Calculus 3 : Divergence

Example Questions

Example Question #47 : Line Integrals

Find  of the vector field:

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:

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:

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:

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:

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

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

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:

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:

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: