new files; more mindmap

9 files changed, 200 insertions, 70 deletions

diff --git a/config/emacs.el b/config/emacs.el
index 773ebff..f8b438d 100644
--- a/config/emacs.el
+++ b/config/emacs.el
@@ -338,16 +338,25 @@ Otherwise, just insert the typed character."
           (lambda () (local-set-key (kbd "y Y") #'eww-copy-page-url)))
 
 (use-package org-roam
-  :init
-  (setq org-roam-db-update-on-save t)
-  (setq org-roam-graph-viewer "firefox")
-  (setq org-roam-directory (file-truename "~/org/website/mindmap"))
-  (setq org-roam-capture-templates '(("d" "default" plain "%?"
-  :target (file+head "${title}.org"
-       "#+title: ${title}\n#+author: Preston Pan\n#+html_head: <link rel=\"stylesheet\" type=\"text/css\" href=\"../style.css\" />\n#+html_head: <script src=\"https://polyfill.io/v3/polyfill.min.js?features=es6\"></script>\n#+html_head: <script id=\"MathJax-script\" async src=\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js\"></script>\n#+options: broken-links:t")
-    :unnarrowed t)))
-  :config
-  (org-roam-db-autosync-mode))
+    :init
+    (setq org-roam-db-update-on-save t)
+    (setq org-roam-graph-viewer "chromium")
+    (setq org-roam-directory (file-truename "~/org/website/mindmap"))
+    (setq org-roam-capture-templates '(("d" "default" plain "%?"
+    :target (file+head "${title}.org"
+	 "#+title: ${title}\n#+author: Preston Pan\n#+html_head: <link rel=\"stylesheet\" type=\"text/css\" href=\"../style.css\" />\n#+html_head: <script src=\"https://polyfill.io/v3/polyfill.min.js?features=es6\"></script>\n#+html_head: <script id=\"MathJax-script\" async src=\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js\"></script>\n#+options: broken-links:t")
+      :unnarrowed t)))
+    :config
+    (org-roam-db-autosync-mode))
+
+(use-package org-roam-ui
+    :after org-roam
+    :hook (after-init . org-roam-ui-mode)
+    :config
+    (setq org-roam-ui-sync-theme t
+	  org-roam-ui-follow t
+	  org-roam-ui-update-on-save t
+	  org-roam-ui-open-on-start t))
 
 (use-package pinentry
   :init (setq epa-pinentry-mode `loopback)
diff --git a/config/emacs.org b/config/emacs.org
index 9c2ce6e..872a7a5 100644
--- a/config/emacs.org
+++ b/config/emacs.org
@@ -420,18 +420,28 @@ to Firefox if I have to:
 ** Org Roam
 For all my mathematics and programming notes:
 #+begin_src emacs-lisp
-  (use-package org-roam
-    :init
-    (setq org-roam-db-update-on-save t)
-    (setq org-roam-graph-viewer "firefox")
-    (setq org-roam-directory (file-truename "~/org/website/mindmap"))
-    (setq org-roam-capture-templates '(("d" "default" plain "%?"
-    :target (file+head "${title}.org"
-	 "#+title: ${title}\n#+author: Preston Pan\n#+html_head: <link rel=\"stylesheet\" type=\"text/css\" href=\"../style.css\" />\n#+html_head: <script src=\"https://polyfill.io/v3/polyfill.min.js?features=es6\"></script>\n#+html_head: <script id=\"MathJax-script\" async src=\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js\"></script>\n#+options: broken-links:t")
-      :unnarrowed t)))
-    :config
-    (org-roam-db-autosync-mode))
+    (use-package org-roam
+      :init
+      (setq org-roam-db-update-on-save t)
+      (setq org-roam-graph-viewer "chromium")
+      (setq org-roam-directory (file-truename "~/org/website/mindmap"))
+      (setq org-roam-capture-templates '(("d" "default" plain "%?"
