So I have a function $u:\mathbb{R} \times (0,\infty) \to \mathbb{R} $ and a constant $a \in \mathbb{R}.$ Define $v:\mathbb{R} \times (0,\infty) \to \mathbb{R}$ by $v(x,t)=u(x+at,t)$.
What is $\frac{\partial}{\partial t}v(x,t)$ in terms of $u$? Is it $\frac{\partial}{\partial t}v(x,t) = u_t(x+at,t) + au(x,t)$? This is my attempt using the chain rule. Thanks for any pointers.
You need an $x$ derivation on $au(x,t)$.
$$\frac{\partial}{\partial t} v(x,t)=\frac{\partial}{\partial t} u(x+at,t)=au_1+u_2$$
Where subscript $i$ means partial derivative w.r.t. $i^{th}$ variable -- the variables separated by commas in the function's input parentheses.
Here is a more general example: $$\frac{\partial}{\partial t} v\left(x(s,t),y(s,t)\right)=v_xx_t+v_yy_t$$
