calculus 3 : Divergence

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #31 : 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:

Example Question #32 : Divergence

Find , where F is given by the following:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector function is given by

where 

So, we must find the partial derivative of each respective component.

The partial derivatives are

The partial derivatives were found using the following rules:

Example Question #33 : Divergence

Find , where F is given by the following curve:

Possible Answers:

Correct answer:

Explanation:

The divergence of a curve is given by

where 

So, we must find the partial derivatives of the x, y, and z components, respectively:

The partial derivatives were found using the following rules:

Example Question #34 : Divergence

Find  of the given function:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector function is given by 

where 

So, we take the respective partial derivatives of the x, y, and z-components of the vector function, and add them together (from the dot product):

The partial derivatives were found using the following rules:

Example Question #41 : Line Integrals

Find  of the vector function below:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector function is given by

where 

So, in taking the dot product of the gradient and the function, we get the sum of the respective partial derivatives. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

 

 

Example Question #42 : Line Integrals

Find  of the vector function below:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector function is given by

where 

So, in taking the dot product of the gradient and the function, we get the sum of the respective partial derivatives. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

Example Question #43 : Line Integrals

Find  where F is given by

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

where 

In taking the dot product of the gradient and the vector field, we get the sum of the respective partial derivatives of F.

The partial derivatives are

Example Question #44 : Line Integrals

Find  where F is given by

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

where 

In taking the dot product of the gradient and the vector field, we get the sum of the respective partial derivatives of F.

The partial derivatives are

Example Question #45 : Line Integrals

Find  where F is given by

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

where 

In taking the dot product of the gradient and the vector field, we get the sum of the respective partial derivatives of F.

The partial derivatives are

Example Question #46 : Line Integrals

Find the divergence of the vector 

Possible Answers:

Correct answer:

Explanation:

To find the divergence of a vector , we use the definition

Using the vector from the problem statement, we get

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