From 5042277a35f3c3393b480b90cb2cd99a7de9c296 Mon Sep 17 00:00:00 2001 From: Sasserisop Date: Fri, 10 Nov 2023 10:38:44 -0700 Subject: [PATCH] fixed a bug that interprets *'s into bold/italic inside equations --- content/Cauchy-Euler equations (lec 10-11).md | 8 ++++---- content/Dirak δ-function (lec 21).md | 2 +- content/Free vibrations (lec 11-12).md | 2 +- content/Linear equations (lec 2-3).md | 4 ++-- .../Method of undetermined coefficients (lec 8-9).md | 10 +++++----- content/Partial differential equations (lec 26).md | 1 + content/Periodic functions (lec 19).md | 1 + content/Power series (lec 22-25).md | 8 ++++---- ...cond order homogenous linear equations (lec 5-7).md | 2 +- ...olving IVP's using Laplace transform (lec 17-18).md | 2 +- themes/zettels/layouts/partials/content.html | 4 ++++ 11 files changed, 25 insertions(+), 19 deletions(-) diff --git a/content/Cauchy-Euler equations (lec 10-11).md b/content/Cauchy-Euler equations (lec 10-11).md index f5c9875..c8c7aa7 100644 --- a/content/Cauchy-Euler equations (lec 10-11).md +++ b/content/Cauchy-Euler equations (lec 10-11).md @@ -1,7 +1,7 @@ We know how to solve second order equations where a, b, c are constants. Even if they're not constant some can be expressed as a linear equation. But not always will they be solvable. However, there is one class of second order equations with non constant coefficients that are always solvable. # Cauchy-Euler equations -*If it has a name in it, its very important, if it has 2 names, its very very important!* +If it has a name in it, its very important, if it has 2 names, its very very important! #cauchy-euler equations are equations in the form: $$ax^2{\frac{d^2y}{dx^2}+bx{\frac{ dy }{ dx }}+cy=f(x)},\ x>0$$ where $a,\ b,\ c$ are still constants and $\in \mathbb{R}$ @@ -50,13 +50,13 @@ substitute: $y=x^r$ after calculating derivatives, plugging in, and simplifying we obtain the polynomial equation: $ar^2+(b-a)r+C=0$ Three cases: -**(i)** $r_1\ne r_{2}$ then: +(i) $r_1\ne r_{2}$ then: $y_{h}(t)=c_{1}e^{rt}+c_{2}e^{rt}$ $y_{h}(x)=c_{1}x^{r_{1}}+c_{2}x^{r_{2}}$ (lousy notation, because the two $y_{h}$ do not equal each other) -**(ii)** $r_{1}=r_{2}=r$ then: +(ii) $r_{1}=r_{2}=r$ then: $y_{h}(t)=c_{1}e^{rt}+c_{2}te^{rt}$ $y_{h}(x)=c_{1}x^r+c_{2}x^r\ln(x)$ (derived by reduction of order.) -**(iii)** $r_{1,2}=\alpha\pm i\beta$ then: +(iii) $r_{1,2}=\alpha\pm i\beta$ then: $y_{h}=e^\alpha(c_{1}\cos(\beta t)+c_{2}\sin(\beta t))$ $y_{h}(x)=x^\alpha(c_{1}\cos(\beta\ln x)+c_{2}\sin(\beta \ln x))$ Now compute your particular solution, $y_{p}$, and combine with $y_{h}$ to obtain your general solution. \ No newline at end of file diff --git a/content/Dirak δ-function (lec 21).md b/content/Dirak δ-function (lec 21).md index 16645e2..9626594 100644 --- a/content/Dirak δ-function (lec 21).md +++ b/content/Dirak δ-function (lec 21).md @@ -104,7 +104,7 @@ $I_{1}=3-2e^{-5t}-e^{-20t}-u(t-1)(3-2e^{-5(t-1)}-e^{-20(t-1)})$ whats $I_{3}$? use: $-10I_{1}+10I_{3}+I_{3}'=0$ (i think) $J_{3}=\frac{10}{s+10}J_{1}$ $J_{3}=300 \frac{1}{s(s+5)(s+20)}(1-e^{ -s })$ -partial fraction it so we can eventuall take the inverse LT: +partial fraction it so we can eventually take the inverse LT: skip some steps: $J_{3}=\left( \frac{3}{s}-\frac{4}{s+5}+\frac{1}{s+20} \right)(1-e^{ -s })$ $I_{3}=3-4e^{ -5t }+e^{ -20t }-(3-4e^{ -5(t-1) }+e^{ -20(t-1) })u(t-1)$ diff --git a/content/Free vibrations (lec 11-12).md b/content/Free vibrations (lec 11-12).md index c394428..baed458 100644 --- a/content/Free vibrations (lec 11-12).md +++ b/content/Free vibrations (lec 11-12).md @@ -48,7 +48,7 @@ $\phi$ is the phase shift in radians. $\frac{\omega}{2\pi}$ is the natural frequency (oscillations/second). Taking the inverse, $\frac{2\pi}{\omega}$ is the period (seconds/oscillation). This behavior is all a result of classical mechanics, but rather beautifully, electronic circuits composed of resistors, inductors, and capacitors can also be described by this equation. You'll see it in biology too. -*"... Nature is beautiful and harmonic."* -Prof (not an exact quotation, I didn't catch what he precisely said.) +"... Nature is beautiful and harmonic." -Prof (not an exact quotation, I didn't catch what he precisely said.) btw we know $A=\sqrt{ c_{1}^2+c_{2}^2 }$ and $\tan \phi=\frac{c_{1}}{c_{2}}$ so we can compute $A$ and $\phi$ from $c_1$ and $c_2$. diff --git a/content/Linear equations (lec 2-3).md b/content/Linear equations (lec 2-3).md index 88b5dcd..2b9e110 100644 --- a/content/Linear equations (lec 2-3).md +++ b/content/Linear equations (lec 2-3).md @@ -1,5 +1,5 @@ - -*The world is non-linear, many solutions, many paths to the solution. It's why linear equations play so nice. We just look down it's path and we will know that it's a straight line for eternity.* +> +The world is non-linear, many solutions, many paths to the solution. It's why linear equations play so nice. We just look down it's path and we will know that it's a straight line for eternity. # Linear equation: $$a(x)\frac{ dy }{ dx }+b(x)y=f(x)$$ diff --git a/content/Method of undetermined coefficients (lec 8-9).md b/content/Method of undetermined coefficients (lec 8-9).md index db86fd2..150b95e 100644 --- a/content/Method of undetermined coefficients (lec 8-9).md +++ b/content/Method of undetermined coefficients (lec 8-9).md @@ -6,9 +6,9 @@ in the case when $f(t)=0$ : 1) $ay''+b'y+cy=0$ then $ar^2+br+c=0$ and solve with quadratic formula general solutions are: -if $r_{1}\ne r_{2}\Rightarrow y_{h}(t)=c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t}$ <- **overdamped** -if $r_{1}=r_{2}\Rightarrow y_{h}(t)=c_{1}e^{rt}+c_{2}te^{rt}$ <- **critically damped** -if $r_{1,2}\in \mathbb{C}\Rightarrow y_{h}(t)=e^{\alpha t}(c_{1}\cos(\beta t)+c_{2}\sin(\beta t))$ <- **underdamped** +if $r_{1}\ne r_{2}\Rightarrow y_{h}(t)=c_{1}e^{r_{1}t}+c_{2}e^{r_{2}t}$ <- overdamped +if $r_{1}=r_{2}\Rightarrow y_{h}(t)=c_{1}e^{rt}+c_{2}te^{rt}$ <- critically damped +if $r_{1,2}\in \mathbb{C}\Rightarrow y_{h}(t)=e^{\alpha t}(c_{1}\cos(\beta t)+c_{2}\sin(\beta t))$ <- underdamped where h means homogenous, (when $f(t)=0$ the equation is homogenous.) But what about the case when $f(t)\ne 0$ ? @@ -92,10 +92,10 @@ final solution to IVP: $$y(t)=e^{-t}(\sin t+2)+\cos t+2\sin t$$ #end of lec 8 #start of lec 9 -*remember in a previous example when we had to guess that $y_{p}=At^2e^{2t}$? Here is a generalized algorithm that can find $y_{p}$ when the RHS falls under the following form. Reducing the guess work to zero:* +remember in a previous example when we had to guess that $y_{p}=At^2e^{2t}$? Here is a generalized algorithm that can find $y_{p}$ when the RHS falls under the following form. Reducing the guess work to zero: # Generalized guesses for undetermined coefficients: case i) $ay''+by'+cy=P_m(t)e^{rt}$ -where $P_{m}(t)=a_{m}t^m+a_{m-1}t^{m-1}+ \dots +a_{0}$ *i.e. P is a polynomial degree m.