Solve the following boundary-value PDE by the method of separation of variables. Note that u is finite at any p and t ди at 1 ди ap K ap2 u(1, t) = 0, u (р, 0) 3D F(р), |u(p, t)} < M

Q: Check it for f(x, y, z) = x²y°z*, that is, compute Vf first, then V x (Vf).

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Q: Prove that if w=f(x, y, z), x=r-s, y=s-t, and z=t-r then dw dw dw + + = 0. dr ds ôt

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Q: Find the exact value of F. dr using any method. b. F(x, y, z)= C is the line segment from (1, 0,…

A: The given vector field is Fx, y, z=yz, xz, xy+2z. We have to find ∫CF·dr.

Q: • Given f(x, y, z) = x² In(x + y- z), find f(2,4, 5). а. 0 b. 1 с. 2 d. 3 O a. b. С. O d.

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A: 1. A. If f(u, v, w) is a function and u = x - y,  v = y - z and w = z - x.   To Prove:…

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Q: Which of the following sets of connectives are expressively adequate? →, ¬, A →, +, V 7,V ^, V

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Q: 6. Let T : R' –→ Rª be the function defined by [f (#1)] f2(r2) f3(r3) 12 T( I3 where f1, f2, f3, f4…

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Q: Let F(x, y, z) = x²yi + 30y3 + yzk Choose all of the following points at which divF equals zero.…

A: Answer and explanation is given below

Q: 4. Let F = (−y, 0, x² + y²), and S = {(x, y, s) : 4x² + y² ≤ 7⁄, z = sin (4x² + y²)}. Compute [ [ (V…

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Q: The joint p.d.f. of the r.v.'s X and Y is given by fx,y (x, y) = (x + y²), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0,…

A: The joint probability density function is, f(x,y)=65(x+y2); 0≤x≤1,0≤y≤1       0       ; otherwise…

Q: x - y -2x Prove by definition that for f(r, y) fy(x, y) = %3D x+ y' (x+ y)2

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Q: 2. Let X, Y c R. Prove that X U (Y – X) = Y if it is true. If not, give counterexamples to explain.

A: Given statement is false.

Q: Let flt.v) = y+3r²e!. Then f satisfies a Lipschitz condition in y with L = 2, given that 1<t<2.…

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Q: 2. For what value of k will x – iky F(x, y) = (x+ y)² + be analytic?

A: Given function is fx,y=x+y2+x-ikyx+iy If it is analytic then it satisfy C-R equation ux=vyuy=-vx

Q: 9. 2 + 1, t2 – 3, t3 + 1 Let C be defined by r(t) = for 0 <t< 2 Evaluate (2x² – 6y)dx + (y – 6x)dy +…

A: The given position vector r(t)→=2t+1, t2-3, 9t3+1 for 0≤t≤2 is the parametric form of the curve C…

Q: Find Vf at the given point. f(x,y,z) = x3 + y2 – 3z2 +z In x, (1,2,4) Vil(1,2,4) =Di+j + ()k ( %3D

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Q: Compute the length of the curve r(t) = (-4t, 3t+1, -7t-6) over the interval 0 ≤ t ≤ 3. (Use symbolic…

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Q: Let f: R² → R be defined by x3 + y3 x2 + y? if (x, y) + (0, 0) f(x, y) : if (x, y) = (0, 0).

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Q: Solve the following P.D.E. , with the following conditions: u(0,y) = 0, u(L,y) = 0, u(x,0) = 0 and…

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Q: 1. Find and sketch the doma in of the function. Ans A f(x,y) =Vy +V64 – x2 – y2 a. ((x,v)| 64 > x? +…

A: Given function is  f(x,y)=y+64-x2-y2 we have to find and sketch the domain.

Q: Explain how the evaluation for each of S. f(x, y) ds, ſ. f(x, y) dx, ſ. f(x,y) dy, and F ( x, y) · d…

A: We have to evaluate each integral. They defines as follows-

Q: with D= 0. Obtain its T.F.

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Q: 5. Find the x-coordinates for all points of discontinuity for f(x)- 「X316, xśl x〉 1 2 c. 1, 32d.…

A: First we check the limits as x approaches to 1 from left side and right side .

Q: = [[ F - a5, I = S F = î4æz – jy² + kyz 1

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Q: 1)Use Lagrange multipliers to find the extreme values of the function f: R? → R, f(1,y) = 1² – y²…

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Q: Show V(X)

A: The joint probability mass function of X and Y is  f(x,y)=(x+y)/21   , x=1,2,3 and y=1,2

Q: Consider F and C below. F(x, y, z) = yzexi + e*²j+xyek, C: r(t) = (t² + 1)i + (t² − 2)j + (t² − 3t)…

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Q: 8.Prove the following relationships: a) I(X,Y)=H(X)-HOX/Y)

A: Solution

Q: Let k e Zt. Let A be a set such that |A| = k + 1, and let a e A (i.e., a is a fixed but unspecified…

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