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The definition of radiance is:

$$L\equiv\frac {\partial^2 \Phi}{\partial A\,\partial\omega\,\cos\theta}$$

where:

$\Phi$ is the radiant flux

$\omega$ is the solid angle

$A\cos\theta$ is the projected surface

Why are partial derivatives used and not full derivatives as in:

$$L\equiv\frac {d^2 \Phi}{dA\,d\omega\,\cos\theta}$$ even though sometimes this formula is also (wrongly?) used ?

I am no mathematician, but $\partial$ and $d$ are not the same and they shouldn't be interchangeable.

The radiance definition is just one example, but I noticed that most physics definitions use partials and not full derivatives. So, if you know why partials are used in other examples, it would maybe help me figure it out for the radiance, which I am particularly interested in.

HDE 226868
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Andrei
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2 Answers2

2

General case is:

$$L\equiv\frac {d^2 \Phi}{dA\,d\omega\,\cos\theta}$$

You must use this (in actual fact) difficult formula when:

$\omega = f(A)$ or $A = f(\omega)$ or
$\omega = f(t)$ and $A = f(t)$ (it can be temperature for example) and so on...

Particular case is:

$$L\equiv\frac {\partial^2 \Phi}{\partial A\,\partial\omega\,\cos\theta}$$ You can use it when $A$ and $\omega$ are not connected.
Usually it is true (I always use it).

Remember, it's always you to decide which pill to eat.

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Oleksii Zhyglov
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0

This is one of the rare cases where the full and the partial derivative are (at least typically) the same. There is a difference if the way you tilt your projection surface influences the solid angle into which you radiate. Whoever uses the full derivative must implicitly assume that such a curious dependency cannot happen (what does the lamp care how you look at it?).

Derivatives in these kinds of definitions of measurement quantities are almost always meant to normalize, in a way that partial derivatives do most naturally. The reason is that full derivatives can only lead to nice, reproducible normalizations if they are determined by a law of nature (think Newton's law, which has a full derivative). Where you have design freedom in how to arrange dependencies, the full derivative would depend on your choices and hence not make for good universal definitions.