Let (X, M, µ) be a measure space such that uX = 1, and let f : X → (0, 1). Show that u) / fdµ 2 1

Q: Let r be a differentiable vector function of t. Show that if r . (dr/dt) = 0 for all t, then | r |…

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Q: Let S be the solid bounded by the planes z = o and y + z = 3, and the cylinder x2+y2 =2y-

A: answer is in next step

Q: (b) Let f be analytic on the unit disk D(0,1). If f (z) +0 for all z E D(0, 1), and set m= min |f(z)…

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Q: LET (X,Y) be a two-dimensional continuous v.a, with density f(x)=x(x+cy), 0<x<1, 0<y<2. So the value…

A: Given:      fx=xx+cy      0&lt;x&lt;1, 0&lt;y&lt;2 To find c, we use…

Q: 2. If f e R[a, b], show using the properties of Riemann Integrability that there exists M > 0 such…

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

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Q: Let E be the region bounded by the hemisphere x² + y? + (z – 2)² = 4, 2 < z < 4, and the plane %3D z…

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Q: Let f(z) = sin z. Use the Maximum Modulus Principle to find the maximum value of f(z)| as %3D z…

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Q: Let the function f is defined on the disk x² + y? < 1, then f has a critical point somewhere on this…

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Q: Find a function v(x, y) such that, given u = 2x(1– y), f(z) = u + jv is analytic in z.

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Q: IJJ is bou bounded and integrable on [a, b] Such that | f(x) | 0).

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Q: Verify the Divergonce Theorem for F=(2x-x)+x*y} -aż? taken over the bounded by x -o, X=1, y=0,…

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Q: Verify the divergence theorem, ffaw F · dS = S[fw, Div FdV, for F(x, y, z) = (z)i + (y)j + (x)k,…

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Q: Calculate FdV where F = 2i+2z j+yk and V is a space bounded by z=0, z = 4 and x² + y² = 9.

A: Put  x=rcosθy=rsinθz=z

Q: Let f: [a, b] → R where f is simply integrable If: a) U(P, f) = L(P, f) b) L(P,f) f(x)dx

A: Let F:[a,b]→R where f is  simply  integrable if

Q: 1. Show that F is a conservative vector field and then find a function f such that F = Vf where F(x,…

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Q: If a vector-valued function r(t) is continuous at t = a, then the limit r(a+ h) – r(a) lim exist.…

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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: If f(x, y)is integrable function on the rectangle R = [a, b] x [c, d] where [a, b] on the x- axis…

A: In the question it is asked to show how the two double integrals are equal to each other.

Q: Find the holomorphic function f (2) = u(x, y) + iv(x, y) (z = x + iy) with f(0) = 0, given that v(x,…

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Q: Exercise 11. Recall that XA denotes the characteristic function of the set A and it is defined by Si…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Let v f(x,y) and w f(x, y). Choose a correct formula %3! to determine where J is a Jacobian. dv aw…

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Q: If V = C[0, 1], the set of continuous functions defined on the interval [0, 1], then (S, 9): r1/2…

A: We use the fact that if f is non zero then &lt;f,f&gt; is positive.

Q: Evaluate the triple integral. SIJE3Y dV, where E is bounded by the planes x = 0, y = 0, : = 0, and…

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Q: If f e R[a, b], show using the properties of Riemann Integrability that there exists M > 0 such that…

A: The given function is f∈Ra, b. Let k and K are bounds of f(x) and f(x)≤M ∀ x∈a,…

Q: find the holomorphic function f(z) = u(x,y) + j*v(x,y)  if  v(x,y) = -ey * sin(x-y), f(0) = 1

A: The general form of a complex number is given by z=a+jb, where a,b are two real numbers and j=-1.…

Q: Let f(x+ y) = f(x).s) Vx, ye R.Prove that f is derivable on R iff it is derivable at one point of R.

A: If the given function is continuous at x=a then it is continuous at x=b.

Q: -dt. dF = (dr · V)F+ at

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Q: Let E = {(x,y, z) | x² + y² < 1 and xy < z < 2} and let S be the boundary of E, oriented outward.…

A: Using Gauss divergence law:  ∯SF.dS=∫∫∫V∇·F.dV                   =∫∫∫V2+3-1.dxdydz…

Q: Show that the vector field F = 2.xyi + (x – 2 y)j is conservative and evaluate the line integral %3D…

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Q: Use the Laplace transform to solve Utt – Uxr = 100, x > 0, t > 0. u (x, 0) = u4 (x, 0) = u (0, t) =…

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Q: If f(x,y)is integrable function on the rectangle R [a, b] x [c, d] where [(a, b] on the x- axis and…

A: On rectangular region. By Fubini's theorem The given Statement is true.

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

A: Based on given data,For Non empty open region R,fx, y is continuous,andfx, y is non -…

Q: Define T: (C¹ [0, 1], || · ||…) → (℃ [0, 1], || ||) by Tf = f'. Show that T is not continuous. Is…

A: T:C'[0,1], .∞→C'[0,1], .∞ T(f)=f' Let f(x)=xn then T(x)=nxn-1 T(x)=nxn-1 as n→∞,   T(x)→∞ So T is…

Q: Calculate fff, V.F dV where V is a closed space bounded by x = 0, x = 1, y = 0, y = 1, z = 0 z = 1…

A: In general, triple integrals involve the integral with respect to x, y, and z. Similar to partial…

Q: Use the Divergence Theorem to evaluate |F.n do where F(x,y,z)=(x² +cos yz, y – e" ,z² +x*) and S is…

A: According to question given :F(x,y,z)=x2+cos yz, y-ez, z2+x2S is the boundary of the solidBounded by…

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: „Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} f A=[0,1], then the interior of A is (0,1) O…

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Q: Let X and Y be jointly continuous with density fxy (x, y) = 2xy for 0 < 2y < x < 2. Then E (XY) is…

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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: Let r be a di¡erentiable vector function of t. Show that if r # (dr/dt) = 0 for all t, then  r  is…

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Q: Let f : C[0,1] → P be given byf (x ) = x (1)Vx e C[0,1]. Show that f is continuous w.r.t the supnorm…

A: f is continuous with respect to sup norm. Let xn be a sequence of continuous functions in 0,1 such…

Q: let J, bessel function , if y=x J, , then dy = .a 2 J'1 dx .b 1 J 2 dx .c x J, dx X .d

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Q: . Assume that f(x, y) is continuous on [a, b) x [c, d] and 9(r, v) = | 5(6,t)dt ds for a <a < b, c <…

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Q: Find of the operstor a norm A(x) = S (2.+st)x(t)dt %3D A: C [o,1] →L [o,I] O a. 2 O b. 7/12 O c.…

A: Option (c) is correct.

Q: Find the limit of the vector-valued function if it exists. lim(e² i + sin(t)j + e¯¹²k) O does not…

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