1. Let d, be a metries on Xo and let the function a: X x XR defined be dr. y) dis. y)= prove that (X,d) is a metrie space.

Q: y dV W

A: Let us consider the triple integral ∫ab∫h(x)g(x)∫p(x,y)q(x,y)f(x,y,z)dzdydx To solve the integral…

Q: 1. Let S C R be the surface defined for all 0 < x < 1, -1 < y < 1 by z = r² + y. Compute ffsa ds

A: It can be solved by partial differentiation and integration

Q: Suppose f(z) is analytic in the unit disk D={|z| &lt; 1} and |f(z)|&lt;= 1 for |z| &lt;1, Prove that…

A: Assume that f(z) is analytic on the unit disk D ={z: |z| &lt; 1}.Also assume that |f(z)|≤1 for…

Q: 3. Prove that if r, y E RP, then ||r + yl|2 = ||x||2 + |ly||² holds if and only if r y = 0. 4. Prove…

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Q: Assume s is a smooth surpace parametrized by e (u,v): <Jv, win' u, SZ-v ). Evaluatk 2x sin y do.

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Q: 18. Suppose that V ·F, = V • F, and V × F, = V × F, over a re- gion D enclosed by the oriented…

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Q: Let f be holomorphic in a domains containing the closed unit disk D(0, 1). Let S = S (0, 1). Show…

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Q: Q4) Let C(R) be a normed space such that || f -sup{ Ifl,x E R}and T: C (R) -R define Tf)-If I. Is T…

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Q: „Consider the space (R,T).´ .where t={R,Ø}U{ACR:[0,1)CA} If A=[1,2], then AS = (1,2] O (1,2) O [1,2]…

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Q: Let (W,p) be a normed space, f: W→F be a non- zero Cinear functional. Then prove that for each x E W…

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Q: 4. Let f(x, y) be a function defined on a disk D not containing the origin which is given by f(r, y)…

A: See the answer below

Q: consider the relative topologies of the usual topology on R on I₁ = (2,4) and I₂ = (8,16) prove that…

A: Here f is a function from I1 to I2 defined as f(x)= 4x for all x in I1. Where I1=(2,4), I2=(8,16).…

Q: (c) Let z(x, y) = xr be a function defined on a disk D in the positive quadrant containing the point…

A: Follow the steps.

Q: Let I = , zdV ,where D is bounded by the planes x = 0, z = 0, x+ y = 2 and the cylinder x2 + z2 = 1…

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Q: 3. Suppose f(z) is continuous in the closed disk {려くR) and analytic in the open disk {|z| < R). Show…

A: To extend Cauchy's theorem to the boundary of the domain of anaylyticity ,under the given conditions

Q: let w- f(xy) and y in Poluy Cordeinate then a+yS %3D or or or 7.

A: Given that, w=fx, y And x and y is in polar form that is, x=rcosθy=rsinθ

Q: (a) Assume that f(x, y, z) and F (x, y, z) are arbitrary differentiable functions such that f(x, y,…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: let X-j m,n; is Z Ž asigma algebra over X ?

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Q: 4.) Verify the Divergence Theorem where F(x, y, z)=(x',xz,3z) given by x' + y +z² <4. and E is the…

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Q: View S' as the boundary of the 2-disk D² = {(x, y) E R² | x² + y? < 1}. Fix To E S', and let X = (S'…

A: Given : D2=x,y∈ℝ2|x2+y2≤1 where x0 belongs to S1 which the boundary of 2-disk. X=(S1×S1)∪(D2×x0) To…

Q: Let : R R be different iable on R. Compute de in terms of r. y and . where y

A: Proceed as shown...

Q: 1. Consider real Sobolev space W².1[–1, 1] and functional f(x) = Ltx(t)dt. Find z E W².1 such that…

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Q: (10) Let the vector space P, has the inner product (p(x), q(x)) = | P(x)q(x)dx (a) Compute ||p(x)||…

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Q: Suppose that (z) is entire and that the harmonic function u(x, y) = Re[fz)] has an upper bound uo;…

A: As requested following is the answer for question 12.

Q: Let 2C R" be open with smooth boundary. Show that (a) Green's function G(x, y), r, y E 2, is…

A: Let Ω∈Rn be open with smooth boundary. (a)To prove the Green's function is uniquely determined:…

Q: Evaluate. where G is a solid that lies in the first octant bounded by x^2 + y^2 = 3z and z = 3.

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Q: Let X = Rand de f ine ||| on X as ||r|| = Va then X is normed linear space

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Q: (b) Show that the function defined by IN if (x, y) (0,0), √√√z²+y² f(x, y) = 0 if (x, y) = (0,0) is…

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Q: - Let f(z) = cosh z. (1) Find the image of S := {z = x+iye C: x < 0,0 < y < } under f.

A: Follow the steps.

Q: Let V be a finite-dimensional inner product space over R. Given u, v E V, prove that ||u + v||? –…

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Q: Let V be an inner product space, and let y, z ∈V. Define T: V →V by T(x) = &lt;x, y&gt;z for all x…

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Q: 2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y themetric d defined byd((x1,…

A: To prove that every Cauchy sequence in the product space XxY converges (under the metric defined in…

Q: ) For the vector space C[0,1], define the inner product (f,g) = , f(x)g(x)dx. Let f = x + 1 and g =…

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Q: Q4 Let V=R, defined the addition by: %3D with slandard scalar multiplication. Is Va valtov space…

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Q: i) True or False: T = {0, R}U {(-x, a] : a € R} is a topology on R.

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Q: Apply Gauss Divergence theorem for F = x²i + y²j + z²k_over the cuboid S bounded by 0 < x< 1, 0< y<…

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Q: et X=R^(2) and C={(x y)in R^(2):x>=0 y=0}. Show that C is not a generating cone.

A: Given: Let   and .  To show that C is not a generating cone.

Q: ) Let z(x, y) = x" be a function defined on a disk D in the positive quadrant the point (1, 2).…

A: As per the question we are given a bivariate function : z(x,y) = xy Defined on a disc D containing…

Q: „Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} :lf A=(0,1), then the closure of A is (0,1)…

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Q: b. Draw the image of the W-plane into which the region bounded by the straight lines in the Z-plane…

A: The solution is given below

Q: Evaluate / (7xi – zk) · ndA over the sphere S : x² + y² + z² = | S a) by divergence theorem, (b)…

A: a) By divergence theorem, ∬SA. ndA=∫∫∫V∇.A dV Now,  ∇.A=∂7x∂x+∂0∂y+∂-z∂z=7-1=6 By changing to…

Q: . Suppose f(a, y) is continuous and non-negative over a non-empty open region R in the plane with a…

A: According to question,f(x, y) is continuous,fx, y ≥ 0        non - negative,∫∫Rfx, y dA = 0

Q: Let D be the portion of z = 2x + 5y inside z? + y? = 1, oriented up, and let %3D F (y, z, –2).…

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Q: Let X = {2,3, 4,..} and de fined metric on X as d (r, y) = |- yl %3D %3D Find орen ball B (2,2) =…

A: Follow the definition: B(2,2) = { x in X : d(x,2) &lt; 2 }

Q: Let X = {1,2,3, 4, ...} and de fined metric on X as d (r, y) |r-yl %3D Find open ballB (2,2) = o {2}…

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