18. Suppose that V ·F, = V • F, and V × F, = V × F, over a re- gion D enclosed by the oriented surface S with outward unit normal n and that F1 • n = F,•n on S. Prove that F¡ = F, throughout D. %3D

Q: Let C be the positively oriented square with vertices 0, 1, 1 + i, i. Evaluate using the definition…

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Q: Compute the line integral (Баллов: 2) Jo ds, Cis the right half of circle x2 + y 16 (counter…

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Q: Let a , b and c be the last three digits of your student ID and let d=max(a,b,c). How many points…

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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: 7.H Jse the divergence theorem to evaluate where S is the sphere x+y? +? = 4 oriented with outward…

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Q: Let D be any surface and let F = (P depends only on x, Q only on y, and R only on z). Then that…

A: We will use the stokes theorem to check if the statement is true.

Q: Let a , b and c be the last three digits of your student ID and let d=max(a,b,c). How many points…

A: Let a,b and c be the last three digits of your statement ID and let d=max(a,b,c) Here we have to…

Q: Consider the solid that is bounded above by x² + y? + z? = 169 and below by z=5. %3D

A: In this question, we have to write the triple integral to find the volume of the solid bounded by…

Q: Q1// Evaluate , F ds where F = yi + xj + 4zk and S is the surface x? + y? = 9 in the first octant…

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Q: Compute the line integral ∫_C F · dr, where F (x, y) = 〈−x^3, xy〉 and C is part of circle x^2 + y^2…

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Q: 21. Show that the volume V of a region D in space enclosed by the oriented surface S with outward…

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Q: Evaluate the surface integral xyzdS where S is the part of the plane z = 1 + 2x + 3y that lies above…

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Q: Assume a to be a nonzero real no.. Determine the area of the portion of surface xy = az in the fir…

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Q: Consider the surface S: z = 4 - x2 ,bounded by y = −5(x2 - 1) and y = 0, just as it is shown in the…

A: We have to find the area of the surface S, and we will use surface integral to do that.

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: (9) Evaluate the surface integral / S, czdS, where o is the part of the plane r+2y+3z = 6 that lies…

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Q: Let S be the part of the sphere r² + y² + z² = 4 which lies above the cone z = 3r² + 3y². Express…

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Q: Evaluate where G is the solid in the first octant bounded by paraboloid : = + y, the cylinder r+ y 9…

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Q: 2. Fnd the area of the sur face in the first octant cut from the cone z2 = x2 y2 by the plane x +y =…

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Q: The area of the surface p = 2, 0sz<5,0<¢< T is 201 Select one: O True O False

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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) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

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Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, z) = (2x²y² + lIn 2) i – ry° j – tanh(x² + y³)…

A: Given function  F(x, y, z) = (2x2y2+lnz)i -xy3j -tanh(x2+y2)k we have to find ∬σF.nds by using guass…

Q: (a) Evaluate the following surface integral yex²+z² dS, where o is the portion of the surface of the…

A: Given the integral ∬σyex2+z2dS we have evaluate the given surface integral where σ is the portion of…

Q: 2. Let S be the part of the cone z = √² + y² that lies between z = 1 and z = 3 with upward pointing…

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Q: Let F = 1xi + 5yj and let n be the outward unit normal vector to the positively oriented circle x² +…

A: To find the flux we have to first compute unit normal vectors by using formulan=Gradient…

Q: 45. If A = (y – 2x )1 + (3x + 2y)j, compute the circulation of A about a circle C in the xy plane…

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Q: State what the Divergence Theorem says, using your own words. You may assume that E is a solid in R³…

A: We can state the Divergence theorem .

Q: Let F(x, y,z) = (-xe² +3x, -ye² + arctan(x? + z), 2e²) and S the portion of the sphere acoording to…

A: Given that Fx,y,z=-xez+3x, -yez+arctanx2+z2, 2ez. Also, S=x,y,z| x2+y2+z2=1, z≥0. To determine the…

Q: Let f(x, y, z) = In (x² + y + z?). The level surface f (x, y, z) = In 4 is a: Elliptical cone around…

A: Given  f (x,y,z) = ln( x2+ y+z2),To find : level surface f= ln4

Q: Consider the surface given by x2 + y? + z? = a². Show that every normal line to this surface passes…

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Q: 3. Please give clear explanations

A: The parametrization of a surface S is given by       G(u, v)=(u, v, uv) where D is u2+v2≤1.

Q: Q4// A surface S is defined by x2 +z = 1 and the bounded by the planes y = 0,y = 2,x 0,z = 0 in the…

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Q: Let S be a square plate with boundary C, having each side of length 2 units, lying in the plane x =…

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Q: Calculate the area of the surface S. x2 y² _ z2 that lies between z = 2 and z = 6. S is the portion…

A: Explained below

Q: If R is a closed bounded region in the xy-plane whose boundary curve is C with arc lengths and…

A: i. Consider that F be a vector field and F→=Mi+Nj and C is a simple closed curves positively…

Q: If F=xi+yj, prove that JF.ñds equals [V.FdA, where R is the disk of radius C R "a" and centre at the…

A: Given Data: F =xi  +yjTo Prove:∫C F . n ds =∫R∇·F dA where R is the disk of radius ''a'' and centre…

Q: Q19// A scalar filed V = xyz exists over a surface S defined by x? + y? = 9 bounded by the plane…

A: Given scalar field, V=xyz This field exists over a surface S defined by x2+y2=9 bounded by the plane…

Q: Compute ,F · dS for F = (xy, 5y,0), S being the cone z² = x + y²,x² + y? 0, with the normal %3D…

A: The vector field is F=⟨xy, 5y, 0⟩ and the parametric equation of the cone is  Rr, θ=rcosθ, rsinθ, r…

Q: 6. Evaluate the surface integral where o is the portion of the cone z = Vr? + y? between the planes…

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Q: Evaluate ||x'dydz + x ydzdx + x*zdxdy over the surface bounded by z= 0 ,z %3D = h, x² + y? = a² %3D

A: here F=x3 , x2y, x2zby divergence theorem ∫∫sF.ds  =∬∫E divF dvhere S is surface bounded by  0 , z=h…

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: The measurements x1 = 10m, x2 = 8m, and x3 = 6m represent the sides of a rectangular parallelepiped…

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Q: Use Stoke's theorem to evaluate the integral xydx+x'dy – 4x’ydz where C is the boundary of the…

A: Here the surface is the plane of the square with vertices A1,0,1, B1,1,1, C0,1,1 and D0,0,1.  To…

Q: 2. Suppose that S is a surface in R3 with unit normal vector N surface area of S is given by (n1,…

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Q: Q19// A scalar filed V = xyz exists over a surface S defined by x? + y? = 9 bounded by the plane…

A: Since you have posted multiple questions. I am answering the first one. Please post the question…