Question

Compute the surface integral over the given oriented surface S defined by the vector field S = y⁹i + 8yj - xk. The surface S is the portion of the plane x + y + z = 1 in the octant where x, y, z are greater than or equal to 0, and it has a downward-pointing normal vector n^. Evaluate the surface integral of F = r · ∇S, where ∇ represents the gradient operator.

Answer 1

To compute the surface integral over the given oriented surface S defined by the vector field S = y⁹i + 8yj - xk, you need to find the dot product of the vector field and the outward-pointing normal vector of the surface. Evaluate the surface integral by using the formula F = r · ∇S and integrating over the surface S.

Step-by-step explanation:

To compute the surface integral over the given oriented surface S defined by the vector field S = y⁹i + 8yj - xk, we need to find the dot product of the vector field and the outward-pointing normal vector of the surface. In this case, the surface S is the portion of the plane x + y + z = 1 in the octant where x, y, z are greater than or equal to 0, and it has a downward-pointing normal vector n^.

To evaluate the surface integral, we can use the formula F = r · ∇S, where ∇ represents the gradient operator. Plug in the given vector field S into this formula to calculate the dot product with the gradient operator, and then integrate over the surface S to find the surface integral.