# Lesson Worksheet: Directional Derivatives and Gradient Mathematics

In this worksheet, we will practice finding a derivative of multivariable functions in a given direction (directional derivative) and finding the gradient vector of the function.

**Q1: **

Find a function so that the vector field is a gradient field.

- A
- B
- C
- D
- E

**Q2: **

Suppose with . Express the gradient (viewed as a matrix) in terms of the matrix , where , and a matrix of partial derivatives of .

- A
- B
- C
- D
- E

**Q3: **

Suppose with and that for a point . Express the gradient (viewed as a matrix) in terms of the matrix and a matrix of partial derivatives of .

- A
- B
- C
- D
- E

**Q4: **

Let where and . Given that there is a line in the - plane for which , find the equation of this line.

- A
- B
- C
- D
- E

**Q5: **

The chain rule for the composition of a curve and a function says that , where is the gradient of . We have been given the function and the curve . At certain points, such as the origin, or points and in the figure, .

Find the parameters for and . Leave your answer in terms of .

- A,
- B,
- C,
- D,
- E,

**Q6: **

Suppose with given by the linear mapping , where are constants.

What is ?

- A
- B
- C

If we consider the domains of and of as consisting of column vectors, then can be written as a multiplication by a matrix. What is this matrix?

- A
- B
- C
- D
- E

There are 6 partial derivatives of . Evaluate .

- A
- B
- C
- D
- E

The gradient of at the point of can also be thought of as a linear map into . What is the corresponding matrix?

- A
- B
- C
- D
- E

Now is a function from to with a gradient represented by a matrix. Writing for , what is this matrix at the point ?

- A
- B
- C
- D
- E

Write a matrix equation relating with .

- A
- B
- C
- D
- E