User:Corkgkagj/draft2

$$ \begin{align} &J(\mathbf{M})=\nabla\otimes\mathbf{M},\\ &\mathbf{g}=J^TJ,\\ &\mathbf{u},\mathbf{v}\in\mathbf{M}:\\ &\langle \mathbf{u},\,\mathbf{v}\rangle=\overline{\mathbf{u}^T}\mathbf{gv} \end{align} $$

$$\begin{align} \left[\int_\mathbb{R}e^{-x^2}\mathrm{d}x\right]^2&=\int_\mathbb{R}e^{-x^2}\mathrm{d}x\int_\mathbb{R}e^{-y^2}\mathrm{d}y\\ &=\iint_{\mathbb{R}^2}e^{-(x^2+y^2)}\mathrm{d}x\mathrm{d}y\\ &=\iint_{\mathbb{R}^2}re^{-r^2}\mathrm{d}r\mathrm{d}\theta\\ &=\int_0^{2\pi}\mathrm{d}\theta\int_0^\infty re^{-r^2}\mathrm{d}r\\ &=2\pi\left[-\frac{e^{-r^2}}{2}\right]_0^\infty\\ &=\pi\\ \int_\mathbb{R}e^{-x^2}\mathrm{d}x&=\sqrt{\pi} \end{align}$$