* +where $P_{m}(t)=a_{m}t^m+a_{m-1}t^{m-1}+ \dots +a_{0}$ i.e. P is a polynomial degree m. Then we guess the particular solution is of the form: $y_{p}(t)=t^s(b_{m}t^m+b_{m-1}t^{m-1}+\dots+b_{0})e^{rt}$ where: s=0, if r is not a root, diff --git a/content/Partial differential equations (lec 26).md b/content/Partial differential equations (lec 26).md index 4d6434e..87ef45e 100644 --- a/content/Partial differential equations (lec 26).md +++ b/content/Partial differential equations (lec 26).md @@ -6,6 +6,7 @@ Heat equation not only describes thermodynamics but it can also model the diffus Strikingly, it can also model option prices in the stock market. However, using it as a strategy to make money is not so simple, because if it worked then everyone would try to use it to make money, which would cause the overall strategy to be less effective as the option prices start to get priced to accommodate for the prediction (🤯). ![[Drawing 2023-11-08 13.07.19.excalidraw.png]] >I'm sorry the image doesn't display properly :( I'm trying to get images to work on my notes. For now you can see the relevant .png files in the github repo under content/drawings/ + We assume that the tube is perfectly insulating along its surface, this helps reduce the problem into a one dimensional problem. Heat can only travel inside and along the x axis. Fourier figured out that: $\text{Heat flux} = -k(x)a\frac{\partial u}{\partial x}(t,x) \Delta t$ diff --git a/content/Periodic functions (lec 19).md b/content/Periodic functions (lec 19).md index b18c2c8..11c7dcd 100644 --- a/content/Periodic functions (lec 19).md +++ b/content/Periodic functions (lec 19).md @@ -1,3 +1,4 @@ + #start of lec 19 This lecture we will learn about periodic functions, specifically, non-sinusoidal periodic functions. # Periodic function diff --git a/content/Power series (lec 22-25).md b/content/Power series (lec 22-25).md index ad9d74a..1da4706 100644 --- a/content/Power series (lec 22-25).md +++ b/content/Power series (lec 22-25).md @@ -149,18 +149,18 @@ $a_{4}=\frac{2^2}{4!}a_{1}$ the pattern leads us to: $a_{3k+1}=\frac{(2\cdot 5 \dots(3k-1))^2}{(3k+1)!}$ where k=1,2,3, ... $a_{3k}=\frac{(1\cdot 4\cdot \dots(3k-2))^2}{(3k)!}a_{0}$ k=1,2,... -$z(x)=a_{0}\left( 1+\sum_{k=1}^\infty \frac{(1*4*\dots(3k-2))^2}{(3k)!}x^{3k} \right)$ +$z(x)=a_{0}\left( 1+\sum_{k=1}^\infty \frac{(1\cdot4\cdot\dots(3k-2))^2}{(3k)!}x^{3k} \right)$ $a_{1}\left( x+\sum_{k=1}^\infty \frac{(2\cdot 5\cdot \dots(3k-1))^2}{(3k+1)!} x^{3k+1}\right)$ there we go, $z$ is a linear combination of those two expressions class done at 1:56 (a lil late but the journey is worth it) #end of lec 23 #start of lec 24 -*midterms have been marked and returned today.