# I don't understand this 'semi colon' notion in regards to PDE solutions

In solving first order PDE's with solution $u(x,y)$, when constructing a graph of $u$ as a union of initial curves $C_s$ emanating from the initial curve $\Gamma$. My lecture notes say, for each $s \in (\alpha,\beta) \subset \mathbb{R}$, $$C_s:= \{(x(t;s),y(t;s),z(t;s)) | t \in (-\epsilon_s, \epsilon_s), \epsilon_s >0\}$$ and $$\mathrm{Graph}_u = \bigcup_{s\in (\alpha,\beta)\subset\mathbb{R}}C_s$$

I don't understand this 'semi-colon' notation given when defining $C_s$ as surely $x,y,z$ are functions of only one variable?

I appreciate any help!

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It denotes the parameter $s$. Maybe better would be to write $x_s(t)$, etc... – David Mitra Jan 18 '13 at 16:36

For each value of $s$, the curve $C_s$ has a parametric equation in which $x,y,z$ are functions of $t$. To distinguish $x(t)$ from the parametric equation of $C_1$ and $x(t)$ from the parametric equation of $C_2$, one can use subscripts: let $(x_s,y_s,z_s)$ be the functions that parameterize $C_s$. Including the argument $t$, the notation becomes $(x_s(t),y_s(t),z_s(t))$.
For various reasons, subscripts and superscripts are not always good places to keep variables in. This is how $(x(t;s),y(t;s),z(t;s))$ becomes an attractive alternative. The semicolon has no mathematical meaning; it's just a separator, like a comma. Using the semicolon emphasizes that $t$ and $s$ are not considered to be "on equal footing". If we did not want to emphasize this, then writing $(x(t,s),y(t,s),z(t,s))$ would be just as correct: after all, $x,y,z$ are functions of two variables.