Evaluate F.ndS for the given F and o. (b) F(x, y, z) = (x² + y) i+ xyj– (2xz + y) k, o: the surface of the plane x + y + z = 1 in the first octant |

Q: Find the surface area for the portion of the surface f(x,y)=1-x² + y that lies above the triangular…

A: Solution :

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: Evaluate ff, F.dS, where F= (4xz, -y², yz) and Sis the surface of the cube bounded by the planes x =…

A: We will use divergence theorem to find given integral  ∫∫SF→dS →= ∫∫∫EdivF→ dV Where E is  solid…

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: Verify: Green's theorem in the plane for f(2x-y³) dx-xydy, where C is the boundary of the region…

A: Given information ∫c2x-y3dx-xydy Here, C is the boundary of the region enclosed by the circle…

Q: Explain briefly how Vø relates to surfaces of constant o(x, y, x). Hence, use Vø to find points in…

A: For the surface of constants ϕx,y,z=k, the gradient vector ∇ϕx0,y0,z0 is orthogonal to the level…

Q: 13. Evaluate . F-di ; where F(x, y, z) = (1,x+ yz, xy – VE) and C is the boundary part of the plane…

A: Using stokes theorem

Q: Calculate F - dS where F (e* sin y, e² cos y, yz²) S is the surface of the box bounded by the planes…

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Q: 25. Evaluate F.N dS where F= yl + 2xj –- zk and S is the surface of the plane 2x +ye 6 in the first…

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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: Calculate F dS if F(x, y, z) = x²yei+ xyj – 2xye k, where S is the surface of the pox bounded by the…

A: Given-F(x,y,z)=x2yezi+xy2j-2xyezk To find- ∬SF.dS ,where S is the surface of the box bounded by the…

Q: וווכu nt that enclosed by the surface z = 12 - x 2 - y 2 and the plane z = A. A=1

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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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Q: 2. Evaluate F.ndS for the given F and o. (a F(x, y, z) = (x² + y)i+xyj- (2xz + y) k, o: the surface…

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Q: 3. Evaluate if G is a solid in the first octant bounded by the plane y+ z = 2 and the surface y = 1–…

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Q: Let G be the solid enclosed by the plane z= y, the xy-plane, and the parabolic cylinder y = 1-x². If…

A: Consider the given information,

Q: The parametric representation of the surface x² – 8x + z? = 0,0 < y < 5 is a) ř(u, v) = (4 cos u , 4…

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

A: Given, F=x2-y2,x,2yz Find the line integral ∮CF·dr in the following. Here C is the boundary of the…

Q: Use Stoke's theorem to evaluate f(V × F) · ndS, where F(x, y, z) = (x², y², z²), n is oriented in…

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Q: Evaluate I zdS when (S) is the part of the plane 2x – 13z = 0 that lies above the domain D bounded…

A: first we will take the projection of s onto xy-plane then calculate the required integral using the…

Q: Let W be the region that is cut out from r² + y² < 4 by the paraboloid z = r² + y² and the plane z =…

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Q: 9. Evaluate F - dS, where F(x, y, z) = xy²i+x²yj+y² k, and S is the positively oriented surface…

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Q: The parametric representation of the surface y2 - 4y + z2 = 0,0 < x < 3 is a) 7(u, v) = (v, 2 + 2…

A: solution:y2-4y+z2=0 , 0≤x≤3considery2-4y+z2=0∴y2-4y+4+z2=4∴y-22+z2=22

Q: Sketch the surface z = f(x,y). f(x.y) = 4x² + 4y² + 2 OC. OD.

A: Given function is fx,y=4x2+4y2+2. We have to sketch the surface of fx,y=4x2+4y2+2.

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

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Q: Calculate S,F.as if F(x,y,z) = xi+ yj and the positive oriented surface o is : The portion of the…

A: To find, ∫∫SF→.dS→ Given that, F→x,y,z=xi^ + yj^   Part i Given portion of plane x+y+z=1 bounded by…

Q: . $F•ds where F = (x, 2y, z+3x) and S is the closed surface (x– 1)? + y² = 4, 0<z< 3 including its…

A: F=x,2y,z+3x and S is the closed surface x-12+y2=4, 0&lt;z&lt;3 including its top and bottom dics.To…

Q: Let I- (VxG) ndo, where G(x.y,z)=(ze*-y,4x,z*) and S is the part ofi the surface z=/x²+6y²+1 that…

A: Given that Gx,y,z=zex-y,4x,z4. The surfaces are given by z=x2+6y2+1⇒z2=x2+6y2+1 that lies inside the…

Q: Evaluate G if G is a solid in the first octant bounded by the plane y +z = 2 and the surface y = 1–…

A: Use the following formulae, to obtain the solution. a-b2=a2-2ab+b2∫xndx=xn+1n+1+C

Q: a) Evaluate Fự ds, where the curve C is the intersection between the surface z = V1- y? and the…

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Q: Evaluate f. F dS where F(x,y, z) = (4x³z, 4y³z, 3z*) and S is the surface of the solid bounded by…

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Q: What is the surface area of the portion of the surface z= x?+y that lays over the triangle on xy-…

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Q: Evaluate ff F.n ds where F(x, y, z) = 12xi +24j+24x²k and is that portion of the paraboloid z = x² +…

A: We will find normal vector n and region of integral on xy plane to find value of integral .

Q: Consider the surface S: z = 4x2 + 2y2 . An equation of the plane tangent to S at the point P (−1, 2,…

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Q: Let P be the tangent plane to the surface z=2x2+y2 at the point Q=(1,1,3). Which of the following…

A: tangent planes to a surface are planes that just touch the surface at the point and are “parallel”…

Q: Find an equation for the plane that is tangent to the surface z = vy-x at the point (1, 2, 1).…

A: To find an equation for the plane that is tangent to the surface at the given point (1,2 1).

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-dS, where F(x, y, z) = (3y, –2x, –), %3D and S is the surface that is composed of the…

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Q: Sketch the surface z = f(x,y). f(x.y) = 4x2 + 4y2 + 2 O A. OB. OC. OD. N

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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: Calculate ⌠⌠F*dS if F(x,y,z)=x^2ye^zi + xy^2j-2xye^zk where S is the surface of the box bounded by…

A: Use projection on xy plane

Q: Which of the following represents a plane surface whose x-, y-, and z-intercepts are a, b, and c…

A: Given questions are solved.

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: The parametric representation of the surface y2 - 4y + z2 = 0,0 <x< 3 %3D is a) F(u, v) = (v, 2 + 2…

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Q: 5. Prove that the surface E, = {(x, y, z)|a² + y² = 4} is locally isometric to the surface E2 = {(x,…

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Q: QA-evaluate the flux of the filed F-xi+yj+zk through the upper part of the cylinder x² + z² = 1 that…

A: Given

Q: The parametric representation of the surface x2 -8x + z2 = 0,0 < y 5 5 %3D is a) (u, v) = (4 cos u ,…

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Q: 15 - Consider the surface defined by z = f(z, y) = x³ + 2x²y +y². Find the equation of a normal line…

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Q: | (VxF)-n ds and (b) O n ds if F = (x + 2y)i – 3z j + xk, $ = 4x +3y - 2z, Evaluate (a) and S is the…

A: (a) The given field is F=x+2y, -3z, x. The curl of the field is ∇×F=i^j^k^∂∂x∂∂y∂∂zx+2y-3zx=3, -1,…