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Cauchy's Hypothesis or Noll's theorem states that $\vec{t}(\vec{X},t;\partial \Omega) = \vec{t}(\vec{X},t;\vec{N})$ where $\vec{N}$ is the outward unity normal to the positively oriented surface $\partial \Omega$. This translates to words as the dependence of the surface interaction vector on the surface on which it acts is only through the normal $\vec{N}$. My question is what is the significance of the semicolon (;)? How does it differ from the comma (,) used to separated the function's first two arguments?

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2 Answers 2

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A semicolon is used to separate variables from parameters. Quite often, the terms variables and parameters are used interchangeably, but with a semicolon the meaning is that we are defining a function of the parameters that returns a function of the variables.

For example, if I write $f(x1,x2,\ldots;p1,p2,\ldots)$ then I mean that by supplying the parameters $(p1, p2,\ldots)$, I create a new function whose arguments are $(x1, x2,\ldots)$.

So the general syntax is $\operatorname{functionName}(\mathrm{variables};\mathrm{parameters})$.

In Noll's theorem it says that the function created by supplying $\partial \Omega$ is the same as that created by supplying $\vec{N}$. That's rather a nice way of saying that the function created only depends on $\vec{N}$.

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    $\begingroup$ I was still a little confused by the answer until I read mathsisfun.com/definitions/parameter.html $\endgroup$
    – Utensil
    Dec 3, 2017 at 11:37
  • $\begingroup$ In programming terms, parameter is a name for a variable in function definition. When using the function, you give arguments. I tend to think that the general syntax in math is funcname(variable parameters; fixed parameters). You need both to calculate the return value for the function, but the second set is usually fixed. $\endgroup$
    – Niko Fohr
    Dec 2, 2020 at 15:52
  • $\begingroup$ What you are describing sounds like what happens when you curry a function but I'm not sure that is how you meant it. $\endgroup$ Dec 7, 2023 at 2:43
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There is no hard mathematical difference between the comma (,) and the semicolon(;).

The semicolon is used sometimes to optically separate some variable group. So the semicolon is not more than a reading aid.

The situation can be compared to the usage of different kind of parentheses, to make complex nestings more readable.

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