* +midterms have been marked and returned today. we consider: $$y''+p(x)y'+q(x)y=0$$ this is in standard form, it's a second order linear equation Definition: -if $p(x)$ and $q(x)$ are **analytic** functions in a vicinity of $x_{0}$ then $x_0$ is **ordinary**. Otherwise, $x_{0}$ is **singular**. +if $p(x)$ and $q(x)$ are analytic functions in a vicinity of $x_{0}$ then $x_0$ is ordinary. Otherwise, $x_{0}$ is singular. we expect that the solution y can be represented by a power series. This is true according to the following theorem: Theorem: If $x_{0}$ is ordinary point then the differential equation above has two linearly independent solution of the form $\sum_{n=0} ^\infty a_{n}(x-x_{0})^n, \qquad\sum_{n=0}^\infty b_{n}(x-x_{0})^n$. The radius of convergence for them is at least as large as the distance between $x_{0}$ and the closest singular point (which can be real or complex). @@ -173,7 +173,7 @@ put it in standard form: $y''-\frac{3xy'}{x+1}+\frac{2y}{x+1}=0$ the only singular point for this equation is $x=-1$ so the minimum value of radius convergence is $\rho=2$ (distance between -1 and x_0) -we are guaranteed that the power series will converge *at least* in $(-1,3)$, possibly more. You can try solving for y as a power series. +we are guaranteed that the power series will converge at least in $(-1,3)$, possibly more. You can try solving for y as a power series. #ex $$y''-\tan xy'+y=0 \quad x_{0}=0$$ diff --git a/content/Second order homogenous linear equations (lec 5-7).md b/content/Second order homogenous linear equations (lec 5-7).md index bac4992..d1a9fc1 100644 --- a/content/Second order homogenous linear equations (lec 5-7).md +++ b/content/Second order homogenous linear equations (lec 5-7).md @@ -1,5 +1,5 @@ most of these "models" in EE are based on these DE. You'll see how important DE are in chemical, electrical, mechanical, engphys, civil (very important for civil!), (mining? idk what's in mining :D -prof) -*DE are important* -prof +DE's are important -prof ## Second order linear equations Second order equations arise from very simple problems many engineers face, for instance a pendulum can be described by a second order equation. #second_order diff --git a/content/Solving IVP's using Laplace transform (lec 17-18).md b/content/Solving IVP's using Laplace transform (lec 17-18).md index 041993e..84b25da 100644 --- a/content/Solving IVP's using Laplace transform (lec 17-18).md +++ b/content/Solving IVP's using Laplace transform (lec 17-18).md @@ -1,6 +1,6 @@ #start of lec 17 -*Lecture starts with a 25 minute long midterm review.* +Lecture starts with a 25 minute long midterm review. He spends most of the time giving us a summary of all the equations (separable, linear, homogenous, exact, ...) The review is available on the eclass page. So, why did we learn all this stuff about Laplace transforms? We will now see how its useful: diff --git a/themes/zettels/layouts/partials/content.html b/themes/zettels/layouts/partials/content.html index 8977608..b2e5f72 100644 --- a/themes/zettels/layouts/partials/content.html +++ b/themes/zettels/layouts/partials/content.html @@ -30,6 +30,10 @@ {{ end }} {{ $content := .Scratch.Get "content" }} + +{{/* $content := $content | replaceRE "(\\$.*)\\*(.*\\$)" "$1$2" */}} +{{ $content := $content | replaceRE "\\*" "\\*"}} +