Evaluate F. dS, where F = < y, – x, z² > and S is the helicoid with vector equation r(u, v) < u cos v, u sin v, v > , 0 < u < 3, 0 < v < 27 with upward orientation. ||

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…

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A:

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A: Sine you have posted a question with multiple subparts we will solve first three subparts for you.…

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A: The complete solutions are given below

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A: Given,

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A: This is a problem of Vector Calculus.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.