Question

Let F=∇f, where f(x,y)=sin(x−6y). Find curves C 1

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and C 2
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that are not closed and satisfy the equation. (a) ∫ C 1
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F⋅dr=0
C 1
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:r(t)=,0≤t≤1
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(b) ∫ C 2
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​
F⋅dr=1
C 2
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:r(t)=,0≤t≤1
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Answer 1

To find curve C1, we use Green's Theorem and solve the integral ∬R (∂F2/∂x - ∂F1/∂y) dA. For C2, we do the same but solve for ∫C2 F⋅dr = 1.

Step-by-step explanation:

For part (a), we have F = ∇f where f(x,y) = sin(x - 6y). To find curve C1, we want ∫C1 F⋅dr = 0. Since F = ∇f, we can use Green's Theorem to rewrite the integral as ∬R (∂F2/∂x - ∂F1/∂y) dA, where R is the region bounded by C1. Solving this integral will give us the equation for C1.

For part (b), we want ∫C2 F⋅dr = 1. Using similar steps as in part (a), we can rewrite the integral as ∬R (∂F2/∂x - ∂F1/∂y) dA and solve it to find the equation for C2.