exercise 3.10 completed. Add an option that amplify the text size by 20%, not sucessfully. Instead, used html tag to style the paragraph headings

Author: govin08

_posts/2024-10-05-Fourier_1.md

@@ -1,6 +1,6 @@
 ---
 layout: single
-title: "1. The Genesis of Fourier Analysis"
+title: "(1) The Genesis of Fourier Analysis"
 categories: mathematics
 tags: [analysis, Fourier analysis, Fourier Series]
 use_math: true
@@ -53,7 +53,7 @@ toc: true
 
 ## 1. The vibrating string
 
-#### Simple harmonic motion
+<p class="text-size-12">Simple harmonic motion</p>
 
 Consider a mass $m$ attached to a horizontal spring.
 Let $y(t)$ be the displacement of the mass at time $t$ where the equilibrium position is set to $y=0$.
@@ -78,7 +78,7 @@ or
 
 $$y(t)=A\cos(ct-\phi).$$
 
-#### Standing and traveling waves
+<p class="text-size-12"> Standing and traveling waves </p>
 
 One dimensional wave can be described as a function $y=u(x,t)$.
 That is, we can specify the vertical displacement $y$ of a wave depending on the horizontal displacement $x$ and time $t$.
@@ -97,7 +97,7 @@ $$u(x,t)=F(x-ct).$$
 Here, $F(x)$ is the initial profile of $u$.
 This wave moves rightward if $c\gt0$ where $c$ can be thought of as the velocity.
 
-#### Harmonic and superposition of tones
+<p class='text-size-12'> Harmonic and superposition of tones </p>
 
 (quote without any modification) The pure tones are accompanied by combinations of overtones which are responsible for the timbre of the instrument.
 
@@ -164,7 +164,7 @@ We can get two kinds of solution to the wave equation ;
 (1) using traveling waves
 (2) using the superposition of standing waves
 
-#### A solution using traveling waves
+<p class='text-size-12'> A solution using traveling waves </p>
 
 We claim that (a) has the general solution of the form
 
@@ -324,7 +324,7 @@ u(x,t)
 \end{aligned}
 $$
 
-#### A solution using superposition of standing waves
+<p class='text-size-12'> A solution using the superposition of standing waves </p>
 
 We claim that (a) has the general solution of the form
 
@@ -568,7 +568,7 @@ Note that the function $f$ in the last expression is the extended version (on $\
 
 ## 3. Exercises
 
-### 3.1
+<p class='text-size-12'> 3.1 </p>
 The absolute value of $z=x+iy$ where $x,y\in\mathbb R$ is
 
 $$|z| = \sqrt{x^2+y^2}.$$
@@ -635,7 +635,8 @@ $$
 \end{align*}
 $$
 
-### 3.2
+<p class='text-size-12'> 3.2 </p>
+
 The complex conjugate of $z=x+iy$ is
 
 $$\bar z = x-iy.$$
@@ -650,7 +651,8 @@ $$z\bar z = (x+iy)(x-iy)=x^2-(iy)^2=x^2+y^2=|z|^2$$
 
 Since $\lvert z\rvert=1$, we have $z\bar z=\lvert z\rvert^2=1$, where we can divide $z$ on both sides.
 
-### 3.3
+<p class='text-size-12'> 3.3 </p>
+
 (a)
 A complex sequence $\\{w_n\\}_{n=1}^\infty$ converges to $w$ or $w_n\to w$ if
 
@@ -758,7 +760,8 @@ $$
 Therefore, the $\\{S_n\\}$ is Cauchy and is convergent.
 It follows that $\sum z_n$ converges.
 
-### 3.4
+<p class='text-size-12'> 3.4 </p>
+
 Define the exponential function as a power series as follows ;
 
 $$
@@ -936,7 +939,8 @@ the third and the first identity follow.
 Substituting $\phi$ by $-\phi$, the second and fourth ones follow.
 Adding and substracting these four identities, the remaining identities follow.
 
-### 3.5
+<p class='text-size-12'> 3.5 </p>
+
 Prove the followings, provided that $m$ and $n$ are integers. 
 
 $$
@@ -1019,7 +1023,8 @@ The first term always vanishes as before.
 The second term does too if $m\ne n$, while if $m=n$, the integrand is $0$ and the second term also vanishes.
 Thus the whole value always equals to 0.
 
