E = {u: [0, 7] - RNu is absolutely continuous. uz (0) = u(7). ù e L²([0. 7]. RN)) with the inner product - f′ (cù(1), v(1)) + (u(1), v(1))]dr AU. VYEM for all u, v € E where (-.-) denotes the inner product in RN. The corresponding norm is defined by

Q: Let F(x, y) = x2 + x² + y2J- (a) Show that F. dr +| F. dr if C and C2 are the semicircular paths…

A: The function given to us is  F(x,y)=yx2+y2i-xx2+y2j

Q: vatty f Z= f(x,y)=4 e In(y), and x = ln (u cosv), yusin v, use chain rule to find: Evaluate : ]]…

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Q: Given f(z)=p(z)/q(z), a meromophic function such that p(z0) is different from 0 , q(z0) = 0 and…

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Q: 4. Perform the indicated multiplication of the two differential operators: (D –x)(x²D– 2x).

A:

Q: 3) y' dx-(xy+y*) dy-o Let the substitution v= i y vx; dy-vdx+xdv %3D )x*dy+(2xyy*) ax -o Let the…

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Q: For the line that has the equation X1 + 2X2 = 10, will pass through: %3D O (5,4) (10,1) O (2,4) O…

A: Given that  The equation of a line is x1+2x2=10 To find a point that will pass through the given…

Q: дz дz Use the chain rule to find and მs at z = ln(3x + 4y), x = s sin(t), y = t cos(s) дz as дz at…

A: z=ln3x+4y , x=s sint , y=t coss

Q: Which functions form the basis of solutions for the ODE y – y" – y" +y' (x) = 0? O {1, e ?, e², æe }…

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Q: Əz Q3.If Z= f(x,y) = 4 e In(y), and x = ln (u cosv), y = u sinv, use chain rule to find : E * ди Q4.…

A: Note: We are entitled to solve only one question at a time. As the specified one is not mentioned,…

Q: с) Let A :1, →1, be defined by Ах, х, ...) %3D (0, 0, х2, Хд, .). Prove that A is self-adjoint,…

A: Please find the answer in next step

Q: 4. Show that both functions x and sin(x) belong to inner product in this space. L² ([0, 1]) and find…

A: L20,1 is a vector space of all functions f:0,1→ℝ satisfying ∫01f(x)2dx<∞. And the inner product…

Q: Use the functions f and g in C[-1, 1] to find (f, g), ||F||, ||9||, and d(f, g) for the inner…

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Q: Prove that if m is a positive interger, l'(-m+) = 1.3.5..(2m -1)

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Q: Use the functions f and g in C[-1, 1] to find (f, g), ||f||, |lg||, and d(f, g) for the inner…

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Q: Compute the Jacobian of the map G(r, s) = (er cosh(s), er sinh(s))

A: The given function is G(r, s) = (er cosh(s), er sinh(s)). Compute the Jacobian as follows.

Q: Consider the following functions: f1(x) = sin (x1) cos(x2) , x ∈ R 2 f2(x, y) = x ⊤ y, x, y ∈ R n…

A: Since you have posted a multiple question according to guidelines I will solve first(Q1) question…

Q: Find a formula for u(x, t) that solves uxx=ut u(x, 0) = 1 (b) Vi (u(x, t) | ≤ ("(x,t)\ -12 for all…

A: Step 1:Separation of Variables: We begin by assuming a separable solution of the form…

Q: əz ди Q3.If Z= f(x,y)=4e* ln(y), and x = ln (u cosv), yusin v, use chain rule to find : (x²) dr du…

A: 3. Given Z=fx, y=4 ex lny and x=lnu cosv, y= u sinv   To Find: ∂Z∂u and ∂Z∂v

Q: Find real differentiable functions s(y) and t(y), such that the complex function f(x + iy) = e"…

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Q: Show that (6.g) = (0,0) is unijomy continuous or not.

A: Uniform continuity

Q: Let f(z) := z5 . Show that f is invertible in a neighbourhood of i and calculate (f-1)'(f(i)).

A:

Q: as f(z) = e e analytic %3D

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Q: _x?yesin (xy2) x2+y2 -Let a function g be defined on R2 at g (x, y) Use: g(x,0), g (x, kx), g (x,…

A: We g(X,y ) first

Q: Consider the function z = f(x, y) implicitly defined by Þ(x, y, z) = x³ + y³ - 2² + cos(xz) = 0. a)…

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Q: Show that at (0,0). f(x, y) = x² sin 0, (²7)(y² +1), (x, y) ‡ (0,0), (x, y) = (0,0). of (0,0)…

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Q: 5. If X any Y are sets, then X-Y= {r€ X: 1¢Y} If X is open set and Y is closed set. Prove that (a)…

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Q: Use the functions f and g in C[-1, 1] to find (f, g), ||f||, |lg||, and d(f, g) for the inner…

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Q: bj f(x,y)= еху cЈ fexiy>= x cosx cобу p² d.j fexiys= (x'y 2) log (x'ty ?).

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Q: Consider F(x, y, z) = x² + y² + z² such that x = r sin q cos 0 ; y = r sin q sin 0; and z = r cos p.…

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Q: Q1-V-{I:[0,1]R}where fare functions. f and f' exists and vanishes at 0 and 1 meaning f(0)=f(1)=0 and…

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Q: 1) Find ôz l ôu and ôzlôv for z= 4e* In(y), x= In(u cos(v)), y=u sin(v) at the point (u,v)=(2,7/4).…

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Q: Use the Chain Rule to find ∂z/∂s and ∂z/∂t. (Enter your answer only in terms of s and t. Please use…

A: From the given data: t = x4 y9 x = S cos (t) y = S sin (t)  dt = d (S cos (t)4 (S sin (t))9…

Q: Compute the Jacobian at the indicated point. (5. 증) G(r, t) = (r sin (t), r – cos (t)), (r, t) = -…

A: Jacobian of the function

Q: Əz 1 Q3.If Z= f(x,y)=4e* ln(y), and x = ln (u cosv), yusin v, use chain rule to find : 201 ди and az

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Q: Let = r, (cos 0, + isin 0, ) and r, (cos 0, + i sin 0, ). Z, = X1 + iy, 1 Z 2 X2 + iy, Then show…

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Q: Compute the dz in the function z = ety sin (xy) A. dz = ze"(cos (zy) + sin (ry) dz+ ye"lcos (ry) +…

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