Show that the linear map (x, y)H (x+ y, x- y) from (R?, || - |1,) to (R?, I| - ||is an isometry.

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y.

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Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. *

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Q: Verify: Green's theorem in the plane for f(2x-y³) dx-xydy, where C is the boundary of the region…

A: Given information ∫c2x-y3dx-xydy Here, C is the boundary of the region enclosed by the circle…

Q: 4. Let L: R² +R? be the linear map defined by L(x, y) = (x + y, I - y). Show that L is invertible.…

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Q: Bounded by the planes z= x, y =x, x+y=2 and z= 0

A: The objective is to find the volume of the region which is bounded by the planes z=x,y=x,x+y=2,z=0.…

Q: Find the image of the semi-infinite strip x ≥ 0, 0 ≤ y ≤ π under the transformation w = exp z, and…

A: w lies in the portion of the closed upper half-plane external to the open unit disk.

Q: Suppose M and N are C manifolds, U an is C. Show that there exists a neighborhoo that F can be…

A: Let us start solving the problem step by step

Q: Evaluate the line integral x'z ds, where C is the line segment from (0, 5, 2) to (4, 6, 5).

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Q: /Evaluate The are bounded by The Semiellipes y =2J5-x' oP The line 6 X = ±1 ,Y=-1 %3D

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Q: Construct a linear map L(z)=az+b that will send the imaginary y-axis to the line v=-u.

A: we have to constract a linear map that maps the y-axis to the line v=-u Basically we have to…

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

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Q: In which of the following cases is the mapping x: R² → R³a patch? (а) x(и, v) — (и, uv, v). (c) x(и,…

A: A mapping defined from x : R3 to R2 is defined a patch if x maps U diffeomorphic ally onto x (U) As…

Q: 3. Show that a reflection a in the x-axis defined by a(x, y) = (x, – y) is an isometry.

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Q: a) F(x,y, z) : planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 1. 3xi + xyj + 2xzk and E is the…

A: Solution of (a) is in given below

Q: Show that fc F dr. is not independent of path. [Hint: Compute and , where  c1 and c2  are the upper…

A: To prove that ∫cFdr is not independent of path. Let C1 be the semi circle which lies above x axis…

Q: If h:(R,+) → (R*,.) is a mapping defined by h(x)=10*, VXER, then (R,+) = (R*,.).

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Q: Evaluate the line integral ∫c x3 z ds, where C is the line segment from (0,8,8) to (5,7,6).

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Q: (6) I. adv, where E is bounded by the planes z=0, y = 0, =0, y+= =1 and +=1, as shown in the figure…

A: Given that,

Q: 5. If мая abstract manifolds, PEM. f a) Define TpM and dfp, where M²₂N is smooth map beton by dfp:…

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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: الموضوع show that if foret A frix by and ; YーZ is a continuous map then gFo relA gf,:X b Z

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Q: Let F = (z, 2x, y) and C is the straight segment from (0,0,0) to (1,2,3). Then ,F dř = %3D %3D

A: Given,                                   F→(x, y, z)=z, 2x, yand C is the straight segment from (0,…

Q: Let H be the plane 2 X- 2y- z=0 in R', that is, H={(x, y, z)eR³ | 2x- 2y- z=0} and let F be the…

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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: Are the following linear maps invertible? a) T: R3 → R³ defined by T(x, y, z) = (y, z, x)?

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Q: A. Let H be the set of all points (x, y) in R2 such that x2 + 3y2 = 12. Show that H is a closed…

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Q: Let f(r, y) = |r|+ lyl- (a) Draw three r-cuts of f(r, y), three y-cuts of f(x,y), and one z-cut.…

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Q: now that the mapping a: E? ->E, (x,y) Coy, x + 2 ) is a transformaton and a collineation by finding…

A: One-One: Let (x1,y1),(x2,y2)∈E2 be such that: α(x1,y1)=α(x2,y2) so,…

Q: 4. Let L: R? H R? be the linear map defined by L(r, y) = (x + y, x - y). Show that L is invertible.…

A: Given problem is :

Q: Find the kernel of linear map L:R R given by L(x,y,z)%3D(0,z,y)?

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: 4. Let f(x, y, z) = e² + zy² and let C be a straight line segment connecting the points (4,2,1) and…

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Q: Which of the following maps is not linear? O A. L-R2-R3 defined by L(x,y) = (x+y,y.x)- OB. K:R R3…

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Q: 3. (a) Show that under the mapping w = 1/z, all circles and straight lines in the =-plane are…

A: (a) We have to show under the mapping w = 1/z, all circles and straight lines in the z plane are…

Q: Prove that R" is connected; Prove that [0, 1] × [0, 1] (with the relative topology from R²) is not…

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Q: Let R be the region in the first quadrant bounded by x = = 1, (x − 2)² + y² = 1, and x² + y² = 4.…

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Q: Construct a linear map L(z) = az + 6 that will send the imaginary y-axis to the line v = - u

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Q: Which of the following maps from R³ to R³ are not linear? 0 f(x, y, z)=(x, y, z + 1). f(x, y, z) =…

A: The solution is given below

Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

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: consider the map given by F(x)=1-x² discuss the charactristics of the orbits starting at x=0

A: As per the question we are given the following map : F(x) = 1 - x2 And we have to explain the…

Q: Use greens theorem to evaluate line interal (x2y)dx+y2dy),where c is the closed path formed be y=x…

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Q: 3. Show that a reflection a in the x-axis defined by o(x, y) = (x, -y) is an isometry.

A: The solution of given problem is followed by

Q: (N) The mapping ||x||(--) := V(x|x) : X → R is a norm on X. CS) |(x\y)| < ||||(1) ||y||41» (x, y € X…

A: We shall solve first three subpart only asper the company guideline- for other kindly post again by…

Q: 1) Given the mapping F(x,y,z)=(3e" ,xzsin 2y ,xy), v=(2,2,-1), using the definition, then the…

A: Let p be a point in ℝn and F:ℝn→ℝm be a map, then the tangent map of F at p is a map between the…

Q: Let F(r, y) = yeyi+xe*vj. Evaluate the line integral | 1 F. dr where C is the closed path consisting…

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Q: Let R have its usual topology and let Y = {x, y, z}. Define f: R → Y by: %3D x 0 (х, f(x) {y, (z,…

A: Quotient topology: If X is a topological space and A is set and if p:X→A is a surjective map, then…

Q: X, Y be topological spaces and f : X Y be a continuous map. If X is completely normal then Y is also…

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Q: Show that a triangle with vertices (x,Y;), (x2,y2), and (x;,y;) has area + x,y3 + X3Y, – X,Y3 – X,Y,…

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Q: (f) Find the length of the projection of i onto ū.

A: Given