Worksheet: Line Integrals of Vector Fields

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In this worksheet, we will practice finding the line integral of a vector field along a curve with an orientation.

Q1:

Calculate ⋅frd for the vector field f(𝑥,𝑦) and curve 𝐶, where fij(𝑥,𝑦)=𝑦−𝑥, 𝐶𝑥=𝑡:cos, 𝑦=𝑡sin, and 0≤𝑡≤2𝜋.

  • A 𝜋 2
  • B − 2 𝜋
  • C − 𝜋 2
  • D0
  • E 2 𝜋

Q2:

Calculate ⋅frd for the vector field f(𝑥,𝑦) and curve 𝐶, where fij(𝑥,𝑦)=𝑥𝑦+𝑥𝑦 and 𝐶 is the polygonal path from (0,0) to (1,0) to (0,1) to (0,0).

  • A 7 1 5
  • B0
  • C − 7 1 5
  • D 1 3 0
  • E − 1 3 0

Q3:

Calculate ⋅frd for the vector field f(𝑥,𝑦) and curve 𝐶, where fij(𝑥,𝑦)=−, 𝐶𝑥=3𝑡:, 𝑦=2𝑡, and 0≤𝑡≤1.

Q4:

Calculate ⋅frd for the vector field fijk(𝑥,𝑦,𝑧)=𝑦−𝑥+𝑧 and the curve 𝐶𝑥=𝑡:cos, 𝑦=𝑡sin, 𝑧=𝑡, 0≤𝑡≤2𝜋.

  • A 2 𝜋 ( 𝜋 + 1 )
  • B 2 𝜋 ( 𝜋 + 1 )
  • C 2 𝜋 ( 𝜋 − 1 )
  • D 2 𝜋
  • E 2 𝜋 ( 𝜋 − 1 )

Q5:

Calculate ⋅frd for the vector field f(𝑥,𝑦) and curve 𝐶, where fij(𝑥,𝑦)=𝑥−𝑦, 𝐶𝑥=𝑡:cos, 𝑦=𝑡sin, and 0≤𝑡≤2𝜋.

Q6:

Calculate ⋅frd for the vector field f(𝑥,𝑦) and curve 𝐶, where fi(𝑥,𝑦)=𝑥+𝑦, 𝐶𝑥=2+𝑡:cos, 𝑦=𝑡sin, and 0≤𝑡≤2𝜋.

Q7:

Calculate ⋅frd for the vector field f(𝑥,𝑦) and curve 𝐶, where fij(𝑥,𝑦)=𝑥−𝑦+𝑥−𝑦; 𝐶𝑥=𝑡:cos, 𝑦=𝑡sin, 0≤𝑡≤2𝜋.

  • A − 2 𝜋
  • B0
  • C 2 3 + 2 𝜋
  • D − 2 3 + 2 𝜋
  • E 2 𝜋

Q8:

Suppose 𝐶 is the path given by r(𝑡)=(𝑡,𝑡) for 0≤𝑡≤1, 𝐶 is the path given by r(𝑡)=(1−𝑡,1−𝑡) for 0≤𝑡≤1, and Fij=𝑥+(𝑦+1)ln. Without calculating the integrals, which of the following is true?

  • A ⋅ > ⋅ F r F r d d
  • B ⋅ = ⋅ F r F r d d
  • C ⋅ < ⋅ F r F r d d

Q9:

In the figure, the curve 𝐶 from 𝑃 to 𝑄 consists of two quarter-unit circles, one with center (1, 0) and the other with center (3, 0). Calculate the line integral ⋅Frd, where F=𝑥2𝑦2sinsinij−𝑥2𝑦2coscos.

  • A 2 1 2 3 2 + 2 1 2 1 2 s i n c o s c o s s i n
  • B − 2 1 2 1 2 − 2 1 2 1 2 s i n c o s c o s s i n
  • C − 2 1 2 3 2 − 2 3 2 1 2 s i n c o s c o s s i n
  • D − 2 1 2 1 2 + 2 1 2 1 2 s i n c o s c o s s i n
  • E 2 1 2 3 2 + 2 1 2 1 2 s i n c o s c o s s i n

Q10:

Let 𝑃 be the arc of a unit circle in the 𝑥𝑦-plane traversed counterclockwise from (0,1) to (1,0). Determine the exact value of the line integral of the vector field Fijk(𝑥,𝑦,𝑧)=3𝑥𝑒+2𝑦𝑧𝑒+𝑦𝑒 over 𝑃.

