39. Show that the graph of r =t sin ti + t cos tj + t'k lies on the paraboloid z = x² + y².

Q: Evaluate L₂Lova (4-3²) dydx -√4-x² by using polar coordinates.

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Q: Use Green's Theorem to compute y dx+(x³+3xy²) dy where C is the closed curve around the triangle…

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Q: Use the IVT to prove that the curves y = x2 and y = cos x intersect.

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Q: Let F = cos(4t) i + 3tj+ 4t°k and G = t cos(4t) i + t j+4e#k Find the cross product F x G =

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Q: Find F. dr for given F and C. (a) (2,4). F = −y sin(x)i + cos(x)] and C is the parabola y = x²…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

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Q: 3 Evaluate the integral cos(x² + y²)dydx by converting to polar coordinates 0 0

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Q: Q1) Evaluate the integral dy dx using polar coordinate

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Q: Compute f cos(Tæ) dæ + 10xy dy, where y is a part of parabola y = 3x? from the point (0,0) to the…

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Q: 19. Find y' in the parametric equations x =, t-1 t+1 A) (f – 1) 7-1 B) (1+ 1)? 2 +1 C) (1+1)? t +1…

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Q: Integrate the function f(r,y, z) = Over the entire 3D coordinate system, i.e., all of R'.

A: We have to integrate the function fx, y, z=1x2+y2+z21+x2+y2+z2 over the entire 3D coordinate system,…

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Q: d the polar coordinates if (x, y) = (-√3, 1), r≥ 0, -2π < E

A: Given query is to find the polar coordinates.

Q: Sketch the bifurcation diagram for y = (A– 4t)y (where A is a parameter, as usual). 2

A: To find- Sketch the bifurcation diagram for y' = λ - 4t2y (where λ is a parameter, as usual)

Q: Find the points of intersection between r =2 cos20 andr=- %3D 1- cose

A: To find the intersection point two curve we have to find r,θ by solving given curve.

Q: 3) Use polar coordinate to evaluate S'S_√x² + y² dxdy.

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Q: 15) Let u = x³y - y³x a. Verify that the function u is harmonic in the whole complex plane b. Find…

A: The given problem is to verify whether the given function u is harmonic or not and also to find the…