Use the Divergence Theorem to evaluate [, F• dS, where F(x, y, z) = z²x i + (y + tanz)j + (x²z + y²)k and S is the top half of the sphere x? + y? + z? = 1. [Hint: Note that S is not a closed surface. First compute integrals over S, and S2, where Si is the disk x? + y² < 1, oriented downward, and S2 = S U Sı.]

Q: Let B be the region in space bounded by the planes z = 0 and z = 3x, and the cylinder x? + y² = 4.…

A: To compute the below surface integral using the Divergence theorem. Here the function is defined as…

Q: (6) Use the Divergence Theorem to evaluate dS, where F(x, y, 2) = xyi + (y² + e*=*)j + sin(xy)k and…

A: Given,         and

Q: 5. Use the Divergence Theorem to calculate the surface integral SF dS, where F(x, y, z) = (xye , xy?…

A: By divergence theorem value of the surface integral is

Q: Evaluate the surface integral. ) ر م(ل 2) ds S is the part of the cylinder x² + y2 = 16 that lies…

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Q: 2. Evaluate the integral // xz? dy dz + (x²y – 2³) dz dx + (2xy+ y²z) dx dy where S is the surface…

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Q: Let (r, y, z) 2 F := + y2 + and let S be the boundary of the sphere x2 + y² + z2 = 1 oriented…

A: Divergence theorem: Let D be a simple solid region and s is the boundary surface of D. Let F→ be a…

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

A: Given vector field is  Divergence theorem

Q: Let E be the solid bounded by the xy-plane and paraboloid z = 4 - x2 - y2 so that S is the surface…

A: According to question, Paraboloid z =4-x2-y2,Fx, y, z = xz sinyz + x3 i + cosyz j + 3zy2-ex2 +y2 k

Q: 1- Calculate the surface integral ffs G(r)dA, where G = (1+z2)} and S is given by the…

A: 1.  G=1+92xz12 ru,v=u,v,2u3 G(r)=1+92u.2u312 o≤u≤2, 0≤v≤2 is the limit of the integration.…

Q: 2. Use the Divergence Theorem to evaluate F.dS where F = 4x i + (1 – 6y) j +k and S is the surface…

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Q: Use the Divergence Theorem to evaluate F. dS where F = (2²a +y?. + tan (2), a?z – 1 2, and S is the…

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Q: Use the Divergence Theorem to evaluate F. ds, where F(x, y, z) = z?xi + (+ tan zj + (x²z + y2)k and…

A: We are given a closed surface integral and we have to use the Divergence Theorem to evaluate that.

Q: Evaluate the surface integral.∫∫S (x2+y2+z2) dS, S is the part of the cylinder x2+y2 = 9 between the…

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

A: Given : F = x,y,z

Q: Let S: x2 + y? = 9,0szS7 be the surface of a closed cyllnder bounding a volume T.F=Ly², x², z²J. Use…

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Q: Use the divergence theorem in R to evaluate the surface integral I of the two-form w = (zry + e5v)…

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Q: Evaluate the triple integral ||//. z°e" dV where E is bounded by the parabolic cylinder z = 16 – y?…

A: The volume of a solid can be calculated using the triple integrals. The unit dV can be written as…

Q: Let S:x+z = 16, 0<y<3 be the surface of a closed cylinder bounding a volume T. F= [z, y, x] Use the…

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Q: Let Q be the solid region x2 + y? + z2 = 16. F(x, y) = xỉ + 4yj+ 2z k . Use the Divergence Theorem…

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Q: Evaluate the surface integral of D enclosed by the rectangular parallelepiped formed by the planes x…

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Q: F. dS, where F(x, y, z) = z²xi + 2) + (x2z + y2)k and S is the top half of the sphere x² + y² + z? =…

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Q: 2- Use Gauss's divergence theorem to evaluate the surface integral / ry dydz + 2y°dxdz + y?zdxdy),…

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Q: Verify the Divergence theorem for the following surface integral A’. dŠ where A (x,y,z) = x²z³i+…

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Q: Use the divergence theorem to calculate the surface integral F. dS; that is, calculate the flux of F…

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Q: Use the Divergence Theorem to calculate the surface integral Sl. F. dS ; that is, calculate the flux…

A: We will find out the required value.

