Let P be the portion of the tilted plane in R' parameterised by r(u, v) = ui + uj + vk where 0 < x < 1 and 0 < z < 1. To calculate the area of P we can use Area(P) = |J(u, v)| dv du. u=0 Jv=0 Here the Jacobian |J(u, v)| is ...

Q: Let F = 4ai + 1yj and let C be the closed path around the circle x² + y² = 9, traversed once…

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Q: Let S be the surface defined by the vector function R(u, v) = (u cos v, u – v, u sin v) with u ER…

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Q: Find the Jacobian ∂(x, y) ∂(u, v)for the indicated change of variables. x = 3uv,  y = 2(u − v)

A: on solving this we get the

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Q: Find a normal vector to the level curve f (x, y) = c at P.                                    f(x,…

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Q: II. Let RVsx, Y be uniform on triangle with vertises (0, 0), (2, 2), (2, 2). Find 1. Joint pdf…

A: Note: As you've posted multiple questions, the first one will be answered as per our policy.Also,…

Q: Let F(r, y,z) = where r2 + y? < 4, oriented by the upward normal. (r,-z,2y(r2-y²)). Find ff, F- dS…

A: Given Fx,y,z=x,-z,2yx2-y2and S is the saddle surface z=x2-y2, where x2+y2≤4, oriented by the upward…

Q: 33. Define T: R³ → R³ by y + 2z 2x +3y - z x +2y - 2z x - y a. Find all vectors in R' that are…

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Q: A normal vector to the tangent plane for the surface z = , at the point (1, 1, 1) is: y - Select…

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Q: Let S be the upper half of the unit sphere x +y +=1 and take n as the upper unit normal. Use Stokes'…

A: Stokes theorem is used to find the line integration

Q: af Let f(x, y, z) = xy + z². Calculate where x = = s², y = 2st, z = t². მs

A: Step 1: Step 2:

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Q: Evaluate the line integral by stokes theorem where F(vector)=[ 0, xyz ,0] , C is the boundary of a…

A: Calculate curl:  curlF=ijk∂∂x∂∂y∂∂z0xyz0         =i-xy+kyz

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A: Given: The portfolio, zs=RTRRT-1esTo find: The payoff of the portfolio zs.

Q: Let F(x, y,z) = (-xe² +3x, -ye² + arctan(x? + z), 2e²) and S the portion of the sphere acoording to…

A: Given that Fx,y,z=-xez+3x, -yez+arctanx2+z2, 2ez. Also, S=x,y,z| x2+y2+z2=1, z≥0. To determine the…

Q: If F(x, y, z) = z - f(x, y) and s is the graph of z = f(x, y) and (a, b,c) is a point on the surface…

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Q: Let f(x, y) = ln (1 + x² + y²), then a) f is not differentiable at (0,0). b) f(xo, yo) = f(xo, yo)…

A: Multivariate calculus

Q: (4) Let S be the part of the plane z = f(x, y) = by (x - 1)² + (y - 3)² ≤ 9 oriented with an upward…

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Q: Use Stokes's theorem to evaluate F.dr, where F(r, y, z) = ry i + x°yj+yz k, where C is a triangular…

A: Stroke's theorem:  ∫Fdr=∬ (Curl F) n^dS Curl F=ijkxy2x2yyzy2x2y=iz n^=∇s∥s∥=i^+k^2

Q: Use Green's theorem for the work done by the force F= (x - y)i + (x + y)j, for the shaded region R…

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