Exercise 7.2.8. Apply Stoke's theorem to find F - dr, where F = yi+zj+ækover the surface S, %3D which completely lies in the plane x + z = a bounded by the curve C of intersection of the hemisphere x? + y² + z² = a² with the plane. upper

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: Consider the rormal line to the surface z= z + y? (parabokoid) at the point P(1,1, 2) Find the point…

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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: 1. Let F(x, y, z) = yi+ xj+ yzk and the surface S be the part of the paraboloid z = 4 – x2 – y? that…

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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: The tangent plane to the graph of the surface z=In(x2+y?) at the point (1,0,0) contains point…

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Q: 1. Let S C R be the surface defined for all 0 < x < 1, -1 < y < 1 by z = r² + y. Compute ffsa ds

A: It can be solved by partial differentiation and integration

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: 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: An equation of the tangent plane to the surface z = 4xy² + 3x²y² at the point (1, 1, 7) is: Select…

A: z= 4xy2+3x2y2F(x,y,z) =…

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: Given F(x, y, z) = -yzi + xzj – xyzk be a vector function that passes through a surface S of z + x2…

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Q: EXERCISES 2.10 1. Use Stokes' Theorem to evaluate E.dr for F=(xz,-y,x²y), where C is the boundary of…

A: Given: F=&lt;xz,-y,x2y&gt; C surface, S is bounded by x=0, y=0 and 2 not include xy-plane Now,…

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: If F = 2y i - zj + x²k and S is the surface of the parabolic cylinder y²= &x in the first octant…

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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: Let (S) be the surface given by the following equation. x²+y? = 1+ z?. An equation of the tangent…

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Q: Determine the projection of the following surface on the xz plane. (xz trace) 4x – + = 0

A: Given equation of surface is 4x-y24+z28=0

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) &amp; (0,1,1) &amp;…

Q: What is the equation of the plane to the surface z = e3y sin(3x) +2 at the point P ,0, 3 O x - y -…

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Q: 4. Evaluate fo F• dr by the Stoke's theorem, where F = -y? î+x j+ z² k, and C' is the curve of…

A: Given :

Q: If F = 3zi + 4xj + 2yk and the boundary of the portion of the paraboloid for which z 2 0. C is the…

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Q: EXERCISES 2.10 1. Use Stokes' Theorem to evaluate F.dr for E = (xz,-y,x²y), where C is the boundary…

A: Applying stokes theorem we solve it

Q: 10. Let D be the solid inside the cylinder r= cos , above the xy-plane and under the paraboloid…

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Q: 59. Find all points on the portion of the plane x+y+z35 in the first octant at which f(x, y, z) = xy…

A: We have to find all points on the portion of the plane x+y+z=5 in the first octant at which f(x, y,…

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: • What is/are the trace(s) of the surface (x – 1)? 22 = 1 on the rz-plane? 36 a. hyperbola b.…

A: For the given surface

Q: Check whether the given surface (x – a)² + (y – b)² + (z – c)? 16 is developable or not. -

A: Given surface(x-a)2+(y-b)2+(z-c)2=16orz=c+16-(x-a)2-(y-b)2If the surface is developable…

Q: 7. Use the Divergence Theorem to evaluate ff(F n) dS, where S is the top half of the sphere r2 +y +…

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Q: Consider D to be the region below the paraboloid z = 3+x² + y² , inside the cylinder x + y² =1 and…

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Q: In exercises 13-18, compute the surface area of the surface obtained by revolving the given curve…

A: Solving part (b) as requested. Given:

Q: Let (S) be the surface given by the following equation. x²+y? = 1+ z?. An equation of the tangent…

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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: 11. Let D be the solid inside the cone z = √²+ y², inside the sphere x² + y² +2²=9 and above the…

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Q: In each of the following cases, draw each solid Q bounded by the surfaces whose equations are given…

A: To graph the below surface that is in the first octant. x2=−z−3z=2y+z=3

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: (b) Use Stoke's Theorem to evaluate the line integral $ (3? – 22) dr + (z² – x²) dy + (y² – x²) dz,…

A: Stokes theorem  ∮CF.dr=∫∫SCurl F .dS Where S be a smooth surface that is bounded by a  closed,…

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: Consider the surface: S:z = 4 – a?, in the first octant and bounded by x² +y°, as shown in the…

A: The equation of the surface given is as follows: z=4-x2 The surface is bounded by the curve x2+y2=4.…

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

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Q: Verify Stoke's Theorem for A = 3yî – xzj + yzk, where S is the surface of the paraboloid 2z = x + y?…

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Q: Let C be the intersection of the surface x² + y² + z² cylinder x² + 3y² = 4, z > 0. = 9 and the

A: As per the question we are given the equation of a sphere and a cylinder (with elliptical cross…

Q: Let h, k, m, neR such that n > m >0. A general equation corresponding to the above surface is: A) (z…

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Q: 7. Consider the Surface S:z = 52² + 5y?. An equation of the plane tangent to Sat the point P (3, 3,…

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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) Evaluate the surface integral ſ 2x²y dS over the surface y² + z² = 1 betwee x = -1 and x = 5.

A: a We have to evaluate the surface integral ∬σ2x2ydSover the surface y2+z2=1 between x=-1 and x=5.…

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

A: Surface integration

Q: The tangent plane to the graph of the surface z=In(x<+y<) at the point (1,0,0) contains point…

A: and point (1,0,0)

Q: 10. Sketch the following surfaces (graphs of functions) for a = 2, b = 1: (a) Elliptic paraboloid.…

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