2. Let U C R³ be the upper hemisphere, i.e., с U = {(x, y, z) € R³: x² + y² + z² = 1, z ≥ 0}. Let D C R² be the closed disk, i.e., D = = {(x, y): x² + y² ≤ 1}. Give an explicit formula for a homeomorphism f:U → D. and its inverse f¯¹:D → U. Draw a simple picture.

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: Evaluate f. F.dr, where F(r, y, z) = z?i+rj and C is the boundary of the square {(r, y, z)|0 < x <…

A:

Q: 3) Let S be the unit sphere x² + y² + z² = 1 and F(x, y, z) = (2x, y²,z²). Use the divergence…

A:

Q: Let G = &lt;x, y | x8 = y2 = e, yxyx3 = e&gt;. Show that |G| ≤16. Assumingthat |G| = 16, find the…

A:

Q: Divergence Theorem

A: and S is the surface of the cube.[-1,1][-1,1][-1,1]

Q: 5. Prove or disprove: Let F : R² → R² be defined as F(r, y) = (r + y, 2r – y). Then F is a rigid…

A:

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

A:

Q: Find a transformation that maps R = {z = x + iy : x, y 2 1} onto the unit disk |2| < 1.

A: please see the next step for solution

Q: Consider an isometry f : E3 → E3 given by f(x, y, z) = (30 − x, z + 22, y). Find a line L in E3 such…

A: Given , f(x,y,z) = (30-x , z+22 , y)

Q: Let T:V U be a homomprphism then ker T={0}if T is one-one.

A:

Q: A normal vector to the tangent plane for the at the point (1, 1, 1) is: y surface x =

A: A(1, 1, 1)z= xyf(x,y,z) =xy-zfx=yfy=xfz=-1Normal vector to the tangent plane for the surface;n^…

Q: 10. Let w =f(e) = (3 + 4) z – 2 + i. (a) Find the image of the disk | z - 1| 1.

A:

Q: Let (t) = (t, t, tº, tª) for all t € R. Define C = Þ(R). Let p = (1, 1, 1, 1). Derive the tangent…

A: Let  for all Let the point be We need to calculate the tangent and normal spaces to C at p.

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

A:

Q: be a vector space

A:

Q: c) Let z(x, y) containing at (1,2). = x² be a function defined on a disk D in the positive quadra:…

A: Given- z(x,y)=xy is a function defined on a disk D in positive quadrant containing point (1,2). To…

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: Compute the curl of the following vector V defined by V(x,y,z) = e(x²+y² +z²) y

A:

Q: Find a continuous function. f:D → R (where R is the real number line) such that f((x1, y1)) = f((x2,…

A:

Q: Is the following a vector space? Let 'V' = 'R'. Defined for 'j', 'k' residing in 'V' and 'm'…

A: Given that V=R, defined for j,k residing in V and m residing in R for the following operations:…

Q: b) The tangent subspace Wp at p ES is the plane through 0 with normal vector p. Find an explicit…

A: b) The tangent subspace at a point p on the unit sphere is the plane through the origin with normal…

Q: Apply Stoke's theorem to compute ||curl F-ds where C is the triangular closed path Toining the point…

A:

Q: The surface S₁ and vector field F are given, in Cartesian coordinates, by S₁ = {(x, y, z) : x² + y²…

A: The surface ,And the vector field  are given in Cartesian co-ordinates,

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 D = {(x, y, z) | x² + y² ≤ 1, z = 0.} C R³ be the unit disk in the xy-plane. Is D a Jordan…

A: We are to answer the following questions:Let D {(x, y, z) | x² + y² ≤ 1, z=0.} in R³ be the unit…

Q: Solve the following differential equation: yde+xdy 1-x²y² +ædr=0. = constant.

A: Given that the differential equation ydx+xdy1-x2y2+x dx=0

Q: Let F = (z, 0, y) and let S be the oriented surface parametrized by Þ(u, v) = (u² – v, u, v²) for 0…

A: From the given information: