Showing that formulae are invariant for all values in the domain

[etotheipi]

I was just thinking about this and couldn't decide whether it was a silly question or not, so naturally I thought I might ask. It was partly prompted by one of the questions asked in the homework section.

Every physical law I can think of is "self-correcting" if you substitute negative values for different quantities. E.g. none of $V=IR$, $x = u_x t + \frac{1}{2}a_x t^2$, $\Delta H = \Delta U + T\Delta S$ change form depending on whether any of those quantities are negative. So long as the value of a quantity is within its domain (e.g. $I \in (-\infty, \infty)$), the equation holds true.

Are all of these equations valid for all real numbers by default? Or do we need to use a symmetry argument of some sort to show that the formulae themselves take the same form no matter whether e.g. $V$ is positive or negative?

And furthermore, are there any examples of physical laws where the domain of a particular variable is restricted (ignoring obvious cases, e.g. lengths must be greater than zero)?

Thanks!

 

[andrewkirk]

$$E = mc^2$$
$m$ cannot be negative, to the best of my knowledge.
Temperature is another measure with bounded domain - on the left - at zero for degrees Kelvin.

Physical laws represented by equations are predicated on the requirement that all quantities lie within their specified domains. In some cases additional requirements apply. For instance Hooke's Law $F=kx$ only applies as long as $x$ is not large enough for the spring to have exceeded its elastic limit. Newtonian and quantum mechanical equations only apply accurately as long as relative speeds and masses are not relativistic. Relativity equations are only accurate as long as distances are not small enough for quantum effects to disrupt them.

Also, how would you classify the equation for current through a diode in terms of the potential difference applied across its terminals? That will be zero for one sign (say negative) and a nonzero for the other sign.

 

[etotheipi]

Thanks! It's perhaps not as black and white as I thought.

For instance, it's not obvious that $V=IR$ will apply if the current is negative, but we know from experience (and the passive sign convention) that the equation has no negative signs if the reference directions are in opposite senses. And current is $\vec{J}\cdot \vec{A}$ so can certainly be negative if the reference direction is in the opposite direction to the current density! And in general to use an equation we need to know the allowed domain of each of the variables and the conditions under which the formula is true.

The diode one is a good example, since it's piecewise and also depends on how you setup your reference directions for current and voltage.