If F = 2y i - zj + x²k and S is the surface of the parabolic cylinder y²= &x in the first octant bounded by the planes y =4 and z = 6, evaluate F.n dS. Ans. 132

Q: 15. If the surface S has projection R on the xy-plane, show that F.īdS = S da dy ה.F provided |ñ -…

A: Question: If the surface S has projection R on the xy-plane, show that ∬SF¯·n¯dS=∬RF¯·n¯dxdyn¯·k¯…

Q: Find the integral surface of the linear PDE xp - yq = z which contains the circle x² + y? = 1, z=1.

A: From Lagrange method

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Q: If A = 2yz i - (x +3y - 2)j + (x²+z)k, evaluate |(V×A) •n dS over the surface of intersection of the…

A: Given:  A=2yzi-x+3y-2j+x2+zk, With the cylinder x2+y2=a2,x2+z2=a2 Now, calculating ∇×A.…

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Q: F = 2yj+z k and S is the surface y? + z? = 4 between x 0 and x = 3 – z. Compute SL.F.ds

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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.

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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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A: To solve the following problem

Q: Evaluate ſ F . n dS if F = (x + y² ™)ī − (2x)j + (2yz)k and $ is the surface of the plane 2x + y +…

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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: F = (x² – y²,x, 2yz); C is the boundary of the plane z = 6 – 2x – y in the first octant.

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A: Let us consider a line integral ∫CP dx+Qdy then it can be written as  ∫D∫∂Q∂x-∂P∂y dA  Where, P and…

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Q: Verify Stoke's for F = (2x – y)i – yz²j – y²zk where S is the upper half of the sphere x? + y? +z² =…

A: Solution is given below

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Q: 2. Evaluate ([.(Vx F). ds, where S is the surface of the paraboloid x2 + y2 + z = 4 above the…

A: We have given F→=(x2+y-4)i+3xyj+(2xz+z2)ksurface is given x2+y2+z=4 and above the xy planediagram of…

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Q: 2 Use Green's Theorem to evaluate Jx*ydx-xy dy, where C is the circle 2+ y? – 2601 with…

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A: The given surface are y=x and y=20-x. The area between the two surfaces is obtained by ∬dx dy.

Q: 3. Evaluate ox?y dx + xy? dy using Green's y 2 Theorem, where C is positively oriented by line y =x…

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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: A solid G is bounded by the paraboloid z = 4 – x2 – y?, the cylinder x2 + y? = 4 and the plane z =…

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Q: O Given the initicl bouridary conditions y(0) =0, y'c0) = 1, letrmine the, solntiem of %3D…

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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: Use Green's Theorem to evaluate (1in z – 2y) dz + (x – ycos y) dy, where C is the circle (x – 4)² +…

A: Given: C is the circle x-42+y-42=9. ∮Cxsinx-2ydx+x-ycosydy. To find: Using Green's theorem, value…

Q: Evaluate the following closed surface integral: Where F = (2x + 3)î + (3y – 4)) + (e*y + 5z)k.

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

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Q: Which of the following is the least distance from the origin to the plane 2x + 2y + z = 1? Lütfen…

A: We have to find

Q: 3.Evaluate || F. ds where F= zi+x j +y k and 'S' is the surface of the cylinder x'+ y=1 contained in…

A: Given that ∬S F→·dS→ where  F→=z i^+x j^+y  k^ .......1and  S is the surface of the cylinder x2+y2=1…

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: 17. F = |r|r, where r = xi + yj + z k, S consists of the hemisphere z = /1 – x² – y2 and the disk x?…

A: The solution are next step

Q: Evaluate SF. NdS where F(x, y, z) = xi + yj + zk__ and the surface S is given by S S: z=4-x² - y²…

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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: Evaluate || 4.nds , where A=zi+xj – 3y°zk and S is the surface of the cylinder x² + y =4 included in…

A: Let ∅=x2+y2-4  Then the normal to the surface S is ∇∅=2xi+2yj The unit normal is…

Q: Evaluate Sſ Ands where A= (x+y*)î – 4xj +4yzk and s is the surface of the plane, 2x+y+2z=8 in the…

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Q: Sketch the reglons enclosed by the glven curves. y = Vx, y= X= 16 y y 6 4 4 3 3 2 10 15 10 15 O-1…

A: Given : y = √x  y = 1/3 x  x = 16

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: Q3// Evaluate ſ. F dS where F = xi + yj + zk and S is the surface x? + y? = 9_in the first two…

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