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I am a bit confused about the difference between variation $\delta u$ and differential $du$.

I saw them in terms of minimizing a functional. The way the variation operates seems very similar to derivatives (chain-rule and so on).

I have a hunch that variations talk about small deviations in function (throughout its domain) whereas differentials talk about small changes in discrete quantity. Am I right about this? An answer in Layman's terms would be much appreciated.

User 10482
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1 Answers1

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In simplistic terms, a differential relates to the increase in the value of a function, an object taking a scalar as argument and returning a scalar, for a "small" variation in the independent variable. A variation relates to the increase in the value of a functional, and object taking a function as argument and returning a scalar, for a small variation in the argument function. Your intuition seems correct, although I am not so sure about the reference to "discrete quantity".

An aedonist
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  • So the only difference is the context? functional vs function? – User 10482 Nov 18 '20 at 14:26
  • The context is certainly different, but they also different objects. They do have analogous meaning inter context, and as you noticed, behave similarly under the chain rule and similar operations. – An aedonist Nov 18 '20 at 16:11