Question About Derivative Notation

[cepheid]

In my radiative processes in astrophysics course I'd often see something like this:

$$I_\nu (\mathbf{r}, \mathbf{k}, t, \nu) = \frac{dE}{dAd\Omega dt d\nu}$$

Where E is energy. So, whatever you want to call it, I_ν is the physical quantity relating to the radiation field that is measured in units of:

W*m^-2*Hz^-1*sr^-1

or, if you like:

ergs*s^-1*cm^-2*Hz^-1*sr^-1

or, if you like:

Jy*sr^-1

Astronomers tend to use the name specific intensity for this quantity.

If I understand what is being said here correctly, we are saying that the energy, E, arriving at the position of an observer due to an incident radiation field is a function of his:

position (x,y) (i.e. where he is)

orientation (θ, φ) (i.e. the direction in which he looks)

time (t) (i.e. when he looks)

frequency band (ν) (i.e. the range of frequencies over which he observes).

If this is true, then here is my question: Aren't we saying that E (energy arriving) is a function of several variables, and that I_ν is a mixed fourth-order partial derivative of this function? If so, why isn't the partial derivative notation used?

Although it may seem to you that my question is frivolous, I am asking because I want to know whether this is just sloppy notation, or whether I am missing something conceptual.

Thank you.

 

[dx]

Isn't that just like writing dq = (charge density)dxdydz? I don't see how this can be written in partial derivative notation like you suggest.

 

[cepheid]

Aha! So I was making a conceptual mistake. But now that you've drawn that analogy, it makes perfect sense. What is being written is not a derivative of E, but a "fraction" of differentials. Okay, thanks.