gitverse/LaTeX: Репозиторий с примерами LaTeX

1. Многомерный интеграл

\iiint\limits_{V} \left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \right) dx\,dy\,dz = \iint\limits_{S} \left( P\,dy\,dz + Q\,dz\,dx + R\,dx\,dy \right)

\int_{\Omega} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dx\,dy = \oint_{\partial \Omega} P\,dx + Q\,dy

\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right) \right) где a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos\left(\frac{n\pi x}{L}\right) dx, \quad b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin\left(\frac{n\pi x}{L}\right) dx

\mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt

\Gamma(z) = \int_0^{\infty} t^{z-1} e^{-t} dt, \quad \text{Re}(z) > 0

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n

\oint\limits_{C} \frac{f(z)}{z - z_0} dz = 2\pi i \cdot \text{Res}_{z=z_0} f(z)

\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc

\text{Пусть } F: \mathbb{R}^{n+m} \to \mathbb{R}^m,\quad F(x_0,y_0)=0,\quad \det \left[ \frac{\partial F_i}{\partial y_j}(x_0,y_0) \right] \neq 0 \\ \Rightarrow \exists g(x): F(x,g(x))=0 \text{ локально}.

i\hbar \frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t) \right] \Psi(\mathbf{r},t)

\phi(x) - \lambda \int_a^b K(x,t)\phi(t)\,dt = f(x)

\eta_{\text{Tr}} = 1 - \frac{1}{\varepsilon^{\gamma - 1}} \cdot \frac{\alpha \rho^{\gamma} - 1}{(\alpha - 1) + \alpha \gamma (\rho - 1)}