Surface Integrals - AP Calculus BC
Card 0 of 540
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
Derivative of an exponential:
Trigonometric derivative:
Note that u may represent large functions, and not just individual variables!
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Derivative of an exponential:
Trigonometric derivative:
Note that u may represent large functions, and not just individual variables!
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
Derivative of a natural log:
Note that u may represent large functions, and not just individual variables!
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Derivative of a natural log:
Note that u may represent large functions, and not just individual variables!
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
Trigonometric derivative:
Note that u may represent large functions, and not just individual variables!
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Trigonometric derivative:
Note that u may represent large functions, and not just individual variables!
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
Derivative of an exponential:
Note that u may represent large functions, and not just individual variables!
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Derivative of an exponential:
Note that u may represent large functions, and not just individual variables!
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
Trigonometric derivative:
Note that u may represent large functions, and not just individual variables!
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Trigonometric derivative:
Note that u may represent large functions, and not just individual variables!
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
Derivative of a natural log:
Note that u may represent large functions, and not just individual variables!
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Derivative of a natural log:
Note that u may represent large functions, and not just individual variables!
Compare your answer with the correct one above
Find the divergence of the function 
at 
Find the divergence of the function
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.
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:
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.
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
What we will do is take the derivative of each vector element with respect to its variable
Then sum the results together:
Compare your answer with the correct one above