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content/Dirak δ-function (lec 21).md
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#start of lec 21 #start of lec 21
From $ma=F$ From $ma=F$
$m\frac{dv}{dt}=f(t)$ $m\frac{dv}{dt}=f(t)$
@ -5,9 +6,9 @@ integrate both sides:
$m\int_{t_{0}} ^{t_{1}} \frac{dv}{dt}dt =\int _{{t_{0}}} ^{t_{1}}f(t) \, dt$ $m\int_{t_{0}} ^{t_{1}} \frac{dv}{dt}dt =\int _{{t_{0}}} ^{t_{1}}f(t) \, dt$
$mv(t_{1})-mv(t_{0})=\int _{t_{0}}^{t_{1}}f(t) \, dt$ $mv(t_{1})-mv(t_{0})=\int _{t_{0}}^{t_{1}}f(t) \, dt$
that is, change in momentum on the LHS equates to an impulse on the RHS. that is, change in momentum on the LHS equates to an impulse on the RHS.
(picture shown, you can have the same impulse, the same area under the graph if you squish down the ) (picture shown, you can have the same impulse, the same area under the graph if you squish down f(t) to be narrower, as long as you make it taller. If we take it to the extreme we get the Dirak delta function.)
the definition of dirak delta function: The definition of the Dirak delta function:
$\delta(t-a)=\begin{cases}0, & t\ne a \\''\infty'', & t=a\end{cases}$ $\delta(t-a)=\begin{cases}0, & t\ne a \\''\infty'', & t=a\end{cases}$
however, a more useful definition is: however, a more useful definition is:
$\int _{-\infty} ^{\infty} \delta(t-a)f(t)\, dt=f(a)$ $\int _{-\infty} ^{\infty} \delta(t-a)f(t)\, dt=f(a)$

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content/_index.md
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@ -21,7 +21,7 @@ Good luck on midterms! <3 -Oct 18 2023
[(Heaviside) Unit step function (lec 18)](heaviside-unit-step-function-lec-18.html) (raw notes, not reviewed or revised yet.) [(Heaviside) Unit step function (lec 18)](heaviside-unit-step-function-lec-18.html) (raw notes, not reviewed or revised yet.)
[Periodic functions (lec 19)](periodic-functions-lec-19.html) (raw notes, not reviewed or revised yet.) [Periodic functions (lec 19)](periodic-functions-lec-19.html) (raw notes, not reviewed or revised yet.)
[Convolution (lec 19-20)](convolution-lec-19-20.html) (raw notes, not reviewed or revised yet.) [Convolution (lec 19-20)](convolution-lec-19-20.html) (raw notes, not reviewed or revised yet.)
[Dirak δ-function (lec 21)](dirak-function-lec-21.html) (raw notes, not reviewed or revised yet.) [Dirak δ-function (lec 21)](dirak-δ-function-lec-21.html) (raw notes, not reviewed or revised yet.)
</br> </br>
[How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.) [How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.)
</br> </br>