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I'm currently taking a PDE course, and would like some help with understanding weak derivatives.

My question is: when computing weak derivative and getting a result like $f'=-2\delta_1 + 4\delta_2 + g$, what difference does the constants infront of Dirac-delta do? When sketching the weak derivative we only show Dirac-delta as an arrow, so I do not really understand what difference the constants makes in this case.

uoiu
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  • A sensible way to distinguish is to draw $\delta$ as an arrow of height $1$ and $2\delta$ as an arrow of height $2$, for example. – Izaak van Dongen Nov 08 '23 at 13:12
  • I understand. But it seems like constants with Dirac-delta must have another purpose later on? Because, so far, Dirac-delta just is "what it is", and not something that we do actual calculations with. – uoiu Nov 08 '23 at 13:38
  • It definitely does have a purpose! $\delta$ is, loosely, "an object such that when you integrate $\delta(x) f(x),,\mathrm dx$ you get $f(0)$". You can be more formal or less formal about this. There is much discussion of this on this site eg here - and I'd expect this to have come up/come up soon in your course, or to have come up in a prerequisite course. Then $2\delta$ is "an object such that when you integrate $2\delta(x) f(x),\mathrm dx$ you get $2f(0)$." – Izaak van Dongen Nov 08 '23 at 13:48
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    Perfect answer, thanks! Then it will probably come up later in the course, and I know that it does make a difference. – uoiu Nov 08 '23 at 13:50

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