Calculus 3 : Divergence

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #67 : Line Integrals

Find the divergence of the vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

, where 

When we take the dot product of the gradient and the vector field, 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 #68 : Line Integrals

Find the divergence of the vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

, where 

When we take the dot product of the gradient and the vector field, 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 #69 : Line Integrals

Find the divergence of the vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

, where 

In taking the dot product of the gradient with the vector field, 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 #70 : Line Integrals

Find the divergence of the vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

, where .

When we take the dot product of the gradient and the vector field, 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 #71 : Line Integrals

Find the divergence of the vector 

Possible Answers:

Correct answer:

Explanation:

To find the divergence of a vector , we use the formula:

Applying to the vector from the problem statement, we get

Example Question #72 : Line Integrals

Find the divergence of the vector 

Possible Answers:

Correct answer:

Explanation:

To find the divergence of a vector , we use the formula:

Applying to the vector from the problem statement, we get

Example Question #73 : Line Integrals

Find the divergence of the function:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

, where 

When we take the dot product of the gradient and the vector field, 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 #74 : Line Integrals

Find the divergence of the vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

, where 

When we take the dot product of the gradient and the vector field, 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 #75 : Line Integrals

Find the divergence of the vector field:

Possible Answers:

Correct answer:

Explanation:

The divergence of a vector field is given by

, where 

When we take the dot product of the gradient and the vector field, 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

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