Evaluate | CurlF nds F(x,v,z) = - yzi + xzj – xyk, and S is that part of the cylinder 4= x² + y² with 0sz s2 and n is outward.

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: Evaluate ∭E sqrt(x2+y2) dV where E is the region that lies inside the cylinder x2+y2=4 and between…

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Q: Cives とz f(xy)= J36-9メ-す i) Find domajn and Ram ge Foy the fuclien

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Q: Integrate G(x, y, z) = xyz over the surface of the cube cut from the first octant by the planes x =…

A: Here, the surface is in the first octant. Therefore, the integration is

Q: If F= 2y I- 3J +x²k and s the surface parabolic cylinder y= 8x in bounded by the planes OF the 2.…

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Q: Determine the area that lies under f (x,y) = 9x² + 4xy + 4 and above the rectangle given by [-1,1] ×…

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

Q: Find curl (7z i + 8xj+2y k). curi (7z i+ &x j+ 2y k) = (1 i+( j +( )k

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Q: The solid bounded by x = y? – 1, z = 1 – y², z = 4 – x, and z = 0.

A: Given that,

Q: Find the sum of maximum and minimum values of the function f(x, y, z) = 6x + 12y+ 18z on the elipse…

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Q: A solid G is bounded by the paraboloid z = 4 – x2 – y², the cylinder 22 + y? = 4 and the plane z =…

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Q: Use Stoke's Theorem to evaluate · F · dĩ where F = (xz, xy, 3xz) where C is the boundary of the…

A: Using the Stoke's theorem we have to evaluate  ∮F.→dr→ where F→=xz,xy,3xz=xzi+xyj+3xzk such that the…

Q: Use Gauss Divergence theorem to calculate (5zi+ 4yj- 2z k).n dA over the sphere S: 2 + y? + 22 = 9.

A: This is a problem related to vector calculus. Based on the general formula of gauss divergence…

Q: find the outward flux of F across the boundary of D.  F = 2xyi + 2yzj + 2xzk D: The cube cut from…

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Q: Considering f(, Y, z) = z + x²y and S as part of the cylinder y² + z² = 25 that is between the plans…

A: It is given that surface S is the part of the cylinder y2+z2=25 lies between the planes x = 0 and x…

Q: Verify the Green's theorem for integral x²y dx + xy dy, where C is the boundary described…

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

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Q: Wrons kiant y" + p(+) y # q (4) j'arlt)g =0 +q(4) j' +r (t)y=0 solotions of fifferential equations…

A: Determinant properties

Q: Evaluate. where G is a solid that lies in the first octant bounded by x^2 + y^2 = 3z and z = 3.

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Q: Compute fS fdS for f(z,v, z) = 2y-z and S is the part of the cone 2-2+ betweenm z 1 and z 4.

A: Integration is the process of finding the anti-derivative function of a given function. The…

Q: Verify Stoke 's Theorem for F = (x² +y – 4) î + 3 xyj + (2 xz + z²) k over the surface of hemisphere…

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Q: Find div F and curl F if F(x, y, z) = 4y³z'i – 11a"z°j – 4xy°k. - -

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Q: FD is the region in the first octant between the spheres x2+y2+z²=1 and x2+y2+z²=16, then dV =

A: Converting into spherical coordinates

Q: (a) Compute the area of the portion of the cone x2 + y2 = z? with z 2 0 that is inside the sphere x2…

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Q: Evaluate both sides of divergence theorem for A = pa, + pcos()a , + za, through the closed surface…

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Q: Evaluate || F.ndS F = xi + yj+(z² –1)k if and o is the curved surface of the cylinder y = \4-x? from…

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Q: Let F(r, y, z) = r²i+ y²j+ z²k and S the surface of the region bounded by 5. the cylinder 1? + y² =…

A: Divergence theorem

Q: -) Let -y M(x, y) = x2 and N(x, y) = + y? x2 + y? how that ƏN ƏM dx ду ) Show that -y F(x, y) = x² +…

A: Given :-  A vector field for conservative field in the plane ,  ∂N∂x = ∂M∂y To Determine :-  M x,y =…

Q: Evaluate fz² In z dz using integration by parts O² Inz-+c Onz-+c Onz-+c 2²lnz - 2 + c 12 21

A: Solution:  step-1: Evaluate ∫z2*lnz dz using integration by parts                Formula for…

Q: (b) Let z(r, y) = exp(-|r|) cos r + y², where r + y /9 = 1. Show that z(r, y) has minimum points…

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Q: 2. Let i = y?x + xŷ + z²ż, and let S be the paraboloid z = x2 + y? with 0 < z < 1. Show that fci. de…

A: Given  v→=y2x^+xy^+z2z^ and S be the paraboloid defined by  z=x2+y2      , 0≤z≤1 The boundary C of…

Q: Find %3D as if F(x, y, z) = 2xi+2yj+2zk and S is the part of the paraboloid z = 4-x-y lying above…

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Q: | Fug kuote the triple integral SSS 3ydv, where =L(ay.) | os xE2, 0sys lu-x?, 0s2sx.

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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 S denote the part of the cylinder x2 + y² = 9 that lies between the planes r + y = 1 and x – y =…

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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: =+ x²y) ds part of the cylinder y² + z² = 9 that lies between the planes x = 0 and x = 3 in the…

A: See the details solution in below

Q: Find div F and curl F if F(x, y, z) = xz°i+ 4y°x²j+ 52?yk. 3: 62. div F= curl F=

A: A detailed solution is given below

Q: Find the maximum value of f(x, y) = 3xy(11 – 132x – 11y) over the triangle with vertices (0,0), (,0)…

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Q: Evaluate f S S12zdV where E is the solid enclosed by z = 1 and z = r + y O 4m O 27 0 3r

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Q: Use the Divergence Theorem to evaluate F. dS where S is the sphere r² + y² + z² = 9 and F = (3r, 2y,…

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Q: se the Divergence Theorem to evaluate (8x + 12y + z2) ds Is here S is the sphere x2 + y2 + z² = 1.

A: To compute the below integral using divergence theorem.

Q: Find div F and curl F if F(x, y, z) = 9y%z®i – 4x°z1°j – 5xy°k. - -

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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: 2y do over the surface S: z = ; 0<x< 2, 0<y<3 Evaluate

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Q: Evaluate both sides of divergence theorem for A = pa,+ pcos(o)a,+ za, through the closed surface…

A: Given A=ρaρ+ρcosϕaϕ+zaz and the closed surface is 0≤ρ≤1, 0≤ϕ≤π, 0≤z≤2. We have to evaluate the both…

Q: Find the analytic functiopn f (z) = u (x,y) + i ( x, y ) such that u ( x, y)=  4xy3-4x3y+x.

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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: Find div F and curl F of F(x, y,z) = (x z³) i + (3x²y1² )j + (3z?y) k

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Q: Evaluate III Vx2 + y² dV E where E is the region that lies inside the cylinder x2 + y2 = 4 and…

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Q: Prave by indretion ŽKczkt) z on n that for all integens aoenzl, 2.

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