+      :target (file+head "${title}.org"
+	   "#+title: ${title}\n#+author: Preston Pan\n#+html_head: <link rel=\"stylesheet\" type=\"text/css\" href=\"../style.css\" />\n#+html_head: <script src=\"https://polyfill.io/v3/polyfill.min.js?features=es6\"></script>\n#+html_head: <script id=\"MathJax-script\" async src=\"https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js\"></script>\n#+options: broken-links:t")
+	:unnarrowed t)))
+      :config
+      (org-roam-db-autosync-mode))
+
+  (use-package org-roam-ui
+      :after org-roam
+      :hook (after-init . org-roam-ui-mode)
+      :config
+      (setq org-roam-ui-sync-theme t
+	    org-roam-ui-follow t
+	    org-roam-ui-update-on-save t
+	    org-roam-ui-open-on-start t))
 #+end_src
+
 ** Pinentry
 Set up pinentry so that I can use emacs as my pinentry frontend:
 #+begin_src emacs-lisp
diff --git a/mindmap/LRC circuit.org b/mindmap/LRC circuit.org
index e28d9b0..b3e8751 100644
--- a/mindmap/LRC circuit.org
+++ b/mindmap/LRC circuit.org
@@ -9,34 +9,37 @@
 #+options: broken-links:t
 
 * Introduction
-LRC circuits are equivalent to mass-spring oscillation systems in terms of the differential equation generated. In other
+LRC circuits are equivalent to mass-spring oscillation systems in terms of the [[id:4be41e2e-52b9-4cd1-ac4c-7ecb57106692][differential equation]] generated. In other
 words, they are an example of a wave generator. First we introduce the LRC circuit without a voltage source. Later,
 another circuit diagram will include a possibly variable voltage source.
-#+name: LRC Circuit
-#+header: :exports results :file lrc_circuit.png 
+#+name: LRC Circuit Without Voltage Source
+#+header: :exports both :file lrc_circuit.png 
 #+header: :imagemagick yes :iminoptions -density 600 :imoutoptions -geometry 400 
 #+header: :fit yes :noweb yes :headers '("\\usepackage{circuitikz}")
-#+begin_src latex :exports results :file 
+#+begin_src latex :exports both :file 
     \documentclass{article}
     \usepackage{circuitikz}
     \begin{document}
     \begin{center}
     \begin{circuitikz} \draw
-    (0,0) to[resistor, l=\mbox{$R$}] (0,4)
-      to[inductor, l=\mbox{$L$}] (4,4)
-      to[capacitor, l=\mbox{$C$}] (4,0)
-      (4,0) -- (0,0)
-      (2,0) -- (2,-1)
-      to (2, -1) node[shape=ground]{};
+    (0,0) to[resistor, l=\mbox{$R$}] (0,12)
+      to[inductor, l=\mbox{$L$}] (12,12)
+      to[capacitor, l=\mbox{$C$}] (12,0)
+      (12,0) -- (0,0)
+      (6,0) -- (6,-3)
+      to (6, -3) node[shape=ground]{};
     \end{circuitikz}
     \end{center}
     \end{document}
 #+end_src
 
-#+RESULTS: LRC Circuit
+#+RESULTS: LRC Circuit Without Voltage Source
 #+begin_export latex
 #+end_export
 
+#+CAPTION: LRC Circuit without voltage source
+[[./lrc_circuit.png]]
+
 * Mass-Spring Equation Equivalence
 We know these relations for the given circuit elements above:
 \begin{align}
@@ -44,7 +47,7 @@ v(t) = L\frac{di}{dt} \\
 i(t) = C\frac{dv}{dt} \\
 v = iR
 \end{align}
-if we analyze the current current signal, Kirchhoff's voltage law tells us that the total voltage
+if we analyze the current signal, Kirchhoff's voltage law tells us that the total voltage
 drop throughout this circuit is zero. We use the capacitor equation in integral form and sum the voltages:
 \begin{align*}
 L\frac{di}{dt} + \frac{1}{C}\int i(t)dt + iR = 0
@@ -54,7 +57,7 @@ We then take a derivative to remove the integral:
 L\frac{d^{2}i}{dt^{2}} + R\frac{di}{dt} + \frac{1}{C}i = 0 \\
 (LD^{2} + RD + \frac{1}{C}) i(t) = 0
