Question

Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above. F(x, y, z) = xyi + yzj + zxk, C is the boundary of the part of the paraboloid z = 1 − x2 − y2 in the first octant.

Answer 1

To evaluate C F · dr using Stokes' Theorem, we need to find the curl of the given vector field and the surface that bounds the given curve. Then, we can use Stokes' Theorem to calculate the line integral.

Step-by-step explanation:

To evaluate C F · dr using Stokes' Theorem, we first need to find the curl of the vector field F(x, y, z) = xyi + yzj + zxk. The curl of F is given by curl F = (0 - z)i + (z - y)j + (y - x)k.

Next, we need to find the surface that bounds the given curve C. The surface is the part of the paraboloid z = 1 - x^2 - y^2 in the first octant.

Finally, we can use Stoke's Theorem to evaluate the line integral. Stoke's Theorem states that the line integral of F · dr over