Lesson Worksheet: Line Integrals of Vector Fields Mathematics

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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:

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⋅>⋅FrFrdd
  • B⋅=⋅FrFrdd
  • C⋅<⋅FrFrdd

Q2:

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 Fij=𝑥2𝑦2−𝑥2𝑦2sinsincoscos.

  • A21232+21212sincoscossin
  • B−21212−21212sincoscossin
  • C−21232−23212sincoscossin
  • D−21212+21212sincoscossin
  • E21232+21212sincoscossin

Q3:

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.

  • A1+𝑒
  • B1+2𝑒
  • C1−𝑒
  • D𝑒−1
  • E1−2𝑒

Q4:

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?

  • A𝑦𝑥+𝑦,−𝑥𝑥+𝑦
  • B−𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • C𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • D−𝑦𝑥+𝑦,−𝑥𝑥+𝑦
  • E−𝑥𝑥+𝑦,𝑦𝑥+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.

  • A𝜃(𝑥,𝑦)=𝜃(−𝑦,𝑥)+𝜋2
  • B𝜃(𝑥,𝑦)=𝜃(−𝑦,𝑥)
  • C𝜃(𝑥,𝑦)=𝜃(−𝑦,−𝑥)+𝜋2
  • D𝜃(𝑥,𝑦)=𝜃(𝑦,−𝑥)+𝜋2
  • E𝜃(𝑥,𝑦)=𝜃(𝑦,−𝑥)

What is ∇𝜃(𝑥,𝑦)?

  • A𝑦𝑥+𝑦,−𝑥𝑥+𝑦
  • B𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • C−𝑦𝑥+𝑦,𝑥𝑥+𝑦
  • D−𝑥𝑥+𝑦,𝑦𝑥+𝑦
  • E−𝑦𝑥+𝑦,−𝑥𝑥+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.

  • A13𝜋12
  • B3𝜋5
  • C𝜋12
  • D𝜋5
  • E12𝜋13

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.

  • A𝜋2
  • B𝜋
  • C𝜋√2
  • D√2𝜋
  • E2𝜋

Q5:

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

Q6:

We will consider two different parameterizations for the line integral ⋅Frd, where 𝐿 is the line segment 𝑦=𝑥 from (0,0) to (1,1) and Fij=6𝑥+(𝑥+𝑦).

Calculate the integral using r(𝑡)=(𝑡,𝑡) over 0≤𝑡≤1.

  • A6
  • B53
  • C3112
  • D1
  • E236

Calculate the integral using r(𝑡)=𝑒,𝑒 over 0≤𝑡≤2ln.

  • A236
  • B12(1−2)ln
  • C13−1222ln
  • D1
  • E182−343ln

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