Calculus: Program 1

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\title{Integration and Differential Equations in Physics}
\author{}
\date{}
\maketitle

\section*{Integral Equations}

\begin{enumerate}
    \item \textbf{Indefinite Integral:}
    \[
    \int x^2 \, dx = \frac{x^3}{3} + C
    \]
    
    \item \textbf{Definite Integral:}
    \[
    \int_0^1 e^x \, dx = e - 1
    \]
    
    \item \textbf{Integration by Parts:}
    \[
    \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x + C
    \]
    
    \item \textbf{Trigonometric Integral:}
    \[
    \int \sin(x) \, dx = -\cos(x) + C
    \]
    
    \item \textbf{Exponential Integral:}
    \[
    \int e^{2x} \, dx = \frac{e^{2x}}{2} + C
    \]
    
    \item \textbf{Improper Integral:}
    \[
    \int_0^{\infty} \frac{1}{x^2+1} \, dx = \frac{\pi}{2}
    \]
    
    \item \textbf{Surface Integral:}
    \[
    \iint_S \mathbf{F} \cdot d\mathbf{S}
    \]
    
    \item \textbf{Volume Integral:}
    \[
    \iiint_V f(x, y, z) \, dV
    \]
    
    \item \textbf{Line Integral:}
    \[
    \int_C \mathbf{F} \cdot d\mathbf{r}
    \]
    
    \item \textbf{Gauss's Theorem:}
    \[
    \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q}{\epsilon_0}
    \]
\end{enumerate}

\section*{Ordinary Differential Equations}

\begin{enumerate}
    \item \textbf{First-Order ODE:}
    \[
    \frac{dy}{dx} = x^2 + y
    \]
    
    \item \textbf{Second-Order ODE:}
    \[
    \frac{d^2 y}{dx^2} - 3 \frac{dy}{dx} + 2y = 0
    \]
    
    \item \textbf{Damped Harmonic Oscillator:}
    \[
    m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0
    \]
    
    \item \textbf{Newton’s Second Law:}
    \[
    F = m \frac{d^2x}{dt^2}
    \]
    
    \item \textbf{Simple Harmonic Motion:}
    \[
    \frac{d^2 x}{dt^2} + \omega^2 x = 0
    \]
    
    \item \textbf{RL Circuit:}
    \[
    L \frac{dI}{dt} + RI = V(t)
    \]
\end{enumerate}

\section*{Partial Differential Equations}

\begin{enumerate}
    \item \textbf{Wave Equation:}
    \[
    \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}
    \]
    
    \item \textbf{Heat Equation:}
    \[
    \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}
    \]
    
    \item \textbf{Laplace's Equation:}
    \[
    \nabla^2 \phi = 0
    \]
    
    \item \textbf{Poisson's Equation:}
    \[
    \nabla^2 \phi = \rho
    \]
    
    \item \textbf{Schrödinger Equation:}
    \[
    i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi + V \psi
    \]
\end{enumerate}

\end{document}

Output 1

Calculus: Program 2

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}

\title{Equations in Physics}
\author{}
\date{}

\begin{document}
	
	\maketitle
	
	\section*{Physics Equations}
	
	\begin{enumerate}
		
		% Algebraic Equations
		\item \textbf{Lorentz Force Law:}
		\begin{equation}
			\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B})
		\end{equation}
		
		\item \textbf{Maxwell’s Equations (Gauss’s Law):}
		\begin{equation}
			\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}
		\end{equation}
		
		\item \textbf{Maxwell’s Equations (Faraday’s Law):}
		\begin{equation}
			\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}
		\end{equation}
		
		\item \textbf{General Relativity (Einstein Field Equation):}
		\begin{equation}
			R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu}
		\end{equation}
		
		\item \textbf{Energy-Mass Equivalence:}
		\begin{equation}
			E = mc^2
		\end{equation}
		
		% Vector and Integral Equations
		\item \textbf{Stokes' Theorem:}
		\begin{equation}
			\int_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}
		\end{equation}
		
		\item \textbf{Divergence Theorem:}
		\begin{equation}
			\iiint_V (\nabla \cdot \mathbf{F}) \, dV = \oint_S \mathbf{F} \cdot d\mathbf{A}
		\end{equation}
		
		
		% Quantum Mechanics
		\item \textbf{Time-Dependent Schrödinger Equation:}
		\begin{equation}
			i \hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi + V(x) \psi
		\end{equation}
		
		\item \textbf{Commutator Relation:}
		\begin{equation}
			[\hat{x}, \hat{p}] = i \hbar
		\end{equation}
		
		\item \textbf{Expectation Value in Quantum Mechanics:}
		\begin{equation}
			\langle \hat{O} \rangle = \int \psi^* \hat{O} \psi \, dx
		\end{equation}
		
		% Classical Mechanics
		\item \textbf{Euler-Lagrange Equation:}
		\begin{equation}
			\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0
		\end{equation}
		
		\item \textbf{Hamilton’s Equations:}
		\begin{equation}
			\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}
		\end{equation}
		
		% Electromagnetism
		\item \textbf{Ampère's Law (Maxwell-Ampère Law):}
		\begin{equation}
			\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}
		\end{equation}
		
		% Fluid Dynamics
		\item \textbf{Continuity Equation (Fluid Dynamics):}
		\begin{equation}
			\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0
		\end{equation}
		
		\item \textbf{Bernoulli's Equation:}
		\begin{equation}
			P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant}
		\end{equation}
		
		% Cosmology and General Relativity
		\item \textbf{Friedmann Equation:}
		\begin{equation}
			\left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G}{3} \rho - \frac{k}{a^2} + \frac{\Lambda}{3}
		\end{equation}
				