 \end{align*}
-it is clear that the characteristic polynomial of this homogeneous linear differential equation is:
+it is clear that the characteristic polynomial of this homogeneous linear [[id:4be41e2e-52b9-4cd1-ac4c-7ecb57106692][differential equation]] is:
 \begin{align*}
 L\lambda^{2} + R\lambda + \frac{1}{C} = 0
 \end{align*}
@@ -63,22 +66,56 @@ which, utilizing the quadratic formula, has the solutions:
 \lambda_{1} = \frac{-R + \sqrt{R^{2} - \frac{4L}{C}}}{2L},
 \lambda_{2} = \frac{-R - \sqrt{R^{2} - \frac{4L}{C}}}{2L}
 \end{align*}
-which implies the general solution to this differential equation is:
+which implies the general solution to this [[id:4be41e2e-52b9-4cd1-ac4c-7ecb57106692][differential equation]] is:
 \begin{align*}
-i(t) = \sum_{n=0}^{\infty} A_{n}e^{\lambda_{1} t} + B_{n}e^{\lambda_{2} t}
+i(t) = Ae^{\lambda_{1} t} + Be^{\lambda_{2} t}
 \end{align*}
 We can make this nicer by setting $-\frac{R}{2L} = m$, $\frac{\sqrt{R^{2} - \frac{4L}{C}}}{2L} = p$,
 then $\lambda_{1} = m + p$, $\lambda_{2} = m - p$. Then:
 \begin{align*}
-i(t) = \sum_{n=0}^{\infty} A_{n}e^{(m + p) t} + B_{n}e^{(m - p) t} \\
-i(t) = e^{m}(\sum_{n=0}^{\infty} A_{n}e^{pt} + B_{n}e^{-pt})
+i(t) = Ae^{(m + p) t} + Be^{(m - p) t}
 \end{align*}
-Then we can just recast our notation for the constants $A_{n}$ and $B_{n}$ to include this $e^{m}$ term:
+** Underdampened Oscillation
+In the case $R^{2} < \frac{4L}{C}$, $p = i\frac{\sqrt{\sigma}}{2L}$ for some $\sigma > 0$. We re-cast $\lambda = \frac{\sqrt{\sigma}}{2L}$ so $p = i\lambda$. Then:
 \begin{align*}
-i(t) = \sum_{n=0}^{\infty} A_{n}e^{pt} + B_{n}e^{-pt}
+i(t) = Ae^{m + i\lambda t} + Be^{m -i\lambda t}
 \end{align*}
-** Dampened Oscillation
-In the case $R^{2} < \frac{4L}{C}$, $p = i\frac{\sqrt{\sigma}}{2L}$ for some $\sigma > 0$. We re-case $\lambda = \frac{\sqrt{\sigma}}{2L}$ so $p = i\lambda$. Then:
+This function $i(t)$ clearly describes an sinusoidal oscillation.
+** Critical Oscillation
+In the case $R^{2} - \frac{4L}{C} = 0$, we have:
+\begin{align*}
+i(t) = Ae^{mt} + Be^{mt} = Ce^{mt}
+\end{align*}
+note that this is actually a decaying solution because $m$ must be negative.
+* AC [[id:951db9ac-3e8b-49a1-b609-2bbb795be834][Voltage]] Source
+Here is the circuit diagram for the LRC circuit with a voltage source:
+#+name: LRC Circuit
+#+header: :exports both :file lrc_circuit_source.png 
+#+header: :imagemagick yes :iminoptions -density 600 :imoutoptions -geometry 400 
+#+header: :fit yes :noweb yes :headers '("\\usepackage{circuitikz}")
+#+begin_src latex :exports both :file 
+    \documentclass{article}
+    \usepackage{circuitikz}
+    \begin{document}
+    \begin{center}
+    \begin{circuitikz} \draw
+    (0,0) to[resistor, l=\mbox{$R$}] (0,12)
+      to[inductor, l=\mbox{$L$}] (12,12)
+      to[capacitor, l=\mbox{$C$}] (12,0)
+      (12,0) to[sinusoidal voltage source] (0,0);
+    \end{circuitikz}
+    \end{center}
+    \end{document}
+#+end_src
+
+#+RESULTS: LRC Circuit
+#+begin_export latex
+#+end_export
+
+#+CAPTION: LRC Circuit
+[[./lrc_circuit_source.png]]
+This new [[id:4be41e2e-52b9-4cd1-ac4c-7ecb57106692][differential equation]] looks like this:
 \begin{align*}
-i(t) = \sum_{n=0}^{\infty} A_{n}e^{i\lambda t} + B_{n}e^{-i\lambda t}
+[LD^{2} + RD + \frac{1}{C}]i(t) = V_{0}sin(\phi + 2\pi\omega t)
 \end{align*}
+where the right hand side of the equation includes the term created by the AC [[id:951db9ac-3e8b-49a1-b609-2bbb795be834][voltage]] source.
