# Worksheet: Line Integrals of Vector Fields

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

Q1:

Calculate for the vector field and curve , where , , , and .

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Q2:

Calculate for the vector field and curve , where and is the polygonal path from to to to .

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Q3:

Calculate for the vector field and curve , where , , , and .

Q4:

Calculate for the vector field and the curve , , , .

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Q5:

Calculate for the vector field and curve , where , , , and .

Q6:

Calculate for the vector field and curve , where , , , and .

Q7:

Calculate for the vector field and curve , where ; , , .

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Q8:

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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Q9:

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 . • A
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Q10:

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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Q11:

Calculate for the vector field and the curve , , , .

Q12:

Calculate for the vector field and the curve , , , .

Q13:

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.  • A
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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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Q14:

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
• BFrom to
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• EFrom to