\[ \boxed{ \mathbf{A} \times \mathbf{B} = \begin{pmatrix} -2y \\ x \\ 0 \end{pmatrix} } \]
\[ \boxed{ \mathbf{A} \times \mathbf{B} \rightarrow カ , \quad \mathbf{B} \rightarrow ア } \]
\[ \oint_C(\mathbf{A} \times \mathbf{B}) \cdot d\mathbf{l} = \oint_C \begin{pmatrix} -2y \\ x \end{pmatrix} \cdot \begin{pmatrix} dx \\ dy \end{pmatrix} = \oint_C(-2ydx+xdy) \] グリーンの定理を使う． \[ \iint_S(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})dxdy = \oint_C(Pdx+Qdy) \] ここでは， \[ P=-2y, \quad Q=x \] \[ \frac{\partial Q}{\partial x} = 1, \quad \frac{\partial P}{\partial y} = -2, \quad \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 3. \] \[ dxdy = 2 \times 4 = 8. \] よって， \[ \oint_C(\mathbf{A} \times \mathbf{B}) \cdot d\mathbf{l} = \oint_C(-2ydx+xdy) = \iint_S(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})dxdy = 3 \times 8 = 24. \] \[ \boxed{ \oint_C(\mathbf{A} \times \mathbf{B}) \cdot d\mathbf{l} = 24. } \]
ガウスの発散定理より， \[ \iint_S(\mathbf{B} \cdot \mathbf{n})dS = \iiint_V (\mathbf{\nabla} \cdot \mathbf{B})dV \] \[ \mathbf{\nabla} \cdot \mathbf{B} = \frac{\partial}{\partial x}\Bigl(\frac{x}{2}\Bigr) + \frac{\partial}{\partial y}(y) + \frac{\partial}{\partial z}(z^3) = \frac{1}{2} + 1 + 3z^2 = \frac{3}{2} + 3z^2 \] よって， \[ \iiint_V (\mathbf{\nabla} \cdot \mathbf{B})dV = \int_{-a}^a \int_{-2}^2 \int_{-1}^1 \Bigl(\frac{3}{2} + 3z^2 \Bigr)dxdydz \] \[ = \int_{-a}^a \int_{-2}^2 \Bigl[\frac{3}{2}x + 3z^2x \Bigr]_{-1}^{1}dydz = \int_{-a}^a \int_{-2}^2 (3 + 6z^2)dydz \] \[ = \int_{-a}^a \Bigl[3y + 6z^2y\Bigr]_{-2}^{2}dz = \int_{-a}^a (12 + 24z^2)dz \] \[ = \Bigl[12z + 8z^3\Bigr]_{-a}^{a}dz = 24a + 16a^3 \] \[ \boxed{ \iint_S(\mathbf{B} \cdot \mathbf{n})dS = 24a + 16a^3 } \]