-### 3.6
+<p class='text-size-12'> 3.6 </p>
+
 Prove that the ODE
 
 $$f''(t)+c^2f(t)=0$$
@@ -1070,7 +1075,8 @@ $$
 a\cos ct + b\sin ct = f(t)\times(\cos^2ct+\sin^2ct)=f(t).
 $$
 
-### 3.7
+<p class='text-size-12'> 3.7 </p>
+
 If $a$ and $b$ are real,
 
 $$a\cos ct+b\sin ct = A\cos(ct-\phi)$$
@@ -1093,7 +1099,8 @@ a\cos ct+b\sin ct
 \end{align*}.
 $$
 
-### 3.8
+<p class='text-size-12'> 3.8 </p>
+
 ([Asked at mathexchange](https://math.stackexchange.com/q/4980904/746048))
 
 Suppose that $F:(a,b)\to\mathbb R$ has a continuous second derivative.
@@ -1209,7 +1216,7 @@ $$
 =\lim_{h\to0}\left(F''(x)+\phi(h)+\phi(-h)\right)=F''(x)
 $$ 
 
-### 3.9
+<p class='text-size-12'> 3.9 </p>
 
 Consider the case of plucked string ; 
 
@@ -1244,9 +1251,9 @@ Note also that neither $p=0$ nor $p=\pi$ as long as $h\ne0$.
 So if $p=\frac\pi2$, then $A_2=A_4=\cdots=0$ and the even harmonics are missing.
 
 $A_3=0$ iff $p=\frac\pi3k$ and $A_6=0$ iff $p=\frac\pi6k$ for an integer $k$.
-If $p=\frac\pi3$ or $p=\frac23p$, then $A_3=A_6=\cdots=0$ and the harmonics of $3k$ are missing.
+If $p=\frac\pi3$ or $p=\frac23p$, then $A_3=A_6=\cdots=0$ and the $3k$-th harmonics are missing.
 
-### 3.10
+<p class='text-size-12'> 3.10 </p>
 
 Show that the expression of the Laplacian
 
@@ -1270,6 +1277,9 @@ $$
 $$
 
 (proof)
+
+(이 계산은 군대에서 전역하고 대학교 2학년으로 복학하던 시점에 했던 기억이 있지만 다시 해봐야지.)
+
 Note that
 
 $$
@@ -1279,58 +1289,113 @@ y&=r\sin\theta
 \end{aligned}
 $$
 
-Let's first calculate the following four derivatives ;
-$\frac{\partial f}{\partial r}$,
-$\frac{\partial^2 f}{\partial r^2}$,
-$\frac{\partial f}{\partial\theta}$,
-$\frac{\partial^2 f}{\partial\theta^2}$,
+Let $u$ be a function from $\mathbb R^2$ to $\mathbb R$
+Let's first calculate the following four derivatives  ;
+$\frac{\partial u}{\partial r}$,
+$\frac{\partial^2 u}{\partial r^2}$,
+$\frac{\partial u}{\partial\theta}$,
+$\frac{\partial^2 u}{\partial\theta^2}$,
 
