Question

Let f = x i y j z3 k and s is the closed surface shown with radius c and height 2c centered at the origin. What is the orientation of the surface?

1) Outward
2) Inward
3) Clockwise
4) Counterclockwise

Answer 1

The orientation of the surface is Outward.option.1)

Step-by-step explanation:

The orientation of the given closed surface can be determined by the right-hand rule. The outward orientation is defined as the direction in which the normal vector to the surface points away from the enclosed region. For the given surface, with f = xi + yj + z^3k, the normal vector points outward, away from the enclosed region. Therefore, the orientation of the surface is outward.

The normal vector to the closed surface can be calculated using the cross product of its partial derivatives. By finding the partial derivatives of f with respect to x, y, and z, and then taking their cross product, we can obtain the normal vector. Evaluating this normal vector at any point on the surface will confirm that it points outward, thus determining the orientation of the surface.

In summary, using the right-hand rule and calculating the normal vector of the given closed surface, we find that its orientation is outward, as the normal vector points away from the enclosed region.

So correct option is Outward.option.1)