# Chapter 9: Chapter 9

Problem 280

Verify Stokes's Theorem for $\mathrm{F}^{-}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\left(3 \mathrm{y},-\mathrm{xz}, \mathrm{yz}^{2}\right)$ where \(\mathrm{S}\) is the surface \(2 \mathrm{z}=\mathrm{x}^{2}+\mathrm{y}^{2}\) bounded by \(\mathrm{z}=2\) and \(\mathrm{C}\) is its boundary.

Problem 281

Let $\mathrm{F}^{\rightarrow}(\mathrm{x}, \mathrm{y}, \mathrm{z})=\left[\left\\{-\mathrm{y} /\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)\right\\},\left\\{\mathrm{x} /\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)\right\\}, 0\right]$ Evaluate \(\oint_{C} F^{-} \cdot d r^{-}\) where \(C\) is the circle \(x^{2}+y^{2}=1 .\) Also evaluate $\int_{\mathrm{S}}\left(\operatorname{curl} \mathrm{F}^{-}\right) \cdot \mathrm{n}^{\rightarrow} \mathrm{d} \mathrm{A}$ and explain the results.

Problem 282

Prove: \(\oint_{C} F^{\rightarrow} \cdot d r^{\rightarrow}=0\) for every closed curve \(C\) if and only if \(\operatorname{curl} \mathrm{F}^{\rightarrow}=0\).

Problem 283

Show that the 2 -form $$ \sigma=\left[(x d y d z+y d z d x+z d x d y) /\left\\{\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}\right\\}\right] $$ satisfies \(\mathrm{d} \sigma=0\) but that \(\sigma\) is not exact. Do this by proving that \(\iint_{\mathrm{S}} \sigma\), where \(\mathrm{S}\) is the unit sphere, is not zero.