8. For the following vector fields and surfaces, calculate the surface integral (a) F(x, y, z) = (z, y, x) S is the part of the parabolic cylinder z = 1-² which lies above the zy-plane and between y = 0 and y = 2, oriented upwards. FdS.

Q: Let S be the surface of revolution of the curve C: z? + (y+ 5)² = 4. (a) Find a parametrization of…

A: Consider the given information: Let the surface is given by, z2+(y+52)=4

Q: Evaluate F. dS , where F = and S is the helicoid with vector equation r(u, v) = 0 < u < 1, 0 < v<T…

A: since, y,-x,z3 and, ru,v=u cosv, u sinv,v So, F⇀ru,v=u cosv,-u sinv, v3 Now the normal upward vector…

Q: Your friends correctly calculate the gradient vector for f(x, y)=x²+y at the point (2,-3) as…

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Q: Write the ratio as a ratio of whole numbers using fractional notation 12 to 1

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Q: 2. Let S be the surface given by r(u, v) = (u, u² + v², v). Find the equation of the tangent plane…

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Q: The surface S is defined by x + y² + z³ = 3. Find a vector normal to S at the point A with position…

A: Given a surface S the normal vector to the surface at the point A is given by

Q: H. Use the gradient to find the equation of the tangent plane to each of the surfaces at the given…

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Q: 1. Find the unit normal vector to the surface z = Vx2 + y? at the point (3, 4, 5).

A: A detailed solution is given below

Q: Let F(x, y, z) = xzi+yzj + xyk, Surface S is the part of the sphere x² + y² + z² = 4 that lies…

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Q: Suppose F 2.xz + 3y²) ay + (4yz²) az. (a) Calculate fLL F dS, where S is the shaded surface in…

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Q: Find the points on the surface ry + yz + zr - r - z = 0 where the tangent (5) plane is parallel to…

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