If S is an arbitrary closed surface covering a V and A = axi + byj + czk, then www prove that f. A.n dS = (a + b + c)V.

Q: Assume S is the surface x1000 + y8000 + z1000 = 1,000,000 and F = [9 + ee^(xyz), x8yz, 3x - 5y -…

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Q: Suppose S is a smooth surface parametrized by r(u, v) over D = [0, 1] x [0, 1] with ru = and rv = .…

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Q: Given: ZWZX ZYZX; ZW = ZY Prove: ZX is a perpendicular bisector of WY

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Q: Verify Stokes' Theorem if o is the portion of the sphere x + y +z² =1 for which z20 and F(x,…

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Q: b) Use Gauss' theorem to evaluate ff F - ndS where F = (r - y)i + y²j+ (ry-y)k and o is the closed…

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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: Let D be any surface and let F = (P depends only on x, Q only on y, and R only on z). Then that…

A: We will use the stokes theorem to check if the statement is true.

Q: Let S be the closed surface composed of the parabaloid z = x2 + y? and the planes z = 2 and z = 4.…

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Q: Let the surface S in R° parametrized by f : R? - → R³, f(u, v) = (v, 2uv, (u – 1)²) Find all (u, v)…

A: Given, that the surface S in R3 parametrized by   f:ℝ2→ℝ3fu,v=v,2uv,u−12 Now, find all the (u, v) at…

Q: Let F=[-x², 3y, 2xz] and S be a sphere of radius R. Using the divergence theorem, the value of the…

A: The answer is option D. Divergence theorem  Let E be a simple solid region and S is the boundary…

Q: A solid G is bounded by the paraboloid z = 4 - x? – y?, the cylinder x² + y? = 4 and the plane z =…

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Q: In RST, U is the Centroid. If SQ=6, find SU and UQ W R

A: Given query is to find SQ and UQ.

Q: Let x = x(4, v) be a regular parametrized surface. A parallel surface to x is a parametrized surface…

A: Let x=x(u,v) be a regular parametrized surface, A parallel surface to x is a parametrized surface…

Q: Let S be the closed surface composed of the parabaloid z = x² + y² and the planes z = 2 and z = 4.…

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Q: Identify the theorem: If S is the boundary surface of a solid E in 3-dimensional space, then SS, F.…

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Q: Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition…

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Q: b) Use Gauss' theorem to evaluate f[ F· ndS where M UTM = (x² – y)i + y°z²j + (xy – y²z³)k and o is…

A: The given problem is to evaluate the given double integral using gauss divergence theorem over the…

Q: Consider the closed surface S consisting of the graph z = 1 - x2 - y2 with z 2 0, and also the unit…

A: Here, F(x,y,z)=(5x,5y,z) z=1-x2-y2 x2+y2+z=1

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

A: Given: ∫Cxydx+x2dy-4x3ydzBoundry square surfaces z=1vertices are (1,0,1),(1,1,1) & (0,1,1) &…

Q: Use Stoke's theorem to evaluate F dr where F(x,y, z)=e-7i+ +e"k and C is the boundary of the part of…

A: Let Fx,y,z be a vector field and the surface S is bounded by a simple closed curve C. The Stoke's…

Q: If S is a surface parameterized by r(u, v) = (3u + 2v)î + (v – u)ĵ+ (8 – v²)k con u € [0,1], v € [2,…

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Q: Let G be the solid enclosed by the plane z = y, the xy-plane, and the parabolic cylinder y = 1-x²…

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Q: b) let R be the disk in the xx-plane centered at the point (0.2) and with radius 2. Calculate JJ₁…

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Q: Let S be the closed surface composed of the parabaloid z = x² + y? and the planes z =2 and z = 4.…

A: Please see the answer below.

Q: Let F = [-x, 3y, 2xz] and S be a sphere of radius R.Using the divergence theorem, the value of the…

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Q: Calculate S,F.as if F(x,y,z) = xi+ yj and the positive oriented surface o is : The portion of the…

A: To find, ∫∫SF→.dS→ Given that, F→x,y,z=xi^ + yj^   Part i Given portion of plane x+y+z=1 bounded by…

Q: Evaluate G if G is a solid in the first octant bounded by the plane y +z = 2 and the surface y = 1–…

A: Use the following formulae, to obtain the solution. a-b2=a2-2ab+b2∫xndx=xn+1n+1+C

Q: (a) Let S be the surface z2 = x² + y? bounded between the planes z = 0 and z = 2. Evaluate (z –…

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Q: In space coordinates, the xy plane consists of all the points of the form O (0, y, z) O (X, y, z) O…

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Q: Surface o is a plane a +y + z = 1 that lies in the first octant. Evaluate

A: Given, surface σ is a plane x+y+z=1 that lies in the first octant. To find              ∬σ(2x+z)dS=…

Q: Exercise 10. Let M be a surface homeomorphic to a sphere in R³, and let TC M be a closed geodesic.…

A: We will apply the Gauss-Bonnet Theorem to prove the given statement.

Q: Sketch the region in the z-plane represented by the following set of points: (i) Iz + il + |z – il <…

A: Given that: z+i+z-i≤3

Q: Let S be the closed surface composed of the parabaloid z = x² + y? and the planes z = 2 and z = 4.…

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Q: Describe and sketch the surfaces in space (R') defined by the following equa- tions: (a) y = -z +1…

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Q: Show that the mapping w = (1 – j)z, where w = u + jv and z = x + jy, maps the region y > 1 in the z…

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Q: let s is the curved surface of the cylinder x²÷a² + y²÷b²=1 0≤z≤c , then evaluate…

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Q: Consider the surface S given by: S :r(u, v) = (uv², u + v, v³), u E [0, 1], v E [0, 2]

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Q: „Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} f A=[0,1], then the interior of A is (0,1) O…

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Q: Consider the solid in xyz-space, which contains all points (x, y, z) whose z-coordinate satisfies 0…

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Q: Let A = 6 zi + (2x + y)j – x k ; Evaluate A.ndS over the entire surface S of the region bounded by…

A: Please see the figure below to comprehend what's going on:

Q: Question 1 Let B = {y: ||y –2|| < r} be an open ball in R. Show from the definition of compactness…

A: Given: B={y: ||y-x||&lt;r} is an open ball in ℝp To show from the definition of compactness that it…

Q: Exercise 7.2.8. Apply Stoke's theorem to find F - dr, where F = yi+zj+ækover the surface S, %3D…

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Q: „Consider the space (R,t). .where t={R,Ø}U{ACR:[0,1)CA} :lf A=(0,1), then the closure of A is (0,1)…

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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: (a) Given that F(x,y,z) = (3x² + 3y²)i – 2xyj + yzk, evaluate S, F·dr for which C is the boundary of…

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Q: 5. Prove that the surface E, = {(x, y, z)|a² + y² = 4} is locally isometric to the surface E2 = {(x,…

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Q: 1. Evaluate the surface integral SL xz² ds ; o is the part of the cone z = Vx2 + y² that lies…

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Q: Every level surface of f(x, y, z) = x + 2y+ 3z is a plane. Select one: O True O False

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