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: 6. (i) Let f: (X, J) → (Y, U), where (X, T) and (Y, U) are two topolog- ical spaces. Show that if…

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Q: (a) Prove that f is a one-to-one correspondence. (b) Prove that ray AF is between rays AB and AE if…

A: (a) To prove that f is a one-to-one correspondence, we need to show that f is both injective and…

Q: 1 ExL =5 (ex-5) f(x). g(»)dx (F-9) ニ Let v be the vector space of oll continous real valued function…

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Q: f(x,y,z) = xe2y + ye3z + ze-x .Give general formulas for ∂n f/∂xn, ∂n f/∂yn , and  ∂n f/∂zn, where n…

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Q: Let r(t) be a positively oriented rectangle with corners (0,0), (0,1), (2,1) and (2,0) and F(r, y)…

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Q: analytic function a domain D. If a zeck. y) + b v(x, y) = C in D are real Constants not all o. #2.…

A: If f(z)=u+iv is analytic in a domain D , then f satiesfies the Cauchy Riemann (C-R) equations that…

Q: Indicate whether the following statement is true or false. Let f(x, y) be any c' real valued…

A: Given fx,y be the C1 real valued function defined on the rectangle R=a,b×c,d. Then we have to check…

Q: Q2) homeomorphism. Prove or disprove: The function f: (R, T1) - (R, 13) defined by f(x) = 3x is a…

A: Clearly the given function f(x)=3x is continuous. Now we want to check whether it is bijective,

Q: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = %3D (x, y) = (0,0) x²y+2xy² Let…

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Q: Assume that u and u are continuously differentiable functions. Using Green's theorem, prove that Uz…

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Q: Let f : X → Y and g : X → Z be two bijective functions. Is the function h: X → Y × Z defined by h(x)…

A: Given: f:X→Y and  g:Y→Z be two bijective functions. To Check: Whether the function h:X→Y×Z defined…

Q: ) Let (x, y) €Rx R. Consider the following functions from R2 to R? r(x, y)) = (-y,x) and s(x, y) =…

A: Since you have asked for multiple parts, solving a) and b) , kindly post the remaining as a separate…

Q: Let C be the counterclockwise boundary of the region enclosed by the x-axis, the line y = x – 2, and…

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Q: Let S be the lateral surface of the cylinder defined by x² + y² = 4 and 0 ≤ z ≤ 3. Let F (x, y, z)…

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Q: Let (f (u, v,w) and (u = u(x, y, z), v = v(x, y, z), w = w(x, y, z) be differentiable functions)…

A: Given function f(u, v, w) and u = u(x, y ,z),  v = v(x, y, z) , w = w(x,y,z) are differentiable…

Q: Consider F and C below. F(x, y, z) = yzex² i + exZj + xyexz k, C: r(t) = (t² + 2)i + (t² − 2)j + (t²…

A: Given vector field is the following :Here Hence given vector field is irrotational .So there must a…

Q: Let f(u,v)=u2−v2, u=xy+z, and v=2x+y2 . Find ∂f/∂z in terms of x, y, and z

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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: 3: Let a function d: R* x R* → R defined by d(x, y) = E-\xi - yıl Vx = (x1,...,x4), y = (y1,...,y4)…

A: Here we will prove that d is a metric function.

Q: Let T: R → R' such that T (x, y, z) = (a²z + 4az, y, 36z + az). What is a value of a so that T is…

A: For Surjection of T, Dim{Range(T)} = Dim{Co-domain}

Q: 1.T.2 Let V be the set of all twice-differentiable functions f : R → R satisfying f"(x) = f(x) – 5.…

A: Let, f and g be two functions in V.  Then,        f"(x) = f(x) - 5       g"(x) = g(x) - 5.

Q: Let X=R2 and defined d2:  R2 x R2 to R by d2((x1, y1)) = max{|x1-x2|, |y1-y2|} Verify that d2 is a…

A: Given a function d2: R2×R2 → R, defined by d2x1,y1,x2,y2=maxx1-x2,y1-y2. To verify that d2 is a…

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

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Q: B). For what values of r, reR is the function [(x+y+z) ƒ(x,y,z) = { x² + y² +z² 0, continuous on R.…

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Q: Consider F and C below. F(x, y, z) = yzexZi + e²j + xye*²k, C: r(t) = (2 + 4)i + (t² – 1)j + (t² -…

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Q: Ex3 i) ii) Let T: C ([0,1]) → R f → f(0) If C ([0,1]) is equipped with the sup-norm, show that T is…

A: As per the policy, we are solving first question. Please repost it and specify which question is to…

Q: Theorem. Let D = = {(x, y) = R² : x² + y² < 1} be the closed unit disk, visualized below: dzumuoan…

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Q: d) Let f be a continuous function. Then 1 [ S f(x,y) dady = f*f* f(x,y) dyda. 0

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Q: Consider the operator Î = k, where k is a scalar, operating in a function space on the interval [a,…

A: we know that the necessary requirements for L^ linear operators is given by- L^(f(x)+α g(x)=…

Q: Let fi(x) E Fun (R,R) be the constant function I, wheret ficx)=1 for all xG RRe Prove that S { sec?…

A: Given: fx∈funℝ, ℝ is a constant function. such that: fx=1……1 S=sec2x, tan2x, fx

Q: Consider F and C below. F(x, y, z) = yzexzi + exzj + xyexzk, C: r(t) = (t2 + 1)i + (t2 − 1)j + (t2…

A: Given data F(x, y,z)=yzexzi+exzj+xyexzkC: r(t)=(t2+1)i+(t2-1)j+(t2-2t)k,  0≤t≤2

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: Let F(x, y, 2) = (z, 1, y). a. Find divF and curlF.

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Q: Let S' denote the set of (x, y) E R × R such that x? + y² This space is called the circle. Define a…

A: S1={(x,y)∈ℝ2/x2+y2=1} and p:ℝ→S1 by pt=cos2πt,sin2πt. To show: p is continuous, open,closed.

Q: c) Prove that a subset W of R’ defined by W ={(x, y, z) e R² / 2x= y = 2z} is a vector space of R'.

A: Given, W = { (x,y,z) in R³ | 2x = y = 2z } is a subset of R³ Let, u = (x,y,z) &amp; v = (a,b,c)…

Q: Let Y be any topological space and y: [0,1] → Y be a continuous function so that y(0) = r(1) Show…

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Q: Consider the function f (z) = ²-sin z z3 The Laurent series of f (z) around 0 is : 1 z2 a) 3! 5! 7!…

A: Given: f(z)=z-sinzz3