# 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.

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Q2:

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

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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 .

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Q4:

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

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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 .

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Q6:

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

What is ?

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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?

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There are 6 partial derivatives of . Evaluate .

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

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

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Write a matrix equation relating with .

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