1. Let S be the part of the sphere x2 + y? + z² plane z = 3, oriented so that the "top" of the surface coincides with the "outside" of the original sphere. Compute ffs(V × F) · dS, where = 25 that lies above the z-y x+Vz-2 F(x, y, z) = \sin(4z)+e²*

Q: Evaluate the surface integral F. dS where F = (4x, -5z, 5y) and S is the part of the sphere x² + y +…

A: The solution is given below:

Q: Which of the following planes is tangent to the sphere x² + y + z' = 17 at the point (2, -2, 3)? %3D…

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Q: 18. Evaluate J (V x F) · N dS where F = yi + (x- 2xz) j- xy k and S is the surface of the sphere x…

A: Stokes theorem is used

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: 5. (a) Evaluate the surface integral /3² dS where o is part of the cone z = VT + y² that lies…

A: As you post multiple questions According to the guideline I only solve the first problem please post…

Q: 16. Let F(x. y, z) = xyi+ xzj+yzk and C be the curve of intersection between the sphere z² +7+z² =…

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Q: Evaluate the surface integral f F. dS where F = (-2x, –5z, 5y) and S is the part of the sphere x2 +…

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Q: 28. F(x, y, z) = x³i+y°j+z°k, and S is the sphere of radius 3 and centered at the origin.

A: divF→=ddxx3+ddyy3+ddzz3        =3x2+3y2+3z2        =3x2+y2+z2  by divergence theorem,…

Q: 1. Verify Stokes' Theorem for F = (2x - y)i - yz²j - y² zk, where S is the upper half surface of the…

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Q: 28. Show that the circulation of F(x, y, z) = (x², y², z(x² + y²)) around any curve C on the surface…

A: Given figure Stoke's Theorem Here take single integral over closed path C and double integral over…

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: Evaluate || f (x,y,z) dS where S is the portion of the sphere centered at the origin of radius two…

A: we have the equation of sphere x2+y2+z2=22s is the portion bounded between xy plane and z=1 in the…

Q: Integrate H(x, y, z) = yz, over the part of the sphere x2 + y2 + z2 = 4 that lies above the cone z =…

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Q: 5. Evaluate the surface integral S_(x+ y) ds ; o is the part of the plane 2x + y +z = 3 that lies in…

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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: (a) Compute directly I = S Ss F-dS, where S is the surface of the sphere x? +y? +z? = 4 and F =…

A: X²+y²+z²=4=2² Therefore radius of sphere is 2.

Q: 2-By revolving the semicircle: X= r cos t, Y=r sin t, (0<t<n) About the x-axis, Show that the…

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Q: 15- Evaluate the surface integral f, xy dswhere x is the part of the plaue y = 1-x-z that lies in…

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Q: Consider the solid E determined by the inequalities x² + y² + z≤ 4, and let S denote the surface…

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

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: Let Y be the portion of the sphere x² + y² + (z − 2)² = 4 which lies below the plane z = 3 with…

A: Given  ∫Cz2+3yi→+2z-2xj→+z2k→.dr→ The portion of the sphere x2+y2+z-22=4 and plane z=3

Q: (b) Verify Stokes' theorem on a region which is a portion of the sphere x? +y? +z² = 16 which lies…

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Q: Suppose f(x, y) satisfies the basic existence and uniqueness theorem in some rectangular region Rof…

A: Consider the given: Let the function f(x,y) satisfies the basic existence and uniqueness theorem in…

Q: (b) Verify Stokes' theorem on a region which is a portion of the sphere r2 +y² +2 16 which lies…

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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 Stokes' Theorem to evaluate , Vx F ·n dS where F(x, y, z) = and C is the part of the sphere…

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Q: Consider the set S given by S = {(x, y, z) ER³ : 2 = 2x² - y²}, where A E R\{0} is constant.…

A: As per the question we are given a quadratic surface as : S = {(x,y,z) ∈ R3 | z =2x2 - λy2} And we…

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: 1. Consider the surface which is the part of the sphere r? + y? + 2? = 4 that lies above the cone z…

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Q: 4. Compute |F- dš, where F = (y, –x,2z) and S is the surface of the sphere of radius 2, centered at…

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Q: 10/ Compute e Fids, where S is the surface z =N16-x²-y² oriented up ward and F = <y,-x, zZ %3D 工=?

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Q: 4. Find the area of portion of the sphere x²+y²+z = 9 between the planes z = 1 and 2 = 0. Draw the…

A: Surface area of a surface z=f(x,y)  A=∫∫D1+fx2+fy2dA Here D is a point on xy plane

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: Evaluate J CurIF. [cariF- nas F(x,v,z) = - yi + xj + zk, and S is part of the sphere z =9-x2 - y?…

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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.…

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Q: Evaluate the surface integral. JI (12 + y2 + 2) ds S is the part of the cylinder x2 + y2 = 9 that…

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Q: 13-Evaluate JJI zdV E where E lies between the spheres x2 + y? + z2 = 1and x2 + y? + z? = 4 in the…

A: The given integration is , Let, I=∫∫∫EzdV change the integral into spherical coordinates Put x=r…

Q: If o is the sphere of radius 2 centered at the origin, then (x² + y² + z³) d$ =

A: Given the sphere

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

A: We need to find the surface integral. We will need basic knowledge of parametrization of vector…

Q: Given that o is the part of the surface z = 4 – x in

A: Given, the surface z = 4 - x in the first octant from y = 0  to  y = 2. If ψx, y, z=xyz, F(x, y, z)…

Q: 6. If S is the surface of the sphere x + y² + z? = 9. Prove that curl F- ds = 0. %3D

A: Solution:-

Q: 1. Find the surface area of the part of the sphere x+ y +z = 4 that lies above the plane z =1.

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Q: 1. Evaluate the surface integral [[' xz² ds ; o is the part of the cone z = x² + y² that lies…

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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: (b) Use Stokes' Theorem to evaluate (f, curlF .n dS, where F(x,y, 2) = x²yzi + yz?j + z³e™Yk and S…

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Q: Let S be the upper surface of the sphere x- 2x+y+z = 0 and let r be the boundary. Find [.curi(F) -…

A: S is upper surface, the equation  x2-2x+y2+z2=0  x-12+y2+z2=1  Hence, is a sphere centered at (1, 0,…

Q: Evaluate the surface integral. v² as S is the part of the sphere x2 + y2 + z2 = 64 that lies inside…

A: A scalar function f(x,y,z) is a rule that is used to associate a number or quantity with a…

Q: Find the extreme values of f(x,y,z) = x-yz + 1 on the intersection of the plane z= 3 with the sphere…

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Q: Evaluate J (V x F) · N dS where F = yi + (x – 2xz) j – xy k and S is the surface of the sphere x +y²…

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