Find the circulation and flux of the field F = -x²i-y²j around and across the closed semicircular path that consists of the semicircular arch r₁(t) = (a cost)i + (a sin t)j, 0≤t≤, followed by the line segment r₂(t)= ti, -astsa. The circulation is (Type an exact answer, using as needed.) The flux is (Type an exact answer, using it as needed.)

Q: 1. Consider the vector field F(r, y) = -y²i+(x+2)j. (a) Evaluate F(0, 1). F(0,1) = (b) Sketch the…

A: Given:

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Q: (b) The square with vertices (0,0), (1, 0), (1, 1) and (0, 1), and the vector field F = 3x²yi + x³j.

A:

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A: ∮C F.T ds  gives circulation of vector field  along the closed curve C  =>∮cF.T ds = ∮cF.dr

Q: 1. Let R and b be positive constants. The vector function r(t) = (R cost, R sint, bt) traces out a…

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Q: Sketch the slope field for the points (ti, y;) where at and dy dt y-2 Z₁ = -1 + 0.25(1-1) = for…

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A: F = (2ycosx , 5xsiny) find curl F =?   curl F  = dF2/dx - dF1/dy here  F1 = 2ycosx  F2 = 5xsiny now…

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A: The solution is given below