3. Solve the Laplace's equation = 0 dx2 dy? d22 in a three dimensional box 0 < x < L, 0 < y < H, 0 < z < G subject to the boundary conditions u(0, y, z) = 0, u(x,0, 2) = 0, u(x, y, 0) = 0, ди (L, y, z) = 0, u(x, H, z) = 0, u(x, y, G) = 0.

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Q: Q. Let a = x1 dx2 + x3 dx4, B = x1 ×2 dx3 dx4 + X3 X4 dx1 dx2 and y = x2 dx1 dx3 dx4 be forms on R4.…

A: given  α=x1dx2+x3dx4 , β=x1x2dx3dx4+x3x4dx1dx2γ=x2dx1dx3dx4 calculate  (i) αβ , αγ…

Q: Sketch the graph of the parametric equations x = sint, y = 2 cos(3t), π ≤ t ≤ 27. Clearly label the…

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Q: 2. f(x, y) = (x² + y²)² — 2(x² - y²) + 1. Ans.: global minimum 0 at (±1,0), saddle at (0, 0).

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Q: Flux of F(x,y) = 5 x î +3 y ĵ across the circle x? + y? = 1 (anticlockwise) is - 8 T Can not find.

A: Use the Gauss’ Divergence theorem,: Use the Gauss’ Divergence theorem,

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Q: 19. Compute the tangent plane at (1.0, 1) for each of the functions 9) f(x, y, z) = x exp(-x² -…

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Q: = (1-2)x' D X where D = 2(c-yd), X = 2λ(yx + y), (1-2)=C+Y where C = 2(d- yc), Y = 2λ(x + YY), where…

A: The analysis considers two scenarios: γ^2&gt;1 and γ^2&lt;1. For each scenario, the behavior of the…

Q: 11. Let X, Y, Z have the joint PDF f(x, y, z)= 6(1+x+y+z) 10 Find the PDF of U=X+Y+Z. if 0 < x,0 <y,…

A: To find the PDF of U = X + Y + Z, we will use the method of transformation of variables. First, we…

Q: (a) Compute the directional derivative of f(x, y) = xy² + x³y at the point (4, -2) in the direction…

A: Since you have posted independent questions, so these are considered as multiple questions.According…

Q: 3. u, = u, - sin x cos t, 1 >0 В.С.: и(0, г) %3D0, = () 1.С.: «(х, 0) %3D 0, u,(x. 0) = 0

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Q: Find a parametrization of the tangent line to r(t) = (In(t))i + t-°j+ 3tk at the point t = 1. (Use…

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Q: If y = f(x) defined as parametrized r(t) = 4t- 2t2, y(t) = 3t - t, then which of the following is…

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Q: 1. Let D = the sales of DVDs in billions of dollars, S = the total sales of streaming videos in…

A: The sales of dvds given as the function D = -0.92t + 8.75 and the total sales of streaming videos…

Q: Q7. Q7. Consider the vector-valued function 3ry ra(r, y. :) = (V=(* + s°), ) z(2² + y°), r2(r, y, z)…

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Q: c) S. cinx *ye= dzdxdy = |z = Inx Ye= dxdy = ye'nx – ye° = y(einx – e°) = y(x – 1) z = 0 y(x - 1)…

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Q: is D.E xy? dx + (x+y) dy= 0 a- linear b- Exact C- non_linear ( Explain that)

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Q: 1. Given f(x,y) = 2x² + y² whose graph is a paraboloid. Fill in the table with the values of the…

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Q: dy (a) Obtain the integral curve for dx y-x passing through the point (1, 0). y+x (b) Find the…

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Q: 1.1: If y2 = x² + 2xy, then dy y+x dx a) y-x b) 2x + 2y x+1 c) y d) - x corre None of the above ||

A: Given y2=x2+2xy

Q: Show that the displacement u(x, t) in an elastic string of length t fireed at both ends which is set…

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Q: Show that the general solution to the two-dimensional Laplace's Equation. et = 0 is (Acosh(tx)…

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Q: Evaluate y = 4x 2 = [{(x² + 2y)dx + xy. dy} from 0(0, 0)to B(2, 4) along the curve

A: Given the integral is :                                  We have to evaluate the integral from…

Q: 4) Solve via separation of variables: u(0, y) =0 ди -(2, y) = 3sin a'u a'u +5 sin =0 ; 6. S0<x<2…

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Q: 2t k, then the distance traveled over 0Sts In 3 is:

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Q: =grating

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Q: 18.) Compute the directional derivative of f in the given ✓ given points P. 3 a) f(x, y, z) = xy² +…

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Q: 1. y = 5u°; u=7x – 4x' + 2x –9 2х-9 u +1. u=Vx? +1 y = u? -1' 3. x=7y -3y +y +8 2.

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Q: Solve via separation of variables: -(x,0) =D0 = 0 ôy l0<y<z B.C.s{ -(x,7)=0 V(0, y) = 0 [V(7, y) =…

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