diff --git a/mindmap/Lagrangian mechanics.org b/mindmap/Lagrangian mechanics.org
index d306be7..c3e078d 100644
--- a/mindmap/Lagrangian mechanics.org
+++ b/mindmap/Lagrangian mechanics.org
@@ -19,7 +19,10 @@ J[f] = \int_{a}^{b}L(t, f(t), f'(t))dt \\
 Defines the actual relationship between $f(t)$ and its level of optimization, where $a$ and $b$ represent the start
 and end points for a certain curve. For example, if you wanted to minimize the surface area of something, $a$ and $b$
 would be the starting and end points of the surface.
-* Euler-Lagrange Equation
+* Euler-Lagrange equation
+:PROPERTIES:
+:ID:       aaba4bf0-3d82-4ede-8cf3-0a1ccddcd376
+:END:
 We first define some function:
 \begin{align*}
 g(t) := f(t) + \epsilon \nu(t)
@@ -69,7 +72,10 @@ must be zero, we get the Euler-Lagrange equation:
 This is because the integral implies that for all selections for this function $\nu(t)$, $\nu(t)(\frac{dL}{df} - \frac{d}{dt}\frac{dL}{dg'}) = 0$. Because $\nu(t)$ can be any
 function assuming it satisfies the boundary conditions, this can only be the case if $\frac{dL}{df} - \frac{d}{dt}\frac{dL}{dg'} = 0$.
 In physics, we re-cast $f$ as $q$ and $f'$ as $\dot{q}$, where $q$ and $\dot{q}$ are the /generalized coordinates/ and /generalized velocities/ respectively.
-* The Hamiltonian
+* Hamiltonian
+:PROPERTIES:
+:ID:       3473dbbe-35b8-4aad-b08f-f02d9929c932
+:END:
 The Hamiltonian represents the total energy in the system; it is the [[id:23df3ba6-ffb2-4805-b678-c5f167b681de][Legendre Transformation]] of the Lagrangian. Applying the Legendre Transformation to the
 Lagrangian for coordinate $\dot{q}$:
 \begin{align*}
diff --git a/mindmap/Maxwell's Equations.org b/mindmap/Maxwell's Equations.org
index c092100..77681a9 100644
--- a/mindmap/Maxwell's Equations.org
+++ b/mindmap/Maxwell's Equations.org
@@ -26,7 +26,7 @@ The divergence of the magnetic field is the same as in [[id:5c36d0f1-06ad-436a-a
 \vec{\nabla} \cdot \vec{B} = 0
 \end{align*}
 ** Ampere's Law with Modifications
-The [[id:5c36d0f1-06ad-436a-a56f-5ecc198b9b3e][magnetostatic]] magnetic field is given by the Bio-Savart Law, which can be derived from [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]] and [[id:5388f4e8-7bb8-452e-b997-fe9892aefcf3][Coulomb's Law]]:
+The [[id:5c36d0f1-06ad-436a-a56f-5ecc198b9b3e][magnetostatic]] magnetic field is given by the Bio-Savart Law:
 \begin{align*}
 \vec{B} = \frac{\mu_{0}}{4\pi}\int_{V}\frac{\vec{J} \times \hat{r}}{r^{2}}d\tau
 \end{align*}
@@ -56,7 +56,8 @@ By definition of electromagnetic induction (and to make Ampere's law consistent
 \vec{\nabla} \times \vec{E} = - \frac{\partial\vec{B}}{\partial t}
 \end{align*}
 Instead of assuming induction as an axiom, it is possible to fix Ampere's equation with the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] first, and then
-assume Lorentz Covariance. This explanation is a work in progress.
