Lesson Worksheet: Directional Derivatives and Gradient Mathematics

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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 𝑄(𝑥,𝑦) so that the vector field F(𝑥,𝑦)=⟨𝑥𝑦,𝑄(𝑥,𝑦)⟩ is a gradient field.

  • A𝑄(𝑥,𝑦)=𝑥𝑦
  • B𝑄(𝑥,𝑦)=𝑥
  • C𝑄(𝑥,𝑦)=𝑥𝑦2
  • D𝑄(𝑥,𝑦)=𝑥2
  • E𝑄(𝑥,𝑦)=𝑦2

Q2:

Suppose 𝑤=𝐹(𝜙(𝑥,𝑦)) with 𝜙=(𝑥+𝑦,𝑥−𝑦,𝑥𝑦). Express the gradient ∇𝑤𝜋,−23 (viewed as a 1×2 matrix) in terms of the 1×3 matrix ∇𝐹(𝑞), where 𝑞=𝜙𝜋,−23, and a matrix of partial derivatives of 𝜙.

  • A∇𝑤𝜋,−23=∇𝐹(𝑞)⎡⎢⎢⎢⎢⎢⎣2𝜋432𝜋−43−23𝜋⎤⎥⎥⎥⎥⎥⎦
  • B∇𝑤𝜋,−23=∇𝐹(𝑞)⎡⎢⎢⎢⎢⎢⎣2𝜋−432𝜋−43−23𝜋⎤⎥⎥⎥⎥⎥⎦
  • C∇𝑤𝜋,−23=∇𝐹(𝑞)⎡⎢⎢⎢⎢⎢⎣2𝜋−432𝜋43−23𝜋⎤⎥⎥⎥⎥⎥⎦
  • D∇𝑤𝜋,−23=∇𝐹(𝑞)⎡⎢⎢⎢⎢⎢⎣𝜋−23𝜋23−23𝜋⎤⎥⎥⎥⎥⎥⎦
  • E∇𝑤𝜋,−23=∇𝐹(𝑞)⎡⎢⎢⎢⎢⎢⎣𝜋−23𝜋−23−23𝜋⎤⎥⎥⎥⎥⎥⎦

Q3:

Suppose 𝑤=𝐹(𝜙(𝑥,𝑦)) with 𝜙=(𝜙,𝜙,𝜙) and that 𝑞=𝜙(𝑝) for a point 𝑝∈ℝ. Express the gradient ∇𝑤(𝑝) (viewed as a 1×2 matrix) in terms of the 1×3 matrix ∇𝐹(𝑞) and a matrix of partial derivatives of 𝜙.

  • A∇𝑤(𝑝)=∇𝐹(𝑞)⋅⎡⎢⎢⎢⎣𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥⎤⎥⎥⎥⎦
  • B∇𝑤(𝑝)=∇𝐹(𝑞)⋅⎡⎢⎢⎢⎣𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦⎤⎥⎥⎥⎦
  • C∇𝑤(𝑝)=∇𝐹(𝑞)⋅⎡⎢⎢⎢⎢⎢⎢⎣𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦⎤⎥⎥⎥⎥⎥⎥⎦
  • D∇𝑤(𝑝)=∇𝐹(𝑞)⋅⎡⎢⎢⎢⎢⎢⎢⎣𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥⎤⎥⎥⎥⎥⎥⎥⎦
  • E∇𝑤(𝑝)=∇𝐹(𝑞)⋅⎡⎢⎢⎢⎢⎢⎢⎣𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦⎤⎥⎥⎥⎥⎥⎥⎦

Q4:

Let 𝑤=𝐹(𝜙(𝑥,𝑦)) where 𝜙=𝑥+𝑦,𝑥−𝑦,𝑥𝑦 and 𝐹(𝑝,𝑞,𝑟)=10𝑝+6𝑞−16𝑟. Given that there is a line in the 𝑥-𝑦 plane for which ∇𝑤=0, find the equation of this line.

  • A𝑦=𝑥
  • B𝑦=2𝑥
  • C𝑦=𝑥2
  • D𝑦=−2𝑥
  • E𝑦=−𝑥

Q5:

The chain rule for the composition of a curve 𝜙(𝑡)=(𝑥(𝑡),𝑦(𝑡)) and a function 𝑓(𝑥,𝑦) says that dd𝑡𝑓(𝜙(𝑡))=∇𝑓⋅𝜙′(𝑡), where ∇𝑓(𝑥,𝑦) is the gradient of 𝑓. We have been given the function 𝑓(𝑥,𝑦)=𝑥+𝑦 and the curve 𝜙(𝑡)=((2𝑡),(𝑡))sincos. At certain points, such as the origin, or points 𝐴 and 𝐵 in the figure, dd𝑡(𝑓(𝜙(𝑡)))=0.

Find the parameters 𝑡,𝑡∈(0,2𝜋] for 𝐴 and 𝐵. Leave your answer in terms of cos.

  • A𝑡=1214cos, 𝑡=2𝜋
  • B𝑡=12cos, 𝑡=𝜋2
  • C𝑡=1212cos, 𝑡=2𝜋
  • D𝑡=1412cos, 𝑡=𝜋
  • E𝑡=14cos, 𝑡=𝜋2

Q6:

Suppose 𝑤=𝐹(𝜙(𝑠,𝑡)) with 𝜙=(𝜙,𝜙,𝜙) given by the linear mapping 𝜙(𝑠,𝑡)=(𝑎𝑠+𝑏𝑡,𝑐𝑠+𝑑𝑡,𝑒𝑠+𝑓𝑡), where 𝑎,…,𝑓 are constants.

What is 𝜙(𝑠,𝑡)?

  • A𝑒𝑠+𝑓𝑡
  • B𝑐𝑠+𝑑𝑡
  • C𝑎𝑠+𝑏𝑡

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

  • A𝑎𝑑𝑏𝑒𝑐𝑓
  • B𝑎𝑐𝑒𝑏𝑑𝑓
  • C𝑏𝑑𝑓𝑎𝑐𝑒
  • D𝑏𝑎𝑑𝑐𝑓𝑒
  • E𝑎𝑏𝑐𝑑𝑒𝑓

There are 6 partial derivatives of 𝜙. Evaluate 𝜕𝜙𝜕𝑡𝜋,−23.

  • A𝑒
  • B𝜋𝑠−23𝑓
  • C𝑓
  • D𝜋𝑠+𝑓
  • E−23𝑓

The gradient ∇𝐹 of 𝐹 at the point (1,2,3) of ℝ can also be thought of as a linear map into ℝ. What is the corresponding 1×3 matrix?

  • A𝜕𝐹𝜕𝑧(1,2,3)𝜕𝐹𝜕𝑦(1,2,3)𝜕𝐹𝜕𝑥(1,2,3)
  • B𝜕𝐹𝜕𝑦(1,2,3)𝜕𝐹𝜕𝑧(1,2,3)𝜕𝐹𝜕𝑥(1,2,3)
  • C⎡⎢⎢⎢⎢⎢⎢⎣𝜕𝐹𝜕𝑥(1,2,3)𝜕𝐹𝜕𝑦(1,2,3)𝜕𝐹𝜕𝑧(1,2,3)⎤⎥⎥⎥⎥⎥⎥⎦
  • D⎡⎢⎢⎢⎢⎢⎢⎣𝜕𝐹𝜕𝑦(1,2,3)𝜕𝐹𝜕𝑧(1,2,3)𝜕𝐹𝜕𝑥(1,2,3)⎤⎥⎥⎥⎥⎥⎥⎦
  • E𝜕𝐹𝜕𝑥(1,2,3)𝜕𝐹𝜕𝑦(1,2,3)𝜕𝐹𝜕𝑧(1,2,3)

Now 𝑤=𝐹(𝑎𝑠+𝑏𝑡,𝑐𝑠+𝑑𝑡,𝑒𝑠+𝑓𝑡) is a function from ℝ to ℝ with a gradient represented by a 1×2 matrix. Writing 𝑝 for 𝜙𝜋,−23, what is this matrix at the point 𝜋,−23?

  • A𝑎𝜕𝐹𝜕𝑥(𝑝)+𝑏𝜕𝐹𝜕𝑦(𝑝)+𝑐𝜕𝐹𝜕𝑧(𝑝)𝑑𝜕𝐹𝜕𝑥(𝑝)+𝑒𝜕𝐹𝜕𝑦+𝑓𝜕𝐹𝜕𝑧(𝑝)
  • B𝑏𝜕𝐹𝜕𝑥(𝑝)+𝑑𝜕𝐹𝜕𝑦(𝑝)+𝑓𝜕𝐹𝜕𝑧(𝑝)𝑎𝜕𝐹𝜕𝑥(𝑝)+𝑐𝜕𝐹𝜕𝑦+𝑒𝜕𝐹𝜕𝑧(𝑝)
  • C𝑎𝜕𝐹𝜕𝑥(𝑝)+𝑐𝜕𝐹𝜕𝑦(𝑝)+𝑒𝜕𝐹𝜕𝑧(𝑝)𝑏𝜕𝐹𝜕𝑥(𝑝)+𝑑𝜕𝐹𝜕𝑦+𝑓𝜕𝐹𝜕𝑧(𝑝)
  • D⎡⎢⎢⎢⎣𝑎𝜕𝐹𝜕𝑥(𝑝)+𝑐𝜕𝐹𝜕𝑦(𝑝)+𝑒𝜕𝐹𝜕𝑧(𝑝)𝑏𝜕𝐹𝜕𝑥(𝑝)+𝑑𝜕𝐹𝜕𝑦+𝑓𝜕𝐹𝜕𝑧(𝑝)⎤⎥⎥⎥⎦
  • E⎡⎢⎢⎢⎣𝑏𝜕𝐹𝜕𝑥(𝑝)+𝑑𝜕𝐹𝜕𝑦(𝑝)+𝑓𝜕𝐹𝜕𝑧(𝑝)𝑎𝜕𝐹𝜕𝑥(𝑝)+𝑐𝜕𝐹𝜕𝑦+𝑒𝜕𝐹𝜕𝑧(𝑝)⎤⎥⎥⎥⎦

Write a matrix equation relating ∇𝑤𝜋,−23 with ∇𝐹(𝑝).

  • A∇𝑤𝜋,−23=∇𝐹(𝑝)⋅𝑎𝑑𝑏𝑒𝑐𝑓
  • B∇𝑤𝜋,−23=∇𝐹(𝑝)⋅𝑎𝑏𝑐𝑑𝑒𝑓
  • C∇𝑤𝜋,−23=∇𝐹(𝑝)⋅𝑏𝑎𝑑𝑐𝑓𝑒
  • D∇𝑤𝜋,−23=∇𝐹(𝑝)⋅𝑎𝑐𝑒𝑏𝑑𝑓
  • E∇𝑤𝜋,−23=∇𝐹(𝑝)⋅𝑏𝑑𝑓𝑎𝑐𝑒

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