  • A 1 + 𝑒
  • B 1 + 2 𝑒
  • C 1 − 𝑒
  • D 𝑒 − 1
  • E 1 − 2 𝑒

Q11:

Calculate ⋅frd for the vector field fijk(𝑥,𝑦,𝑧)=−+ and the curve 𝐶∶𝑥=3𝑡, 𝑦=2𝑡, 𝑧=𝑡, 0≤𝑡≤1.

Q12:

Calculate ⋅frd for the vector field fijk(𝑥,𝑦,𝑧)=𝑥+𝑦+𝑧 and the curve 𝐶𝑥=𝑡:cos, 𝑦=𝑡sin, 𝑧=2, 0≤𝑡≤2𝜋.

Q13:

We explore an example where a vector field F=⟨𝐹,𝐹⟩ satisfies 𝜕𝐹𝜕𝑦−𝜕𝐹𝜕𝑥=0 but does not come from a potential. On the plane with the origin removed, consider the vector field F(𝑥,𝑦)=−𝑦𝑥+𝑦,𝑥𝑥+𝑦.

On the (open) half-plane 𝑥>0, we can define the angle function 𝜃(𝑥,𝑦)=𝑦𝑥arctan. This is well defined and gives a value between −𝜋2 and 𝜋2. What is the gradient ∇𝜃?

  • A 𝑦 𝑥 + 𝑦 , − 𝑥 𝑥 + 𝑦
  • B − 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦
  • C 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦
  • D − 𝑦 𝑥 + 𝑦 , − 𝑥 𝑥 + 𝑦
  • E − 𝑥 𝑥 + 𝑦 , 𝑦 𝑥 + 𝑦

Using the figure shown, use 𝜃 above to define the angle function 𝜃(𝑥,𝑦) on the region 𝑦>0 by a suitable composition with a 𝜋2 rotation.

  • A 𝜃 ( 𝑥 , 𝑦 ) = 𝜃 ( − 𝑦 , 𝑥 ) + 𝜋 2
  • B 𝜃 ( 𝑥 , 𝑦 ) = 𝜃 ( − 𝑦 , 𝑥 )
  • C 𝜃 ( 𝑥 , 𝑦 ) = 𝜃 ( − 𝑦 , − 𝑥 ) + 𝜋 2
  • D 𝜃 ( 𝑥 , 𝑦 ) = 𝜃 ( 𝑦 , − 𝑥 ) + 𝜋 2
  • E 𝜃 ( 𝑥 , 𝑦 ) = 𝜃 ( 𝑦 , − 𝑥 )

What is ∇𝜃(𝑥,𝑦)?

  • A 𝑦 𝑥 + 𝑦 , − 𝑥 𝑥 + 𝑦
  • B 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦
  • C − 𝑦 𝑥 + 𝑦 , 𝑥 𝑥 + 𝑦
  • D − 𝑥 𝑥 + 𝑦 , 𝑦 𝑥 + 𝑦
  • E − 𝑦 𝑥 + 𝑦 , − 𝑥 𝑥 + 𝑦

Since 𝜃 and 𝜃 agree on the quadrant 𝑥>0, 𝑦>0, we can define the angle 𝑇(𝑥,𝑦) at each point of the union with values between −𝜋2 and 3𝜋2. Use the functions 𝜃 and 𝜃 to find ⋅Frd, where 𝐶 is the arc of the unit circle from 12,−√32 to −1√2,1√2. Answer in terms of 𝜋.

  • A 1 3 𝜋 1 2
  • B 3 𝜋 5
  • C 𝜋 1 2
  • D 𝜋 5
  • E 1 2 𝜋 1 3

In the same way, we can define 𝜃 on the half-plane 𝑥<0 and 𝜃 on 𝑦<0. Hence, evaluate the line integral ⋅Frd around the circle of radius √2, starting from 𝑃(1,−1) and going counterclockwise back to 𝑃, stating your answer in terms of 𝜋.

  • A 𝜋 2
  • B 𝜋
  • C 𝜋 √ 2
  • D √ 2 𝜋
  • E 2 𝜋

Q14:

Suppose that F is the gradient of the function 𝑓(𝑥,𝑦)=2𝑥−𝑦, and we are given points 𝑃(0,0),𝑄(1,0),𝑅(0,1),𝑆(1,1), and 𝑇(−1,−1). Choose a starting and an end point from this set so as to maximize the integral ⋅Frd, where 𝐶 is the line between your chosen points.

  • Afrom 𝑆 to 𝑄
  • Bfrom 𝑅 to 𝑇
  • Cfrom 𝑇 to 𝑄
  • Dfrom 𝑃 to 𝑅
  • Efrom 𝑄 to 𝑇

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