+assume Lorentz Covariance. This explanation is a work in progress. Though, a possible explanation is through the fact
+that this is the simplest way for Maxwell's equation to describe waves in a vacuum.
 * Implications
 Maxwell's Equations can be used to calculate all electromagnetic phenomena on the macro scale all the way down to the atom.
 In practice, solving Maxwell's Equations can be analytically impossible, so several simplifying assumptions are often made.
@@ -77,30 +78,88 @@ It is known that Maxwell's Equations are consistent with [[id:e38d94f2-8332-4811
 in terms of curved spacetime. In fact, if relativity is taken as an axiom, it can be proven that the electric
 and magnetic fields are descriptions of the same phenomena; this can be taken as a specific example of a [[id:1b1a8cff-1d20-4689-8466-ea88411007d7][duality]].
 The result is a Lorentz Invariant theory of Electromagnetism.
-** Relativistic Electrodynamics
-The [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] has a Lorentz Invariant counterpart. We unify the charge density and the current density
-under a four-vector quantity ~j~:
-\begin{align*}
-j = (\rho , J)
-\end{align*}
-Where ~J~ is the current density, and $$ \rho $$ is the charge density. The continuity equation can be reformulated
-like so:
+** Gauss-Ampere Equation
+The [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] has a Lorentz Invariant counterpart, as predicted by [[id:e38d94f2-8332-4811-b7bd-060f80fcfa9b][special relativity]]. We take a look at
+Maxwell's equations (specifically Gauss' law and Ampere's law) in the uncondensed form:
+\begin{align}
+\partial_{x}E_{x} + \partial_{y}E_{y} + \partial_{z}E_{z} = \frac{\rho}{\epsilon_{0}} \\
+\partial_{x}E_{x} + \partial_{y}E_{y} + \partial_{z}E_{z} = \mu_{0}c^{2}\rho \\
+\frac{1}{c}(\partial_{x}E_{x} + \partial_{y}E_{y} + \partial_{z}E_{z}) = \mu_{0}j_{t}
+\end{align}
+Where $j_{t} = c\rho$. Once we formulate a covariant form of the continuity equation, this will become more clear.
+And now the unexpanded version of Ampere's law:
+\begin{align}
+\partial_{y}E_{z} - \partial_{z}E_{y} = \mu_{0}j_{x} + \frac{1}{c}\partial_{t}E_{x} \\
+\partial_{z}E_{x} - \partial_{x}E_{z}  = \mu_{0}j_{y} + \frac{1}{c}\partial_{t}E_{y} \\
+\partial_{x}E_{y} - \partial_{y}E_{x}  = \mu_{0}j_{z} + \frac{1}{c}\partial_{t}E_{z}
+\end{align}
+(where $\partial_{t} = \frac{1}{c}\frac{\partial}{\partial t}$). We can now construct a 4-dimensional rank-2 tensor equation for these equations:
+\begin{align}
+D \cdot
+\begin{pmatrix}
+0 && \frac{1}{c}E_{x} && \frac{1}{c}E_{y} && \frac{1}{c}E_{z} \\
+-\frac{1}{c}E_{x} && 0 && E_{z} && -E_{y} \\
+-\frac{1}{c}E_{y} && -E_{z} && 0 && E_{x} \\
+-\frac{1}{c}E_{z} && E_{y} && -E_{x} && 0 \\
+\end{pmatrix}
+= \mu_{0}
+\begin{pmatrix}
+j_{t} \\
+j_{x} \\
+j_{y} \\
+j_{z}
+\end{pmatrix}
+\end{align}
+where the right hand side is an emergent four-vector $(c\rho, j)$, and $D$ is an operator that takes the t, x, y,
+and z derivative of each respective column and then sums the rows to make a vector. We can reformulate
+the [[id:a871e62c-b4a0-4674-9dea-d377de2f780b][continuity equation]] in terms of this four-vector:
 \begin{align*}
-\partial_{\alpha} j^{\alpha} = 0
+\vec{\nabla} \cdot \vec{j} = 0
 \end{align*}
-Likewise, the electric scalar potential and the magnetic vector potential create an invariant four-vector:
+Where $\vec{\nabla}$ in this case represents the del operator but with a forth time dimension.
