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: 

Derivative of a natural log: 

Note that u may represent large functions, and not just individual variables!

At the point 

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 v may represent large functions, and not just individual variables!

At the point 

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:

Note that for the complexity of the function, the derivatives are rather simple, given that when deriving with respect to one variable, we treat the other as constant!

At the point 

Using chain rule, we get 

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

 at 

x:

y:

The slope is 

To consider finding the slope, let's discuss the topic of the gradient.

For a function , the gradient is the sum of the derivatives with respect to each variable, multiplied by a directional vector:

It is essentially the slope of a multi-dimensional function at any given point

Knowledge of the following derivative rules will be necessary:

Derivative of an exponential: 

Note that u may represent large functions, and not just individual variables!

The approach to take with this problem is to simply take the derivatives one at a time. When deriving for one particular variable, treat the other variables as constant.

 at 

x:

y:

The slope is 

Let 

By applying the chain rule, we get, 

implies,

By substituting back we get,

Implicit differentiation allows us to extend the power rule to rational powers. So,

If and are integers and then

 by chain rule, so using the information from the table, we can find

. 

Using chain rule and product rule, we get 

When , we get