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authorPrefetch2022-10-27 20:40:09 +0200
committerPrefetch2022-10-27 20:40:09 +0200
commit6e70f28ccbd5afc1506f71f013278a9d157ef03a (patch)
treea8ca7113917f3e0040d6e5b446e4e41291fd9d3a /source/know/concept/dirac-delta-function
parentbcae81336764eb6c4cdf0f91e2fe632b625dd8b2 (diff)
Optimize last images, add proof template, improve CSS
Diffstat (limited to 'source/know/concept/dirac-delta-function')
-rw-r--r--source/know/concept/dirac-delta-function/index.md20
1 files changed, 6 insertions, 14 deletions
diff --git a/source/know/concept/dirac-delta-function/index.md b/source/know/concept/dirac-delta-function/index.md
index 518eba1..0185b78 100644
--- a/source/know/concept/dirac-delta-function/index.md
+++ b/source/know/concept/dirac-delta-function/index.md
@@ -65,11 +65,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-scale"/>
-<label for="proof-scale">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-scale">Proof.</label>
+
+{% include proof/start.html id="proof-scale" -%}
Because it is symmetric, $$\delta(s x) = \delta(|s| x)$$.
Then by substituting $$\sigma = |s| x$$:
@@ -77,9 +74,8 @@ $$\begin{aligned}
\int \delta(|s| x) \dd{x}
&= \frac{1}{|s|} \int \delta(\sigma) \dd{\sigma} = \frac{1}{|s|}
\end{aligned}$$
+{% include proof/end.html id="proof-scale" %}
-</div>
-</div>
An even more impressive property is the behaviour of the derivative of $$\delta(x)$$:
@@ -89,11 +85,8 @@ $$\begin{aligned}
}
\end{aligned}$$
-<div class="accordion">
-<input type="checkbox" id="proof-dv1"/>
-<label for="proof-dv1">Proof</label>
-<div class="hidden" markdown="1">
-<label for="proof-dv1">Proof.</label>
+
+{% include proof/start.html id="proof-dv1" -%}
Note which variable is used for the
differentiation, and that $$\delta'(x - \xi) = - \delta'(\xi - x)$$:
@@ -102,9 +95,8 @@ $$\begin{aligned}
&= \dv{}{x}\int f(\xi) \: \delta(x - \xi) \dd{x}
= f'(x)
\end{aligned}$$
+{% include proof/end.html id="proof-dv1" %}
-</div>
-</div>
This property also generalizes nicely for the higher-order derivatives: