8.6 ベクトル積分定理

1.

(a) $8$(b) 0(c) $${\frac{1}{4}}$$

2.

(a) $${\frac{36 \sqrt{3}}{5}\pi}$$(b) 0(c) $${\frac{672}{5}}$$(d) $81\pi$

3.

Greenの定理において， $P = -y , Q = x$とおくと，

$$\oint_{c}xdy - ydx = \iint_{\Omega}\left(\frac{\partial(x)}{\partial x} - \frac{\partial(-y)}{\partial y}\right)dx dy = 2 \iint_{\Omega}dx dy = 2A$$

4.

$\pi ab$

5.

発散定理において ${\bf F} = f {\rm grad}g$とおくと

$${\bf } \cdot {\hat{\bf n}} = f ({\rm grad}g \cdot {\hat{\bf n}} = f\frac{\partial g}{\partial n}$$

また，

$${\rm div} {\bf F} = {\rm div}(f {\rm grad}g)$$

$$= \nabla \cdot (f \nabla g) = (\nabla f)\cdot \nabla g + f\nabla \cdot \nabla g$$

$$= {\rm grad}f \cdot {\rm grad}g + f \nabla^2 g$$

よって

$$\iint_{S}f \frac{\partial g}{\partial n} dS = \iiint_{V}(f \nabla^2 g + {\rm grad}f \cdot {\rm grad}g)dV$$