# Lesson Worksheet: Line Integrals of Vector Fields Mathematics

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

**Q1: **

Suppose is the path given by for , is the path given by for , and . Without calculating the integrals, which of the following is true?

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**Q2: **

In the figure, the curve from to consists of two quarter-unit circles, one with center (1, 0) and the other with center (3, 0). Calculate the line integral , where .

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**Q3: **

Let be the arc of a unit circle in the -plane traversed counterclockwise from to . Determine the exact value of the line integral of the vector field over .

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**Q4: **

We explore an example where a vector field satisfies but does not come from a potential. On the plane with the origin removed, consider the vector field .

On the (open) half-plane , we can define the angle function . This is well defined and gives a value between and . What is the gradient ?

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Using the figure shown, use above to define the angle function on the region by a suitable composition with a rotation.

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What is ?

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Since and agree on the quadrant , , we can define the angle at each point of the union with values between and . Use the functions and to find , where is the arc of the unit circle from to . Answer in terms of .

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In the same way, we can define on the half-plane and on . Hence, evaluate the line integral around the circle of radius , starting from and going counterclockwise back to , stating your answer in terms of .

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**Q5: **

Suppose that is the gradient of the function and we are given points , and . Choose a starting and an end point from this set so as to maximize the integral , where is the line between your chosen points.

- AFrom to
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**Q6: **

We will consider two different parameterizations for the line integral , where is the line segment from to and .

Calculate the integral using over .

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Calculate the integral using over .

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