Given that f(x, y) = 8x(y + 2)² +8y(x − 2)² is a potential function for F = (8(y + 2)² + y (16x — 32), z (16y +32) + 8(x − 2)²), calculate the integral [F. dr, where C is the lower half 1 of the unit circle oriented counterclockwise with parameterization r(t) = (cos(t), sin(t)), π ≤ t ≤ 2π. Enter an exact answer. Provide your answer below: ScF. dr =

Q: | cos zdr + sen ydy + x?dz

A: The solution are next step

Q: Evaluate the line integral f(2x+4y)ds for C: r(t)= ⟨3t2, 4t2+3⟩, 0 =< t =< 2

A: Now, we can compute this integral. We'll need to multiply  (22t2+12)(10t),  integrate with respect…

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Q: Let f (x, y) = 2x3 + x2y - y2. Compute d2f/dx2, d2f/dy2, and d2f/dxdy at (x, y) = (1, 2).

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Q: Compute V · Fand V x F and verify explicitly V - (V x F) = 0. F(x, y, z) = sinh(x – z)i + 2yj+ (z –…

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Q: suppose cX) is valued function on such that f?E R Faib] Does it fallows bounded real taib] that…

A: In the given question given that f(x) is a bounded real valued function on [a,b] then we have to…

Q: Let F(x, y) = (3x, 2y) and let y(t) be the integral curve of F passing through the point (x, y) =…

A: Given: F(x,y)=3x,2y andγ(t) be the integral curve of F passing through the pointx,y=8,5 at t=0. To…

Q: B- Let r(t) = (²+2) i + (3t) j .Find (t) such that r(t) parallel to r'(t).

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Q: Find r(t) if r' (t) = 5t+i + 7tºj + √√/ tk and r(1) = i + j.

A: As per our guideline, we are supposed to solve only first question. Kindly repost other question as…

Q: Verify that the function u, defined by u(t, x) = t sin(x – t²), is a solution to the equation ,2 U +…

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Q: 2. (a) Show that F = (e² cos(y), –E sin(y), 2) is conservative using partial derivatives. (b) Find…

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Q: -yi + xj 2. Find the flux of F across the curve C given by r(t) = a cos(t)i+a sin(t)j, VI² + y² for…

A: F=-y i+x jx2+y2,    r(t)=a cos(t) i +a sin(t) j 0≤t≤2π x(t)=-a sin(t) ,     y(t)=a cost    ,  F=-y…

Q: Let F = (2x³y² + 2x, 2x¹y³ + y). Given that f(x, y) = x¹y¹ + ² + y² is a potential function of F.…

A: The given function is F→=2x3y4+2x,2x4y3+y, and its potential function is f(x,y)=12x4y4+x2+12y2. To…

Q: Compute the discriminant D(x, y) of ƒ(x, y) = x2y2.

A: fx, y=x2y2 Partially differentiating with respect to x, we get ∂f∂x=2xy2 Partially differentiating…

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Q: Calculate r'(t) and T(t), where r(t) = (2t – 5, 4 – 4t, 1 + 4t). %3D r'(t) = { %3D T(t) = {

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Q: yi + aj 2. Find the flux of F = across the curve C given by r(t) = a cos(t)i+a sin(t)j, ²+y² for 0…

A: ∵ Given thatF=-yi^+xj^x2+y2;           0≤t≤2π r(t)=acosti^+asintj^ As we know that Flux=∬ F·nds…