Q: Use Stoke's theorem to compute the integral SL(V × F) · n dS, where F(x, y, z) = (xz, yz, xy) and S…

A: We have to compute ∬(∇×F)·n dS , where F(x,y,z)=(xz,yz,xy) and S is the portion of the sphere…

Q: O Use the divergence theorem to evaluate F-dS, where F (x, y, z) = zxi – e"j+ 4zk and S is the…

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Q: Evaluate the triple integral (* + 1y) dV where E is bounded by the parabolic cylinder y = 32² and…

A: We have to find the triple integral  ∫∫∫E(x+y)dVwhere E is bounded by the parabolic cylinder y=3x2…

Q: 5. Compute / | F. aš, where F = (y, -x, 2z) and S is the surface of the sphere of radius 2, centered…

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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: .Use the Divergence Thm to calculate the surface integral F.nds (a) S is the boundary of the cube…

A: We will find out the required value.

Q: Evaluate the triple integral. SSS 5x²e" dv I where E is bounded by the parabolic cylinder z = 1 - y²…

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Q: Use the Divergence Theorem to evaluate the surface integral |a? – 3y + 62) ds, S. here S is the…

A: Given the surface integral ∬Sx2-3y+6zdS, where S is the sphere x2+y2+z2=9. The integrand is a scalar…

Q: 2- Use Gauss's divergence theorem to evaluate the surface integral || «ry°dydz +2y°dxdz + y°zdzdy),…

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Q: 4. Evaluate the surface integral ff. F.dS by the divergence theorem where F = (3xy2, yx? -…

A: S is the surface x2+y2≤25, 0≤z≤2 So surface S is a cylinder with radius 5 as shown in below figure.…

Q: Evaluate the surface integral. ∫∫s(x2 + y2 + z2) dS, S is the part of the cylinder x2 + y2 = 9…

A: To evaluate :  ∫∫s(x2 + y2 + z2) dS, S is the part of the cylinder x2 + y2 = 9 between the planes z…

Q: Let S be the piece of the surface z = 9– x2 that lies inside the cylinder x2 +y? = 9, and let F =…

A: According to question, Given Surface, z = 9 -x2 ,This surface lies Inside the cylinder x2 + y2 = 9,F…

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

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Q: Assume that S is the disk 2 + y² <4 in the plane z 0 oriented upwards with positively oriented…

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Q: Verify the formula F.ndS = div F dV G in the Divergence Theorem by evaluating the surface integral…

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Q: Use the Divergence Theorem to evaluate  F · dS,  where  F(x, y, z) = z2xi +  + ((1/3)y3 + cos z)j +…

A: Given data:

Q: Let S:x+z? = 16,0sys3 be the surface of a closed cylinder bounding a volume T.F=[z?, y, x²]. Use the…

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Q: Use the Divergence Theorem to find F- ndS, where F = (y°z,x²z,z') and the surface S is the sphere x…

A: See the details solution below

Q: Evaluate the triple integral zdV, where E is bounded by the cylinder y² + 2² = 4 and the planes z =…

A: Given problem:- Evaluate the ∫∫∫E z dV, where E is bounded by the cylinder y2+z2=4 and the planes z…

Q: Verify the formula F-nds = | div F dv in the Divergence Theorem by evaluating the surface integral…

A: Given that Fx,y,z=2xi→-yzj→+z2k→ Where σ is the paraboloid z=x2+y2 capped by the disc x2+y2≤16 in…

Q: Use the Divergence Theorem to determine the value of f f, F•dS where F (x, y, z) = (3x – x² )i+…

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Q: Verify the formula F-n dS : div F dV in the Divergence Theorem by evaluating the surface integral…

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Q: Verify the Divergence Theorem by evaluating F.N dS as a surface integral and as a triple integral.…

A: Here, using the Divergence Theorem, ∯F·Nds=∫∫∫v∇·Fdv Here, F=xy,z,x+y So, ∇·F=∂F∂x,∂F∂y,∂F∂z=y,0,0

Q: Let F(x, y, 2) = 32?ai+ (y³ + tan(z))j+ (3x²z- 4y?)k. Use the Divergence Theorem to evaluate fs F dS…

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Q: Verify the Divergence Theorem for the F(x,y,z) = 4xi +4yj +2z? k over the surface S of x² +y² and z…

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