Question

Question: Use Stokes’ Theorem to evaluate

\({\rm{F(x,y,z) = 2ycoszi + }}{{\rm{e}}^{\rm{x}}}{\rm{sinzj + x}}{{\rm{e}}^{\rm{y}}}{\rm{k,}}\)

\({\rm{S}}\)is the hemisphere \({{\rm{x}}^{\rm{2}}}{\rm{ + }}{{\rm{y}}^{\rm{2}}}{\rm{ + }}{{\rm{z}}^2}{\rm{ = 9,z}} \ge {\rm{0,}}\)oriented upward

Short answer

The required integral is=\( - 18\pi \).

Step-by-step solution

Concept Introduction

The surface integral of a function's curl over any surface limited by a closed path is equivalent to the line integral of a given vector function around that path, according to a theorem.

Explanation of the solution

When applying the Stokes's theorem the first thing you need to do is to identify the correct boundary of the surface. In our case, this boundary is the circle of radius 3 at the \({\rm{xy}}\) plane.

We immediately parametrize it as:

\(\begin{array}{l}\gamma {\rm{(t) = (3cost,3sint,0)}}\\\gamma '{\rm{(t) = ( - 3sint,3cost,0)}}\end{array}\)

Next, recall that the Stokes's theorem states that:

\(\begin{array}{c}\int {\int_{\rm{S}} {{\rm{curl}}} } {\rm{F \times dS = }}\int_{\partial {\rm{S}}} {\rm{F}} {\rm{ \times ds}}\\ = \int_{\rm{0}}^{{\rm{2\pi }}} {\rm{F}} {\rm{(}}\gamma {\rm{(t)) \times }}\gamma '{\rm{(t)}}\end{array}\)

The required integral is=\( - 18\pi \).