Use the Divergence Theorem to evaluate  F · dS,  where  F(x, y, z) = z2xi +  + ((1/3)y3 + cos z)j + (x2z + y2)k  and S is the top half of the sphere  x2 + y2 + z2 = 4.  (Hint: Note that S is not a closed surface. First compute integrals over S1 and S2, where S1 is the disk  x2 + y2 ≤ 4,  oriented downward, and S2 = S1 ∪ S.)

Q: Evaluate the line integral using Green's Theorem and check the answer by evaluating it directly.…

A: Here , we have given an integral and we have to integrate over a rectangle whose vertices are given…

Q: 4. Use the divergence theorem to evaluate the surface integral F dS where F - xzi + yzj + z?k and S…

A: we will use gauss divergence theorem here.

Q: |Use the Divergence Theorem to evaluate || F•ndS where F(x, y,z)=(x+cos z)i+(2y+sin z)j+(z+e*)k .n…

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Q: According to the divergence theorem, one of the following integrals represents the surface integral…

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Q: Evaluate the triple integral. I/| (x - y) dv, where E is enclosed by the surfaces z = x2 - 1, z = 1…

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Q: Calculate the line integral [(of).do where f(x, y, z) = x ² ¹² + 4 xy² z yz с + е хуча and C is the…

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Q: Evaluate the integralE F•n dA directly or, if possible, by the divergence theorem if F= y?, x², z²],…

A: In this question, we evaluate the surface integral by using gauss divergence theorem.

Q: 4. Use the divergence theorem to evaluate the surface integral F. dS where F = xzi + yzj + z²k and S…

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Q: Evaluate the triple integral (x + 2y) dV where E is bounded by the parabolic cylinder y = 7x² and…

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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 we use divergence theorem .

Q: Evaluate the following line integral by two methods: (a) directly; (b) using Green's Theorem. $y²dx…

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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: Evaluate double integral f.ns where F=xi-yj+(z2-1)k and s is closed surface bounded by the planes…

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

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Q: 387. Use the divergence theorem to calculate surface integral F. dS when F(x, y, z) = x²z³i+ 2xyz³j+…

A: Given that: F(x,y,z)=x2z3i+2xyz3j+xz4k and S is the surface of the box with vertices (±1,±2,±3).

Q: Use Divergence Theorem to evaluate the flux integral ∫∫F·ndS, where F = 〈x^2, y^2, z^2〉, S is the…

A: Divergence theorem:∫∫sF.⇀  N^dS =∬∫DdivF⇀dV , here D is the solid bounded by the closed surface S.

Q: Evaluate the surface integral. (z + x2y) ds S is the part of the cylinder y2 + z2 = 16 that lies…

A: Given ∫∫S(z+x2y)dS where S is the part of the cylinder y2+z2=16 that lies between the planes x=0 and…

Q: Use Green's Theorem to evaluate the line integral ſ (y + e√x) dx + (2x + cos y²) dy, where the given…

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Q: Evaluate the triple integral ∭E(x+2y)dV where E is bounded by the parabolic cylinder y=2x2 and the…

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Q: Calculate the integral of f(x, y, z) = 3x² + 3y2 + z4 over the curve r(t) = (cos(t), sin(t), t) for…

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Q: Evaluate the triple integral. SIJE3Y dV, where E is bounded by the planes x = 0, y = 0, : = 0, and…

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Q: Evaluate the integral F F•n dA directly or, if possible, by the divergence theorem if F= [y?, x²,…

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Q: (b) Evaluate the surface integral S.S x² dS over the surface x² + y² = 1 between z = 1 and z = 4.

A: x=cos(u), y=sin(u), z=v  0≤u≤2π, 1≤v≤4 S(u,v)=< cos(u), sin(u), v >

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

A: Here we use divergence theorem

Q: Set up an iterated double integral for the flux integral S ſ, F•dS where F(x, y, z) = ri+ yj+2k and…

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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 F.dS, where F (x, y, z) = zxi – e"´j+ 4zk and S is the…

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Q: According to the divergence theorem, one of the following integrals represents the surface integral…

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Q: Evaluate the integral // Fn ds where F = (2r+32)i-(rz+y)3+(y² +2)& and S is the surface of the…

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Q: Compute the line integral of the scalar function f(x,y,z)=2x^2+8zf(x,y,z)=2x2+8z over the curve…

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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 triple integral /// IdV where E is the solid bounded by the paraboloidz 7y +7 and z=7.

A: To find the triple integral

Q: (a) Compute the surface integral || a xy dS, where S is the quarter cylinder x2 + y? = 1; x, y > 0…

A: Given that The surface integral is ∬Sxy dS, where S is the quarter cylinder x2+y2=1; x,y≥0 between…

Q: 2. Compute the scalar line integral xy ds where C is the part of the circle of radius 2, centered at…

A: Put x=2cost , y=2sint

Q: Let S: x+y =9,0<z<7 be the surface of a closed cylinder bounding a volume T. F= Ly, x², z²]. Use the…

A: Q - 1 is  a simple formula based. Requires no discussion.

Q: Let R:x²<ys 3 be the closed region having boundary C with counter-clockwise orientation. F=[X…

A: We are given that the closed region R: x2 ≤ y ≤ x/3, is having boundary C with counter-clockwise…

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A: The point of intersection of the cylinder x=y2 and y=x2 is 0 and 1.

Q: Evaluate the triple integral -p (x + 6y) dV where E is bounded by the parabolic cylinder y = 8x² and…

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

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Q: Use the Divergence Theorem to evaluate Double intergal Z Z of S  −→F . d−→S Where −→F (x, y, z) =…

A: Given that, F=3xy2i^+xezj^+z3k^ and the surface S:y2+z2=1,x=-1,x=3

Q: Evaluate the line integral F dr, where C is given by r(t) = (t², t, vT), 0<ts2 and (x, y, z) = (z,…

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Q: Use Green's Theorem to evaluate the line integral along the positively oriented closed curve C.…

A: In the given question we have to choose the correct options from the given options. we have to…

Q: A. Evaluate and sketch the line integral :1 = f yx dx + (2X – Y)dy around the region bounded by…

A: Greens theorem

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.