Calculus 3 : Divergence

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #47 : Line Integrals

Find  of the vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

where 

In taking the dot product, we are left with the sum of the respective partial derivatives of the vector function. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

Example Question #48 : Line Integrals

Find  of the vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

where 

In taking the dot product, we are left with the sum of the respective partial derivatives of the vector function. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

Example Question #49 : Line Integrals

Find , where F is the following vector field:

 

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

where 

In taking the dot product, we get the sum of the respective partial derivatives of the vector field. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

Example Question #50 : Line Integrals

Find , where F is the following vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

where 

In taking the dot product, we get the sum of the respective partial derivatives of the vector field. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

Example Question #51 : Line Integrals

Find the divergence of the following vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

where 

In taking the dot product, we end up with the sum of the respective partial derivatives of the vector field. To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

Example Question #52 : Line Integrals

Find the divergence of the vector 

Possible Answers:

Correct answer:

Explanation:

To find the divergence of the vector , we use the following formula

Applying to the vector from the problem statement, we get

Example Question #53 : Line Integrals

Find the divergence of the vector 

Possible Answers:

Correct answer:

Explanation:

To find the divergence of the vector , we use the following formula

Applying to the vector from the problem statement, we get

Example Question #54 : Line Integrals

Find the divergence of the following vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

where 

When we take the dot product, we end up with the sum of the respective partial derivatives of the vector field.

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

 

Example Question #55 : Line Integrals

Find the divergence of the following vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

where 

When we take the dot product, we end up with the sum of the respective partial derivatives of the vector field.

To find the given partial derivative of the function, we must treat the other variable(s) as constants.

The partial derivatives are

Example Question #56 : Line Integrals

Find the divergence of the following vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of the vector field is given by

where 

Taking the dot product gives us the sum of the respective partial derivatives of the vector field. For higher order partial derivatives, we work from left to right for the given variables.

The partial derivatives are

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