+If we condense the notation, we can write:
 \begin{align*}
-a = (V, A)
+D \cdot M = \vec{j}
 \end{align*}
-We notice that Gauss' Law:
+** Gauss-Faraday Equation
+By the same logic, we can use the remaining two equations (the divergence and curl of $B$) to construct this tensor:
+\begin{align}
+D \cdot
+\begin{pmatrix}
+0 && -B_{x} && -B_{y} && -B_{z} \\
+B_{x} && 0 && -B_{z} && B_{y} \\
+B_{y} && B_{z} && 0 && -B_{x} \\
+B_{z} && -B_{y} && B_{x} && 0 \\
+\end{pmatrix}
+= \vec{0}
+\end{align}
+we can write this in compact form:
 \begin{align*}
-\nabla^{2} V = \frac{\rho}{\epsilon_{0}}
+D \cdot M' = \vec{0}
 \end{align*}
-needs a more general formulation, as the potential and the charge density need to be replaced with Lorentz Invariant
-quantities. We then use Ampere's law:
+$M'$ is the dual tensor of $M$, where all the $E_{n}$ are swapped with $B_{n}$, and it is negative. Note that
+all these tensors are antisymmetric; they are symmetric under reflection across their diagonals and by flipping the signs.
+** Covariant Form
+The covariant form can be reached by gauge-fixing. Gauge fixing is a process wherein we utilize gauge invariance
+(invariance to a scalar addition to a potential) in order to fix a single scalar and remove a redundancy. Remember that
+in [[id:63713308-0ff7-433f-8103-8b64ba9bdbe1][electrostatics]]:
+\begin{align}
+\vec{E} = -\vec{\nabla}V
+\end{align}
+and in [[id:5c36d0f1-06ad-436a-a56f-5ecc198b9b3e][magnetostatics]]:
+\begin{align}
+\vec{B} = \vec{\nabla} \times \vec{A}
+\end{align}
+but in electrodynamics:
+\begin{align}
+\vec{E} = -\vec{\nabla}V - \frac{\partial \vec{A}}{\partial t} \\
+\vec{B} = \vec{\nabla} \times \vec{A}
+\end{align}
+because any [[id:6f2aba40-5c9f-406b-a1fa-13018de55648][conservative force]] field added to $\vec{A}$ preserves the properties of $\vec{B}$ when you take the [[id:b25e0e44-c764-4f0a-a5ad-7f9d79c7660d][curl]].
+You can verify that if we take it for granted that $(\frac{V}{c}, A)$ is covariant, we can formulate the above two tensor
+equations in terms of a single equation:
 \begin{align}
-\vec{\nabla} \times \vec{\nabla} \times \vec{A} = \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial(\nabla V)}{\partial t} \\
-\vec{\nabla} \cdot \vec{\nabla} \times \vec{\nabla} \times \vec{A} = \nabla \cdot \mu_{0}\vec{J} + \mu_{0}\epsilon_{0}\frac{\partial(\nabla^{2} V)}{\partial t} \\
-\vec{\nabla} \cdot \vec{\nabla} \times \vec{\nabla} \times \vec{A} = \mu_{0} \nabla\cdot\vec{J} + \mu_{0}\frac{\partial\rho}{\partial t} \\
-\vec{\nabla} \cdot \vec{\nabla} \times \vec{\nabla} \times \vec{A} = \mu_{0} (\nabla\cdot\vec{J} + \frac{\partial\rho}{\partial t}) \\
+D \cdot (M + M') = \mu_{0}\vec{j} \\
+\Box A = \mu_{0}\vec{j}
 \end{align}
+where $\Box = \partial^{2}_{t} - \vec{\nabla}$. This single equation is the covariant Maxwell's equation. Truly a beautiful sight.
