2.* Consider the following heat system with U c R" open and bounded и, — Ди %3 f(x, 1) хEU, 1> 0 ди и + а— — () dv on ðU, и(х, 0) %3D 9(х), x € U where v is the outward unit normal to the boundary. Suppose that a(x) > 0 for any x e ôU (recall that u e C;(U × (0, 0)) U C(U × [0, 0)) to the above system. := Vu · v). Use the energy method to show that there is at most one solution dv

Q: Q. If z = f(x, y), where x = u cos 0 – v sin 0 and y = u sin 0 + v cos 0, with 0 a constant, show…

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Q: = f(x,y, z) = x + sin y and Calculate if w ди x = x(u, v) = In (u + v), y = y(u, v) = e".º and z =…

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Q: 20. Show that a²F(x, y) -dx dy дх ду over the rectangle xo < x < x1, Yo < y < y1, is F(x, у) — F(x0.…

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Q: The adjoint operator of the operator L(u): d'u ;xE[0, 1] %3D dx2 with boundary conditions u(1)=2…

A: We are given   The boundary conditions at x=1 is given as u(1)=2u(0); u'(1)=u'(0)

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A: GIven, ω=F(x,z)dy+G(x,y)dz ⇒dω=∂F∂xdx∧dy+∂G∂xdx∧dz+∂G∂y-∂F∂zdy∧dz Required dω=zdx∧dy+y dx∧dz on…

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A: Note : As per our guidelines we are supposed to answer only first 3 subparts if there are multiple…

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A: As per the question we have to change the variable of integration to get it in the form of modified…

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A: Given that fx,y,z=xy+ez and gx,y,z=y2sinz. We know that if f:ℝ3→ℝ is a function, then…

Q: 13. Let f(x, y, z) = xy+z², with x(s,t) = s², y(s, t) = 2st, and z(s, af/ds and df/dt %3D %3D

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Q: 10 if the equations f(x, y, z)= 0, g(x, y, z)=0 can be solved for y and z as differentiable…

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Q: 2. Compute Ja F · ds for F c(t) = (t , In(1+ t²), e*) for 0 <t < 1 VV, where V(r, y, z) = 4x² In(1+…

A: Find your answer below

Q: 10. f (x, y) = 1 sec(xy) in(rv)

A: Solution:-

Q: 3 13. If x = u + v³ and y = uv – v² are solved for u and v in terms of x and y, evaluate ди ди dv dv…

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Q: 2. Find the temperature distribution inside a solid parallelepiped with sides (a, b, c) whose faces…

A: Here, uxx + uyy +uzz = 0                                  .......1 with conditions, ux,y,z,0 = A…

Q: 59. Let P(x, y, z) = real number. Let a be a real number such that the rate of change of P at the…

A: Let Let  be a space curve such that  and Given the directional derivative is at the point  in the…

Q: 1. Compute ¡ E px 2 ` y 2 ` z 2 q 1{2 dV , where E “ tpx, y, zq | x ď 0, y ď 0, z ě 0, 4 ď x 2 ` y 2…

A: Use the spherical coordinates to evaluate the integral as follows.

Q: dV where V = dt y- x x= 2t, y = 5t, and z = t. Use the Chain Rule to find x+z' av dx (Type an…

A: Please see the explanation below

Q: 23. Let f(x, y) = xy² and let x(s, t) = t² ln(s) and y(s, t) = = s-t Compute of woivos Əs (s,…

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Q: Suppose f(x, y) = x²y + 6. Compute the following quantities: f(0, 0) f(4, -3) • f(-3, 4) . . = =…

A: NOTE: Refresh your page if you can't see any equations. . substitute (x,y)=(0,0)

Q: I'm a bit confused about the graphs. which part of the question do they belong to, and how can I…

A: Given system of linear inequalities are in the following : Here region bounded by these inequalities…

Q: 11. Integrate the function. y? dy (49 – y²) 7y y + C V49 – y2 – sin y+C -1 - V49 - y2 | O A. O B.…

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Q: If z = f(r*y,2x) + g (E with f and g differentiable, then if u= r*y, w= 2r and t == Then: dz A) 2ry…

A: Given : z = fx2y, 2x + gxy with f and g differentiable. Also if u = x2y, w = 2x and t = xy.   To…

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A: To check whether an equilibrium point is unstable we have to find wheather that point corresponds to…

Q: 1. The gradient of the function f(x, y) = 2xy' + In Vx + y² is a. Vf =| 4xy' + 2.x i+ 6x y + 2y lj x…

A: Given function: fx,y=2x2y3+lnx2+y2 The formula to find the gradient is given by ∇fx,y=fxx,yi+fyx,yj…

Q: 5. Let f(x, y) = sin(2.xy). Find f.(x,y) A) f,(x,y)= 2xcos(xy) B) f,(x,y) = 2y cos(xy) C) f.(x,y)= 2…

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Q: 7. Let f(x,y)=, 2 0, y0, and f(0,0) = 0. a) Is f continuous at (0,0) ? . • b) Calculate the partial…

A: Answer given below.As per the guidelines I have solved one question.

Q: d) Find for w = xy + yz + xz , x = u + v , y = u – v , z = uv dv %3D at the point (u, v) = (-2,3).…

A: Given, The  functions  W=xy + yz + xzx=u+v,y=u-v,z=uv and the point is (u, v) = (-2, 3).