Use Stokes' Theorem to evaluate ∫
C
F � dr
where F(x, y, z) = x2yi + 1/3x3j + xyk and C is the curve of intersection of the hyperbolic paraboloid z = y2 ? x2
and the cylinder x2 + y2 = 1 oriented counterclockwise as viewed from above.
Find parametric equations for C, Let x and y be in terms of t where
0 ? t ? 2?

Step-by-step explanation:

Given that:

Here C is the curve of intersection of the hyperbolic parabolic and the cylinder

Using Stokes' Theorem

From above ;

S = the region under the surface and above the circle

Suppose, we consider

therefore, S will be the level curve of f(x,y,z) = 0

Recall that:

is always normal to the surface S at the point (x,y,z).

∴

This implies that the unit vector

So

Also,

Similarly ;

Then:

converting the integral to polar coordinates

This implies that:

⇒

Therefore, the value of

The parametric equations for the curve of intersection of the hyperbolic paraboloid can be expressed as the equations of the plane and cylinder in parametric form . i.e

Set them equal now,

the Parametric equation of

at least two

step-by-step explanation: