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,𝑥𝑦,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.

  • A𝑤𝜋,23=𝐹(𝑞)2𝜋432𝜋4323𝜋
  • B𝑤𝜋,23=𝐹(𝑞)2𝜋432𝜋4323𝜋
  • C𝑤𝜋,23=𝐹(𝑞)2𝜋432𝜋4323𝜋
  • D𝑤𝜋,23=𝐹(𝑞)𝜋23𝜋2323𝜋
  • E𝑤𝜋,23=𝐹(𝑞)𝜋23𝜋2323𝜋

Q3:

Suppose 𝑤=𝐹(𝜙(𝑥,𝑦)) with 𝜙=(𝜙,𝜙,𝜙) and that 𝑞=𝜙(𝑝) for a point 𝑝. 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.

  • A𝑤(𝑝)=𝐹(𝑞)𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥
  • B𝑤(𝑝)=𝐹(𝑞)𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑦
  • C𝑤(𝑝)=𝐹(𝑞)𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦
  • D𝑤(𝑝)=𝐹(𝑞)𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥𝜕𝜙𝜕𝑦𝜕𝜙𝜕𝑥
  • E𝑤(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,𝑥𝑦,𝑥𝑦 and 𝐹(𝑝,𝑞,𝑟)=10𝑝+6𝑞16𝑟. 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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