auto-update(nvim): 2025-02-11 16:58:54

documents/by-course/math-6a/course-notes/main.typ

@@ -1,11 +1,21 @@
 #import "@youwen/zen:0.1.0": *
 #import "@preview/cetz:0.3.1"
 
+#set math.equation(numbering: "(1)")
+#show math.equation: it => {
+  if it.block and not it.has("label") [
+    #counter(math.equation).update(v => v - 1)
+    #math.equation(it.body, block: true, numbering: none)#label("")
+  ] else {
+    it
+  }
+}
+
 #show: zen.with(
   title: "Math 6A Course Notes",
   author: "Youwen Wu",
   date: "Winter 2025",
-  subtitle: [Taught by Nathan Scheley],
+  subtitle: [Taught by Nathan Schley],
 )
 
 #outline()
@@ -99,3 +109,95 @@ speed over $t$.
 $
   s(t) = integral^t_0 ||arrow(c)'(u)|| dif u
 $
+
+= