(1 point) use stokes' theorem to evaluate (v x f). ds where f(x, y, z) = -13yzi + 13xzj +4(x2 + y2)zk and s is the part of the paraboloid z = x2 + y2 that lies inside the cylinder x2 + y2 = 1, oriented upward. js

By using Stokes's theorem to the required (VxF).dS  is 26π.

Given that,

Function F(x, y, z) = -13yzi + 13xzj +4(x2 + y2)zk .

And S is the part of the paraboloid z = x2 + y2 that lies inside the cylinder x2 + y2 = 1, oriented upward. JS.

We have to determine,

Use Stokes' theorem to evaluate (V x F).dS.

According to the question,

By using Stokes's theorem to evaluate (V x F).dS .

F(x, y, z) = -13yzi + 13xzj +4(x2 + y2)zk .

Stokes' theorem says the integral of the curl of over F is s equal to the integral of F along the boundary of S , with counterclockwise orientation (when viewed from above).

This boundary is the circle set in the plane .

Then,

The parameterize path is given by the,

Then,

Hence, By using Stokes's theorem to the required (VxF).dS  is 26π.

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Stokes' theorem equates the integral of across with the integral of along the boundary of , which is the intersection of the paraboloid and the cylinder , or the circle in the plane centered at (0, 0, 1) with radius 1.

Parameterize this intersection (call it ) by

with . Then

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