Problem

7. (a) $C$ is a smooth curve in the upper half plane going from $(1,1)$ to $(0,\sqrt{2})$.

$\displaystyle\mathbf{F}(x,y)=\frac{x-3y}{x^2+y^2}\mathbf{i}+\frac{3x+y}{x^2+y^2}\mathbf{j}$. Then, $\displaystyle\int_C \mathbf{F}\cdot d\mathbf{r} = \underline{\quad(17)\quad}$.

(b) $S$ is the part of the sphere $x^2+y^2+z^2=9$ that lies above the cone $z=\sqrt{x^2+y^2}$ with upward orientation. $\displaystyle\mathbf{F}(x,y,z)=\frac{x\mathbf{i}+y\mathbf{j}+z\mathbf{k}}{(x^2+y^2+z^2)^2}$. Then, $\displaystyle\iint_S \mathbf{F}\cdot d\mathbf{S} = \underline{\quad(18)\quad}$.