 $$
 \begin{aligned}
-\frac{\partial f}{\partial r}
+\frac{\partial u}{\partial r}
 &=
-\frac{\partial x}{\partial r}\frac{\partial f}{\partial x}
+\frac{\partial x}{\partial r}\frac{\partial u}{\partial x}
 +
-\frac{\partial y}{\partial r}\frac{\partial f}{\partial y}\\
-&=\cos\theta\frac{\partial f}{\partial x}
-+\sin\theta\frac{\partial f}{\partial y}\\
-\frac{\partial^2 f}{\partial r^2}
+\frac{\partial y}{\partial r}\frac{\partial u}{\partial y}\\
+&=\cos\theta\frac{\partial u}{\partial x}
++\sin\theta\frac{\partial u}{\partial y}\\
+\frac{\partial^2 u}{\partial r^2}
 &=\frac{\partial}{\partial r}\left(
-\cos\theta\frac{\partial f}{\partial x}
-+\sin\theta\frac{\partial f}{\partial y}
+\cos\theta\frac{\partial u}{\partial x}
++\sin\theta\frac{\partial u}{\partial y}
 \right)\\
 &=\cos\theta\left(
-  \frac{\partial x}{\partial r}\frac{\partial^2f}{\partial x^2}
-  +\frac{\partial y}{\partial r}\frac{\partial^2f}{\partial x\partial y}
+  \frac{\partial x}{\partial r}\frac{\partial^2u}{\partial x^2}
+  +\frac{\partial y}{\partial r}\frac{\partial^2u}{\partial x\partial y}
 \right)
 +\sin\theta\left(
-  \frac{\partial x}{\partial r}\frac{\partial^2f}{\partial x\partial y}
-  +\frac{\partial y}{\partial r}\frac{\partial^2f}{\partial y^2}
+  \frac{\partial x}{\partial r}\frac{\partial^2u}{\partial x\partial y}
+  +\frac{\partial y}{\partial r}\frac{\partial^2u}{\partial y^2}
 \right)\\
-&=\cos^2\theta f_{xx}
-+2\sin\theta\cos\theta f_{xy}
-+\sin^2\theta f_{yy}\\
-\frac{\partial f}{\partial\theta}
+&=\cos^2\theta u_{xx}
++2\sin\theta\cos\theta u_{xy}
++\sin^2\theta u_{yy}\\
+\frac{\partial u}{\partial\theta}
 &=
-\frac{\partial x}{\partial\theta}\frac{\partial f}{\partial x}
+\frac{\partial x}{\partial\theta}\frac{\partial u}{\partial x}
 +
-\frac{\partial y}{\partial\theta}\frac{\partial f}{\partial y}\\
-&=-r\sin\theta\frac{\partial f}{\partial x}
-+r\cos\theta\frac{\partial f}{\partial y}\\
-\frac{\partial^2 f}{\partial\theta^2}
+\frac{\partial u}{\partial\theta}\frac{\partial u}{\partial y}\\
+&=-r\sin\theta\frac{\partial u}{\partial x}
++r\cos\theta\frac{\partial u}{\partial y}\\
+\frac{\partial^2 u}{\partial\theta^2}
 &=\frac{\partial}{\partial\theta}\left(
--r\sin\theta\frac{\partial f}{\partial x}
-+r\cos\theta\frac{\partial f}{\partial y}
+-r\sin\theta\frac{\partial u}{\partial x}
++r\cos\theta\frac{\partial u}{\partial y}
 \right)\\
-&=-r\cos\theta\frac{\partial f}{\partial x}
+&=
+-r\cos\theta\frac{\partial u}{\partial x}
 -r\sin\theta\left(
-  \frac{\partial x}{\partial\theta}\frac{\partial^2 f}{\partial x^2}
-  +\frac{\partial y}{\partial\theta}\frac{\partial^2 f}{\partial x\partial y}
-\right)
--r\sin\theta\frac{\partial f}{\partial y}
+  \frac{\partial x}{\partial\theta}\frac{\partial^2u}{\partial x^2}
+  +\frac{\partial y}{\partial\theta}\frac{\partial^2u}{\partial x\partial y}
+\right)\\
+&\phantom{=}
+-r\sin\theta\frac{\partial u}{\partial y}
 +r\cos\theta\left(
-  \frac{\partial x}{\partial\theta}\frac{\partial^2 f}{\partial x\partial y}
-  +\frac{\partial y}{\partial\theta}\frac{\partial^2 f}{\partial y^2}
-\right)
+  \frac{\partial x}{\partial\theta}\frac{\partial^2u}{\partial x\partial y}
+  +\frac{\partial y}{\partial\theta}\frac{\partial^2u}{\partial y^2}
+\right)\\
+&=-r\cos\theta u_x-r\sin\theta u_y
++r^2\sin^2\theta u_{xx}
++r^2\cos^2\theta u_{yy}
+-2r^2\sin\theta\cos\theta u_{xy}
+\end{aligned}
+$$
+
+Therefore,
+
+$$
+\begin{aligned}
+\left(\frac{\partial^2}{\partial r^2}+\frac1r\frac{\partial}{\partial r}+\frac1{r^2}\frac{\partial^2}{\partial\theta^2}\right) u
+&=\frac{\partial^2u}{\partial r^2}+\frac1r\frac{\partial u}{\partial r}+\frac1{r^2}\frac{\partial^2u}{\partial\theta^2}\\
+&=\cos^2\theta u_{xx}+2\sin\theta\cos\theta u_{xy}+\sin^2\theta u_{yy}\\
+&\phantom{=}
++\frac1r\cos\theta u_x+\frac1r\sin\theta u_y
+-\frac1r\cos\theta u_x-\frac1r\sin\theta u_y\\
+&\phantom{=}
++\sin^2\theta u_{xx}+\cos^2\theta u_{yy}-2\sin\theta\cos\theta u_{xy}\\
+&=u_{xx}+u_{yy}\\
+&=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}\\
+&=\left(
+  \frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}
+  \right)u\\
+\end{aligned}
+$$
+
+Since $u$ was arbitrary,
+
+$$
+\frac{\partial^2}{\partial r^2}+\frac1r\frac{\partial}{\partial r}+\frac1{r^2}\frac{\partial^2}{\partial\theta^2}
+=
+\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}
+$$
+
+as desired.
+Moreover
+
+$$
+\begin{aligned}
+\left|\frac{\partial u}{\partial r}\right|^2
++
+\frac1{r^2}\left|\frac{\partial u}{\partial\theta}\right|^2
+&=
+\left(\cos\theta u_x+\sin\theta u_y\right)^2
++
+\left(-\sin\theta u_x+\cos\theta u_y\right)^2\\
+&={f_x}^2+{f_y}^2\\
+&=
+\left|\frac{\partial u}{\partial x}\right|^2
++\frac1{r^2}
+\left|\frac{\partial u}{\partial y}\right|^2
 \end{aligned}
 $$

_sass/minimal-mistakes/_utilities.scss

@@ -660,4 +660,9 @@ a.reversefootnote {
   display: block;
   margin-left: auto;
   margin-right: auto;
-}
+}
+
+.text-size-12 {
+  font-size: 1.2rem; /* Adjust the size as needed */
+  font-weight: bold;
+}