		
	\end{enumerate}
	
\end{document}

Output 2

Calculus: Program 3

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{bm}   % For bold vectors
\usepackage{tensor} % For tensor notation

\title{Tensor and Physics Equations}
\author{}
\date{}

\begin{document}
	
	\maketitle
	
	\section*{Tensor and Advanced Physics Equations}
	
	\begin{enumerate}
		
		% Tensor Equations
		\item \textbf{Stress-Energy Tensor (General Relativity):}
		\begin{equation}
			T^{\mu \nu} = (\rho + p) u^\mu u^\nu + p g^{\mu \nu}
		\end{equation}
		
		\item \textbf{Einstein Field Equation (General Relativity):}
		\begin{equation}
			R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R = \frac{8 \pi G}{c^4} T_{\mu \nu}
		\end{equation}
		
		\item \textbf{Riemann Curvature Tensor:}
		\begin{equation}
			R^\rho_{\sigma \mu \nu} = \partial_\mu \Gamma^\rho_{\nu \sigma} - \partial_\nu \Gamma^\rho_{\mu \sigma} + \Gamma^\rho_{\mu \lambda} \Gamma^\lambda_{\nu \sigma} - \Gamma^\rho_{\nu \lambda} \Gamma^\lambda_{\mu \sigma}
		\end{equation}
		
		\item \textbf{Ricci Tensor:}
		\begin{equation}
			R_{\mu \nu} = R^\lambda_{\mu \lambda \nu}
		\end{equation}
		
		\item \textbf{Ricci Scalar:}
		\begin{equation}
			R = g^{\mu \nu} R_{\mu \nu}
		\end{equation}
		
		\item \textbf{Christoffel Symbols of the Second Kind:}
		\begin{equation}
			\Gamma^\lambda_{\mu \nu} = \frac{1}{2} g^{\lambda \sigma} \left( \partial_\mu g_{\sigma \nu} + \partial_\nu g_{\sigma \mu} - \partial_\sigma g_{\mu \nu} \right)
		\end{equation}
		
		% Quantum Field Theory
		\item \textbf{Dirac Equation (Relativistic Quantum Mechanics):}
		\begin{equation}
			(i \gamma^\mu \partial_\mu - m) \psi = 0
		\end{equation}
		
		\item \textbf{Yang-Mills Field Strength Tensor:}
		\begin{equation}
			F^a_{\mu \nu} = \partial_\mu A^a_\nu - \partial_\nu A^a_\mu + g f^{abc} A^b_\mu A^c_\nu
		\end{equation}
		
		% Electromagnetism
		\item \textbf{Electromagnetic Field Tensor:}
		\begin{equation}
			F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu
		\end{equation}
		
		\item \textbf{Maxwell's Equations in Tensor Form:}
		\begin{equation}
			\partial_\mu F^{\mu \nu} = \mu_0 J^\nu
		\end{equation}
		
		% Fluid Dynamics
		\item \textbf{Navier-Stokes Equation (Fluid Dynamics):}
		\begin{equation}
			\rho \left( \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{v} \cdot \nabla) \mathbf{v} \right) = - \nabla p + \mu \nabla^2 \mathbf{v} + \rho \mathbf{g}
		\end{equation}
		
		% Statistical Mechanics
		\item \textbf{Partition Function (Statistical Mechanics):}
		\begin{equation}
			Z = \sum_i e^{-\beta E_i}
		\end{equation}
		
		\item \textbf{Boltzmann Distribution:}
		\begin{equation}
			f(E) = \frac{1}{Z} e^{-\beta E}
		\end{equation}
		
		% Cosmology
		\item \textbf{Robertson-Walker Metric (Cosmology):}
		\begin{equation}
			ds^2 = -c^2 dt^2 + a(t)^2 \left( \frac{dr^2}{1 - kr^2} + r^2 d\Omega^2 \right)
		\end{equation}
		
		\item \textbf{Friedmann Equation (Cosmology):}
		\begin{equation}
			\left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G}{3} \rho - \frac{k}{a^2} + \frac{\Lambda}{3}
		\end{equation}
		
		% Quantum Electrodynamics
		\item \textbf{QED Interaction Lagrangian:}
		\begin{equation}
			\mathcal{L} = \bar{\psi}(i \gamma^\mu D_\mu - m) \psi - \frac{1}{4} F_{\mu \nu} F^{\mu \nu}
		\end{equation}
		
		% Thermodynamics
		\item \textbf{Clausius-Clapeyron Equation (Phase Transitions):}
		\begin{equation}
			\frac{dP}{dT} = \frac{L}{T \Delta V}
		\end{equation}
		
		% Generalized Force in Lagrangian Mechanics
		\item \textbf{Lagrange's Equation of Motion:}
		\begin{equation}
			\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = Q_i
		\end{equation}
		
		% Nonlinear Dynamics
		\item \textbf{Lorenz System (Nonlinear Dynamics):}
		\begin{equation}
			\frac{dx}{dt} = \sigma (y - x), \quad \frac{dy}{dt} = x(\rho - z) - y, \quad \frac{dz}{dt} = xy - \beta z
		\end{equation}
		
		% Advanced Electromagnetism
		\item \textbf{Liénard-Wiechert Potentials (Electrodynamics):}
		\begin{equation}
			\mathbf{A}(\mathbf{r}, t) = \frac{\mu_0}{4 \pi} \int \frac{J(\mathbf{r}', t_r)}{|\mathbf{r} - \mathbf{r}'|} d^3r'
		\end{equation}
		
	\end{enumerate}
	
\end{document}

Mathematical Representations

Calculus: Program 1

Output 1

Calculus: Program 2

Output 2

Calculus: Program 3

Output 3