# Calculus 3 : Divergence

## Example Questions

### Example Question #31 : Divergence

Compute the divergence of the following vector function:

Explanation:

For a vector function ,

the divergence is defined by:

For our function:

Thus, the divergence of our function is:

### Example Question #32 : Divergence

Find , where F is given by the following:

Explanation:

The divergence of a vector function is given by

where

So, we must find the partial derivative of each respective component.

The partial derivatives are

The partial derivatives were found using the following rules:

### Example Question #33 : Divergence

Find , where F is given by the following curve:

Explanation:

The divergence of a curve is given by

where

So, we must find the partial derivatives of the x, y, and z components, respectively:

The partial derivatives were found using the following rules:

### Example Question #34 : Divergence

Find  of the given function:

Explanation:

The divergence of a vector function is given by

where

So, we take the respective partial derivatives of the x, y, and z-components of the vector function, and add them together (from the dot product):

The partial derivatives were found using the following rules:

### Example Question #41 : Line Integrals

Find  of the vector function below:

Explanation:

The divergence of a vector function is given by

where

So, in taking the dot product of the gradient and the function, 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 #42 : Line Integrals

Find  of the vector function below:

Explanation:

The divergence of a vector function is given by

where

So, in taking the dot product of the gradient and the function, 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 #43 : Line Integrals

Find  where F is given by

Explanation:

The divergence of a vector field is given by

where

In taking the dot product of the gradient and the vector field, we get the sum of the respective partial derivatives of F.

The partial derivatives are

### Example Question #44 : Line Integrals

Find  where F is given by

Explanation:

The divergence of a vector field is given by

where

In taking the dot product of the gradient and the vector field, we get the sum of the respective partial derivatives of F.

The partial derivatives are

### Example Question #45 : Line Integrals

Find  where F is given by

Explanation:

The divergence of a vector field is given by

where

In taking the dot product of the gradient and the vector field, we get the sum of the respective partial derivatives of F.

The partial derivatives are

### Example Question #46 : Line Integrals

Find the divergence of the vector

Explanation:

To find the divergence of a vector , we use the definition

Using the vector from the problem statement, we get