TeXipedia

partial

Denotes partial derivatives and boundaries in mathematical expressions, particularly in multivariable calculus and differential equations.

Overview

Essential in advanced mathematics and physics for expressing rates of change with respect to specific variables while holding others constant.

  • Commonly used in thermodynamics to represent partial derivatives of state functions
  • Appears frequently in complex analysis and differential geometry
  • Used to denote boundaries of sets and manifolds in topology
  • Critical in expressing partial differential equations (PDEs) in physics and engineering applications
  • Often appears in optimization problems involving multiple variables

Examples

Expressing a partial derivative of function f with respect to x.

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

Writing the heat equation in partial differential form.

∂u∂t=α∂2u∂x2\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} 
\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}

Showing the gradient in multiple variables using partial derivatives.

∇f=(∂f∂x,∂f∂y,∂f∂z)\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) 
\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)