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What I mean is that, say we start with a 'starting point' equation, in this example Ficks law of diffusion (wikipedia:):

[tex] J = -D \frac{\delta \phi}{\delta x} [/tex]

In this case, what is the physical meaning of multiplying a scalar (D) by the derivative?

Can this be generalized to things like, multiplying the gradient of something by the gradient of something else, taking the gradient of a tensor, then multiplying that by the gradient of something else?

What I am trying to get at is in a lot of physics derivations, equations become huge and complex, but I want to know how they break down, and what physical meaning the 'propagation' of all of these multiplication and additions means?

And then when you have things like integrals of the gradient of a function

Essentially what I would like to know is, given some complex equation such as (eq from the navier stokes page on wikpedia:)

If I was just looking at the equation without knowing its roots, how can I take an educated guess about what It might be describing?

I hope this question makes sense.

Thanks