# Calculus 3 : Divergence

## Example Questions

### Example Question #57 : Line Integrals

Find the divergence of the following vector field:

Explanation:

The divergence of the vector field is given by

where

Taking the dot product gives us the sum of the respective partial derivatives of the vector field. For higher order partial derivatives, we work from left to right for the given variables.

The partial derivatives are

### Example Question #58 : Line Integrals

Find the divergence of the vector

Explanation:

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

Using the vector from the problem statement, we get

### Example Question #59 : Line Integrals

Find the divergence of the vector

Explanation:

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

Using the vector from the problem statement, we get

### Example Question #60 : Line Integrals

Find the divergence of the following vector field:

Explanation:

The divergence of the vector field

where

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

The partial derivatives are

### Example Question #61 : Line Integrals

Find the divergence of the vector field:

Explanation:

The divergence of the vector field

where

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

The partial derivatives are

### Example Question #62 : 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 of the gradient with the vector field, 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 #63 : Line Integrals

Determine the divergence of the 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.

The partial derivatives are

### Example Question #64 : Line Integrals

Find the divergence of the 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.

The partial derivatives are

### Example Question #65 : 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 of the gradient with the vector field, we are left with the sum of the respective partial derivative of the vector field.

The partial derivatives are

### Example Question #66 : Line Integrals

Find the divergence of the vector field: