stylistic:tone
Notation

When writing PDEs, it is common to denote partial derivatives using subscripts. For example: u x = ∂ u ∂ x, u x x = ∂ 2 u ∂ x 2, u x y = ∂ 2 u ∂ y ∂ x = ∂ ∂ y (∂ u ∂ x). {\displaystyle u_{x}={\frac {\partial u}{\partial x}},\quad u_{xx}={\frac {\partial ^{2}u}{\partial x^{2}}},\quad u_{xy}={\frac {\partial ^{2}u}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right).} In the general situation that u is a function of n variables, then u denotes the first partial derivative relative to the i -th input, u denotes the second partial derivative relative to the i -th and j -th inputs, and so on.

The Greek letter Δ denotes the Laplace operator; if u is a function of n variables, then Δ u = u 11 + u 22 + ⋯ + u n n. {\displaystyle \Delta u=u_{11}+u_{22}+\cdots +u_{nn}.} In the physics literature, the Laplace operator is often denoted by ∇; in the mathematics literature, ∇ u may also denote the Hessian matrix of u.