Differentiating with respect to a constant

[unchained1978]

Quick question on typical integration/differentiation. What is the justification for differentiating some integrals with respect to constants in order to obtain results for them, i.e. \frac{∂}{∂α}\int e^{-αx^{2}}dx=\int-x^{2}e^{-αx^{2}}dx? It seems to me α is a constant, so it seems a little confusing to even talk about the partial derivative. I guess if you treat the entire integral to be a multivariable function of both x and α, f(x,α) then it's somewhat justified, though I could just as well replace alpha with some number and we'd be stuck with something like \frac{∂}{∂8} which seems very sketchy.

 

[Simon Bridge]

I guess if you treat the entire integral to be a multivariable function of both x and α, f(x,α) then it's somewhat justified

... that is exactly what is happening.

though I could just as well replace alpha with some number

... you can say the same about x. There is nothing special about the letter x that says it has to be a variable. It may be that alpha is something like -i p_{x}/\hbar in which case your "constant" has physical significance.

 

[unchained1978]

... you can say the same about x. There is nothing special about the letter x that says it has to be a variable.

... that is exactly what is happening. ... you can say the same about x. There is nothing special about the letter x that says it has to be a variable. It may be that alpha is something like -i p_{x}/\hbar in which case your "constant" has physical significance.

The problem I'm having with this though is that x is understood in this sense to be a variable, and therefore \frac{d}{dx} is a measure of how sensitive some function f(x) is to a change in x. But when dealing with partials with respect to constants, this intuitive definition fails, because functions can't change with respect to changes in constants, otherwise they wouldn't be constants! I'm not questioning the mathematical machinery behind this, more so I'm curious as to the reasoning you have to provide when doing so. It seems it would have to go further than simply assuming f to be a multivariate function to differentiate in that sense.

 

[HallsofIvy]

How are you determining that "\alpha" is a constant? In the example you give,
\frac{\partial}{\partial\alpha}\int e^{-\alpha x^2}dx= -\int x^2e^{-\alpha x^2}dx
\alpha is definitely NOT a constant.

 

[Simon Bridge]

What he said: I think you have to give up this idea about what makes a constant and what makes a variable. d/da tells you how the function is sensitive to changes in parameter a. The role a plays in the equation depends on where it came from.

In physics x is usually a parameter representing 1D space, and t is the same for time. But it does not have to be. These letters are just labels for ideas.
In math, x often stands for the parameter of interest in a function and has no special meaning. Others tend to be coefficients and parameters which may or may not be held constant. It is perfectly valid to ask what would happen if a parameter were allowed to vary.

You could as easily write:\frac{\partial}{\partial x}\int e^{-x\alpha^2}d\alpha...