Calculus 3 : Divergence Theorem

Study concepts, example questions & explanations for Calculus 3

varsity tutors app store varsity tutors android store

Example Questions

← Previous 1 3

Example Question #1 : Divergence Theorem

Find the divergence of the function  at 

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

 at 

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Example Question #2 : Divergence Theorem

Find the divergence of the function  at 

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

 at 

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Example Question #3 : Divergence Theorem

Find the divergence of the function  at 

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

 at 

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Example Question #4 : Divergence Theorem

Find the divergence of the function  at 

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

 at 

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Example Question #5 : Divergence Theorem

Find the divergence of the function  at 

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

 at 

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Example Question #6 : Divergence Theorem

Find the divergence of the function  at 

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

 at 

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Example Question #7 : Divergence Theorem

Find the divergence of the function  at 

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

 at 

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Example Question #8 : Divergence Theorem

Find the divergence of the function  at 

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

 at 

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Example Question #9 : Divergence Theorem

Find the divergence of the function  at 

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

 at 

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

Example Question #10 : Divergence Theorem

Find the divergence of the function  at 

Possible Answers:

Correct answer:

Explanation:

Divergence can be viewed as a measure of the magnitude of a vector field's source or sink at a given point.

To visualize this, picture an open drain in a tub full of water; this drain may represent a 'sink,' and all of the velocities at each specific point in the tub represent the vector field. Close to the drain, the velocity will be greater than a spot farther away from the drain.

Vectorfield

What divergence can calculate is what this velocity is at a given point. Again, the magnitude of the vector field.

We're given the function

 at 

What we will do is take the derivative of each vector element with respect to its variable 

Then sum the results together:

← Previous 1 3
Learning Tools by Varsity Tutors

Incompatible Browser

Please upgrade or download one of the following browsers to use Instant Tutoring: