2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y themetric d defined byd((x1, y1),(x2, y2)) = dX(x1, x2) + dY (y1, y2)Show that (X × Y, d) is complete.

Q: „Consider the space (R,T). where t={R,Ø}U{ACR:[0,1)CA} :Then (R,T) is not T, O T, but not T, O T2 O…

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Q: 7. D is bounded by the surface (r2 + y² + z²)² = a²(x² + y² – 22). %3D

A: We have to find volume where D is bounded by the surface (x² + y² + z²)² = a²(x² + y² — z²).

Q: Verify Divergence Theorem for F = (x+ y²) î – 2 xj + 2 yzk and the volume of a tetrahedron bounded…

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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: Verify Green's theorem in a plane with respect to S. [(x2 - y²)dx + 2xy dy], where C is the boundary…

A: Using greens theorem

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

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Q: Calculate F - dS where F (e* sin y, e² cos y, yz²) S is the surface of the box bounded by the planes…

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Q: If E is the solid shape bounded by x 2 0,y 20 ,z 2 0 and the plane 2x + 3y +z = 6 evaluate (x + y)…

A: The given question is solved below

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

A: Here we can use spherical coordinate but we use peoperty of triple and double integration that it…

Q: IF F= 3yI - Xy I+ 9ZK and s is the Surface of the Pasabolied 2X =x2 fy Bounded By Evaluate %3D I=2…

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Q: Let R denote the solid in the first octant bounded by the planes 2x + 6y + 3z = 6, x = 0, y = 0, z =…

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Q: Let A = { (x,y)e RXR; 0<max{}x-31. ly-21)<1}, then: %3D None of the choices OA is not compact…

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Q: 4. Show that the set of points (z, w) e C with w2 = sin z is naturally a Riemann surface. %3D

A: Let, S=z,w∈ℂ2|w2=sinz be the set in equation. The multivalued analytic function fz=sinz can be…

Q: (c) By using the Green's theorem, evaluate $(2xy) dx + (4x²y*) dy if C is the boundary of a region…

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Q: Let I = , zdV ,where D is bounded by the planes x = 0, %3D 6. z 0, x+y = 2 and the cylinder x2 + z2…

A: Given: I=∫∫∫D z dVwhere D is bounded by the planes x=0, z=0, x+y=2 and the cylinder x2+z2=1 in the…

Q: 6- Evaluate the line integral ScF · dr, where F = [xy, yz, xz] and C is the boundary of the…

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Q: Verify Green's theorem for the disc D with center (0, 0) and radius R and the following functions.…

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3.Please resubmit the question and…

Q: If F = 2y i - zj + x²k and S is the surface of the parabolic cylinder y²= &x in the first octant…

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Q: Consider 2-dimensional hyperbolic space with metric ds ds'+dy and y > 0. The length of a line…

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Q: Calculate S," (2x+y) dV where V is a closed space bounded by the cylinder z = 4 – x2 and the planes…

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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: Use the Divergence Theorem to evaluate (5x + 10y + z) ds where S is the sphere x2 + y2 + z2 = 1.

A: In the given problem, S is the sphere x2+y2+z2=1Divergence Theorem:∬SF·n dS=∭Ediv F dVNow, the…

Q: Suppose D is the region in the plane bounded by the lines x + y = 0, x +y = 3, x – 2y = 0, and x –…

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Q: Calculate . (VxF). n dS if F = (x + 2y)i – 3zj + xk and S is a 2x+y+2z=6 plane surface bounded by x…

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Q: Let I = [[, ydV ,where D is bounded by the planes y = 0, z = 0 , x+ y = 5 and the cylinder y2 + z2 =…

A: We have to solve given problem:

Q: Suppose that D is the unit disk in the xy-plane, which is contained in the unit square S = [0, 1] ×…

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Q: evaluate $ A. dš over the sphere R = 3 and show whether Divergence theorem holds or not. %3D

A: Calculate surface integral over R=3 or dr=0 ∮A→×ds→=∮R cos2θ,π,0×r2sinθdθdϕ,rdϕdr,rsinθdrdθ…

Q: If E is the solid shape bounded by x 20,y 20,z 20 and the plane 6x + 4y + 3z = 12 evaluate %3D (y +…

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Q: At what points (x,y.z) in space are the tunctions continuous? a. h(x.y.z) = In (4z² - 4x – 2y³ - 5)…

A: This is a problem of multi variable calculus.

Q: Evaluate .n dS and .ndS if F = (x + 2y)i– 3zj +xk, , and S is the surface of 2x + y + 2z = 6 bounded…

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Q: 4. Use Green's Theorem of the Plane to evaluate the following closed path integral. xy2 dx+ 2x2y dy…

A: According to green’s theorem closed path integral over the curve C is calculated by,

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

A: Put  x=rcosθy=rsinθz=z

Q: Calculate ( (2x + y) dV , where V is a closed space bounded by the cylinder z = 4 – x? and the…

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Q: b) Use Gauss' theorem to evaluate RR F. ndS where F = (x² - y)i + y³z²j + (xy – y²z³)k UTM UTM and o…

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Q: 4. Evaluate //F FdS, where F(x, y, z) = (2x + 2xz, -4xy², y - z2) and R R surface of the solid…

A: The given problem is to evaluate the given surface integral over the given solid surface bounded by…

Q: Let C denote the positively oriented boundary of the square whose sides lie along the lines x =+ 2…

A: The general form of a complex number is given by z=x+iy, where x,y are two real numbers and i=-1.…

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: Evaluate x dV , where E is the E region in the first octant bounded by the sphere x² + y² + z² = 1…

A: given region is in the first octant. using spherical coordinates x=ρsinϕcosθy=ρsinϕsinθz=ρcosϕ and…

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: 1) Evaluate the triple integral SS, f(x,y,z)dv : B) f(x,y,z) = 3y² – 2z , Q is the tetrahedron…

A: given f(x,y,z) = 3y2-2z, Q=3x+2y-z=6 The region bounded by the equation Q is a tetrahedron cutting…

Q: Use the Divergence Theorem to evaluate (4x + 7y + z²) dS where S is the sphere x2 + y2 + z² = 1.

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Q: An _____ is a set of points (x, y) in a plane such that the sum of the distances between (x, y) and…

A: The sum of the distances to any point on the ellipse (x,y) from the two foci (c,0) and (-c,0) is a…

Q: Use Gauss' theorem to evaluate F.ndS for the given F and o. (b) F(x, y,z) = x³i+x²yj+xyk, o : the…

A: Gauss theorem states that ∫∫σ F.ndS=∬∫V div(F) dVwhere V is the volume inclosed by the closed…

Q: A solid G is bounded by the paraboloid z = 4 – r - y², the cylinder 2+ y = 4 and the plane z = 5.…

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Q: Let M be the region bounded by the parabola y = x² and the line y = x +2. %3D 6. y = r + 2 5. 4.…

A: Consider the given information and diagram: Let y=x2 and the line is y=x+2.

Q: Use the Divergence Theorem to evaluate (4x + 3y + z) dS where S is the sphere x2 + y2 + z2 = 1.

A: The divergence theorem establishes the equality between surface integral and volume integral.

Q: Compute s F=

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Q: Use Stoke's theorem to evaluate the integral xydx+x'dy – 4x’ydz where C is the boundary of the…

A: Here the surface is the plane of the square with vertices A1,0,1, B1,1,1, C0,1,1 and D0,0,1.  To…

Q: Let X = R², x, y EX where x = (x₁, x2), y = (y₁.Y2) with the metric function (4 pts) (x, y) =…

A: This is a question of topology and metric space.