Worksheet: Directional Derivatives and Gradient

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

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