diff --git a/mindmap/electrostatics.org b/mindmap/electrostatics.org
index 6625d4c..d865f86 100644
--- a/mindmap/electrostatics.org
+++ b/mindmap/electrostatics.org
@@ -1,5 +1,5 @@
 :PROPERTIES:
-:ID:       32f0b8b1-17bc-4c91-a824-2f2a3bbbdbd1
+:ID:       63713308-0ff7-433f-8103-8b64ba9bdbe1
 :END:
 #+title: electrostatics
 #+author: Preston Pan
@@ -113,10 +113,16 @@ field:
 \oint\vec{E} \cdot d\vec{l} = \vec{0}
 \end{align*}
 
-* Electrostatic Potentials
+* electrostatic potential
+:PROPERTIES:
+:ID:       951db9ac-3e8b-49a1-b609-2bbb795be834
+:ROAM_ALIASES: "potential difference" voltage
+:END:
 Because \(\vec{E}\) is a [[id:6f2aba40-5c9f-406b-a1fa-13018de55648][conservative field]]:
 \begin{align*}
 \vec{E} = -\vec{\nabla}V \\
 \nabla^{2}V = -\frac{\sigma(\vec{r_{1}})}{\epsilon_{0}} \\
-V(\vec{r}) = \frac{1}{4\pi \epsilon_{0}}\int\frac{\sigma(\vec{r_{2}})}{r}dr
+V(\vec{r}) = -\frac{1}{4\pi \epsilon_{0}}\int\frac{\sigma(\vec{r_{2}})}{r}dr
 \end{align*}
+Note that the difference of two points in the potential scalar field is also known as potential difference
+between those two points, or the voltage between those two points.
diff --git a/mindmap/everything.org b/mindmap/everything.org
index 57321af..b33f754 100644
--- a/mindmap/everything.org
+++ b/mindmap/everything.org
@@ -6,7 +6,6 @@
 #+html_head: <link rel="stylesheet" type="text/css" href="../style.css" />
 #+date: <2023-06-10 Sat>
 
-#+OPTIONS: tex:dvipng
 * Everything?
 This article is about everything. No, really, it is.
 Every possible thought you could have is encapsulated
diff --git a/mindmap/index.org b/mindmap/index.org
index 98144d6..1177694 100644
--- a/mindmap/index.org
+++ b/mindmap/index.org
@@ -66,13 +66,16 @@ set +f
 - [[file:mass-spring system.org][mass-spring system.org]]
 - [[file:magnetostatics.org][magnetostatics.org]]
 - [[file:ltximg][ltximg]]
+- [[file:lrc_circuit_source.png][lrc_circuit_source.png]]
 - [[file:lrc_circuit.png][lrc_circuit.png]]
 - [[file:LRC circuit.org][LRC circuit.org]]
+- [[file:#LRC circuit.org#][#LRC circuit.org#]]
 - [[file:Lorentz Force.org][Lorentz Force.org]]
 - [[file:logic.org][logic.org]]
 - [[file:limit.org][limit.org]]
 - [[file:Legendre Transformation.org][Legendre Transformation.org]]
 - [[file:Lagrangian mechanics.org][Lagrangian mechanics.org]]
+- [[file:Kirchhoff's Laws.org][Kirchhoff's Laws.org]]
 - [[file:inverse square.org][inverse square.org]]
 - [[file:inheritance.org][inheritance.org]]
 - [[file:infinity.org][infinity.org]]
@@ -89,6 +92,7 @@ set +f
 - [[file:duality.org][duality.org]]
 - [[file:divergence theorem.org][divergence theorem.org]]
 - [[file:dirac delta.org][dirac delta.org]]
+- [[file:differential equation.org][differential equation.org]]
 - [[file:derivative.org][derivative.org]]
 - [[file:del operator.org][del operator.org]]
 - [[file:continuity equation.org][continuity equation.org]]
diff --git a/mindmap/lrc_circuit.png b/mindmap/lrc_circuit.png
index af03da3..d512172 100644
--- a/mindmap/lrc_circuit.png
+++ b/mindmap/lrc_circuit.png