# Additional exercises 2 ​

## Exercise ​

Let and . Identify the domain and codomain of , . Determine if and are well defined. If it is well defined then write the function explicitly and determine the domain and codomain.

## Exercise ​

Let . Compute the partial derivatives , .

## Exercise ​

Let . Calculate , the gradient of .

## Exercise ​

Determine the directional derivative, for in the direction of .

## Exercise ​

Compute the Jacobian matrix of the transformation

## Exercise ​

Consider the functions,

Let . Calculate both by using the chain rule and by first calculating and then differentiating and confirm that the answer is the same using either method. Out of curiosity, try to understand what curve traces out as varies.

## Exercise ​

Let be defined, for , as

and . Calculate the 4 missing values in the following table, assuming that .

linevalue
?
?
?
?

What does this say about when approaching along these different lines?

## Exercise ​

Consider the surface . Verify that the point is contained in the surface. Find the tangent plane to this surface at this point.

Hint: write this surface as a level set , calculate at the specified point and use the known connection between gradient and tangent plane.