LaTex symbol partial derivative

The \partial Command in LaTeX

The \partial command is used in LaTeX to represent a partial derivative.

Basic Usage

The \partial command can be used as follows to display the partial derivative of a function $f$ with respect to a variable $x$:

\frac{\partial f}{\partial x}

$$ \frac{\partial f}{\partial x} $$

Subscript under the Partial Derivative

It is possible to add a subscript under the partial derivative to specify with respect to which variable the derivative is taken. For example, to represent the partial derivative of a function $f$ with respect to the variable $x$ while specifying that the variable $y$ is constant, the following command can be used:

\frac{\partial f}{\partial x}_y

$$ \frac{\partial f}{\partial x}_y $$

Higher Order Partial Derivatives

It is also possible to represent higher order partial derivatives. For example, to represent the second partial derivative of a function $f$ with respect to the variable $x$, the following command can be used:

\frac{\partial^2 f}{\partial x^2}

$$ \frac{\partial^2 f}{\partial x^2} $$

The notations for subscripts and higher order partial derivatives can be combined. For example, to represent the second partial derivative of a function $f$ with respect to the variables $x$ and $y$, the following command can be used:

\frac{\partial^2 f}{\partial x \partial y}

$$ \frac{\partial^2 f}{\partial x \partial y} $$

Example of partial differential equations

Partial differential equations are equations that involve partial derivatives of unknown functions. They are often used to model physical phenomena. Here are some examples of famous partial differential equations:

– The heat equation:

The heat equation is a partial differential equation that models the diffusion of heat in an object. It is given by:

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

$$ \frac{\partial u}{\partial t} = \alpha \nabla^2 u $$

where $u(x, t)$ is the temperature at location $x$ and time $t$, $\alpha$ is the thermal diffusivity, and $\nabla^2$ is the Laplace operator.

– The Stokes equations:

The Stokes equations are a system of partial differential equations that model the motion of an incompressible viscous fluid. They are given by:

\begin{align}
\nabla \cdot \mathbf{u} &= 0\\
\rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) &= - \nabla p + \mu \nabla^2 \mathbf{u}
\end{align}

$$ \begin{align} \nabla \cdot \mathbf{u} &= 0\\ \rho \left( \frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} \right) &= - \nabla p + \mu \nabla^2 \mathbf{u} \end{align} $$

where $\mathbf{u}$ is the fluid velocity, $p$ is the pressure, $\rho$ is the fluid density, $\mu$ is the dynamic viscosity, and $\nabla^2$ is the Laplace operator.

– The Navier-Stokes equations:

The Navier-Stokes equations are partial differential equations that describe the motion of an incompressible viscous fluid. They are of great importance in fluid mechanics, engineering, and physics.

The Navier-Stokes equations in $d$ dimensions are:

\begin{align}
\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} &= - \frac{1}{\rho} \nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{f}, \quad \mathrm{for} \quad \mathbf{x} \in \Omega, \\
\nabla \cdot \mathbf{u} &= 0, \quad \mathrm{for} \quad \mathbf{x} \in \Omega,
\end{align}

$$ \begin{align} \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} &= - \frac{1}{\rho} \nabla p + \nu \nabla^2 \mathbf{u} + \mathbf{f}, \quad \mathrm{for} \quad \mathbf{x} \in \Omega, \\ \nabla \cdot \mathbf{u} &= 0, \quad \mathrm{for} \quad \mathbf{x} \in \Omega, \end{align} $$

where $\mathbf{u}(\mathbf{x},t)$ is the velocity of the fluid at position $\mathbf{x} \in \Omega$ and time $t$, $p(\mathbf{x},t)$ is the pressure, $\rho$ is the mass density of the fluid, $\nu$ is the kinematic viscosity, $\mathbf{f}(\mathbf{x},t)$ is the external force applied to the fluid, and $\nabla$ is the gradient operator.

The first equation is the conservation of momentum equation, which describes the evolution of the fluid velocity in response to applied forces. The second equation is the continuity equation, which expresses the conservation of mass.

The Navier-Stokes equations are nonlinear and can be very difficult to solve. They are often solved numerically using methods such as finite element method or finite difference method.