d set): Let (M ,d) be a (SUT)' = S' U T'.

Q: Use špheričal čoördiñatės. Evaluate tet dv, where E is the portion of the unit ball x2 + y² + z² s 1…

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Q: KLM is a kite. If mZJKN = 28° and mZKLM = 103°, find mZ JML. * %3D %3D K J M

A:

Q: Let G = Z31, S = {1,5, 6, 25, 26, 30}. %3D %3D Prove that Cay(G, S') is normal.

A: It is given that

Q: Let X (2,3. 4) and de fined metrie on X as d(x, y) = |-y Find open ball B (2.2) = (0) O (2) 0 2.0…

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Q: (15) Let D be the half-disk x² + y? 0. Compute / | (a + y) dA D

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Q: Evaluate | Vtan a sec" zdx. O a. 2 z tani z +tani +C 2 O b. 2 tani æ + tan +C 2 Ос. 2 O c. 2 2 t…

A: To evaluate the integral,  ∫tanxsec4xdx Rewrite the term sec4x=sec2x·sec2x…

Q: For z € C, define Tz: CC by Tz(u) = zu. Characterize those z for which Tz is normal, self-adjoint,…

A: Given that: For z∈C, Define Tz:C→C by Tzu=zu. The objective is to verify the given statement. Now…

Q: The functon T: RsR3 i dyenis ar follous tC") = AV = ]M] %3D 2 -2 os funis Tcr), where v= (2,) as…

A:

Q: Ex: © Suppise that the fiowing Aransferuntin Conneclğs the lei) and (ti) coordnate Syshems; a4ス?) 2…

A: Given that, ζ=x¯1=ex1+x2x¯2=ex1-x2 To find: a) Jacobian matrix J and J-1 b) Compute J¯

Q: Let r = xi + yj + zk and let r = ||r||. Prove that

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: et T E L(V,W), and U e L(W, X). Prove that UT : V → X is also linear

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Q: Let r = ri+ yj + zk and let r = ||r||. Prove that

A: As per our guidelines, we are supposed to solve only first question. Kindly repost other question as…

Q: Given the bases B={u , u2} and B'={u'1, u'z} for R2, where , U2 and u'2 = The transition mat rix…

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Q: Find the family of orthogonal trageclarnes fo he for

A: Given that            y3 = k x5 Here we have to find the family of orthogonal trajectories to the…

Q: Q2: In usual topological space (R.T), answer the following: 4- Is [0,1] is arcwise in R? Why?

A: Given that ℝ, Tu is usual topological space. To show: 0,1 is arcwise in ℝ.

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

A:

Q: Let r(t) = ⟨t, t, t⟩. Compute r'(t) and interpret the result.

A: rt=t, t, t

Q: if x=u(1-v),y=(uv) then verify that a(u,v) a(x,y). -=D1 a(x, y) a(u,v)

A:

Q: The orthogonal projection of y onto u is the same as the orthogonal projection of y onto cu whenever…

A: solution

Q: 3. Find the circulation of F 2xi + 2zj +2yk around closed contour %3D

A:

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

A: Proceed as shown...

Q: Let w(z, y, z) = 1 + y² + z?, where z = 3', y = Int, and z = = et. Find I using the Chain Rule.

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Q: Let r, y EN such that y # 0, then y+r#0.

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Q: Prove that f, uv v • dỉ = – §̟ vũu • dĩ

A: Whenever the force is conservative in nature the work done over the closed loop is always zero. i.e…

Q: f (lnx + y) dx − a² dy, where & is the rectangle with vertices (1, 1), 1. Use Green's Theorem to…

A: Let I =∮(ln x+y)dx-x2 dy, where the region is C which is a rectangle with vertices…

Q: Q2/ Show that {1,cosx, sinx,cos2x, sinx,...} Form an orthogonal set on [-Tt,t] *

A: A set of vector B is orthogonal if every vector in the set is orthogonal. Therefore, the dot product…

Q: Z is the centrold of AUVW. W What is ZX if WZ = x? A. 5 O B. В. ос. 2т O D.

A: Answer is mentioned below

Q: For z ∈C, define Tz: C →C by Tz(u) = zu. Characterize those z for which Tz is normal, self-adjoint,…

A: Firstly, find out Tz*.

Q: Let f(x,y,z) = zIn(x³ycos(z)), find Z 3x² O In(x3ycos(z)) - ztan(z) 0 2 Y о 3z X af ila

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Q: Let P = {(z, y, 2) E R° | az + by + cz = 0} where (a, b, c) # (0,0,0). (a) Prove that (a, b, c) ¢ P.…

A: Given that P=x,y,z∈ℝ3 : ax+by+cz=0 where a,b,c≠0,0,0 (a) The objective is to prove that a,b,c∉P Let,…

Q: of the b) nd y- Fons c) with point.

A:

Q: Use the Chain Rule to find ôz/əs and əz/ðt. z = tan(u/v), u = 4s + 7t, v = 7s – 4t Əz as az II

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Q: . Show that the function u = p(ry) + Vry v(y/x) provides the quation = 0. ду? %3D dx²

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Q: Let z, w E C. Show that zw = 0 implies z = 0 or w = 0. %3D

A:

Q: Let, T: R' → R';T(x, y,z) = (x+ y+ z,2y + z,2y +3z) Find the dimension of KerT and ImT

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Q: b- Let X₁, X2,... X (a) Find a comp (b) Find the UM

A: Solution:   (a):

Q: Use Green's Theorem to evaluate x dx + xy dy where C is the triangle with vertices (0,0), (1,4),…

A:

Q: Evaluate /// xe²+y²+z² dV, where E is the portion of the unit ball x2 + y? + z2 < 1 hat lies in the…

A: Given The given integral is ∫∫∫Exex2+y2+z2dV E is the portion of unit ball x2+y2+z2≤1.

Q: Let E be the hemisphere described by Oszs V1-x? -y2 and F=Vz? + x2 + y² i+Vz? + x² +v² j+Vz? + x²…

A: We can solve this by gauss divergence theorem

Q: Ex: © Suppise that the fuowing trausfruntiun Cowneciğa the (I') and (xi) coarduate رک سهاور؟ てス4ス)…

A: Given  x¯1=ex1+x2x¯2=ex1-x2 Required Jacobian is…

Q: $ ((x,y) (E,6) )e product Prove 2 xt + 3yt on e?. Gompute inner IL(,2), where the norm i's…

A: We have to prove that f( (x, y) , (t, t) ) = 2xt + 3yt is an inner product on R2 . 1.) &lt; ( 0, 0 )…

Q: Use Green's Theorem to evaluate x' dx + xy dy where C is the triangle with vertices (0,0), (1,4),…

A:

Q: 3.use Greens theorem s (1+x5 dx + 2xy dy ,where C is triangle With vertices (0,0),(1,0),(1,3)…

A: Apply greens theorem

Q: Use Green's Theorem to evaluate S. axy dy + cxy dy, where C is a counterclockwise oriented…

A: Given : a=2, c=8 We have to use Green's Theorem to evaluate            ∫C axy dy+cxy dy where C is a…

Q: which of the following are inner product on V, CR) where u=(d1,de] , V = ( P1» P2) %3D 2 2 2 3) =…

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Q: Let E be the upper half of unit ball E = {x2 + y2 + z2 ≤ 1, z ≥ 0}. Compute ∫∫∫Ez dV.

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Q: let F= Mit Nj + Pk show that Zt)zde +. p人(2人 gYG,Z)ydf t JM(W, Y, 2)dw

A: .

Q: + Let U= -Zit V=a Deteronine sich Shat cthogonal paralle! a) u v ae

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

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

A: Given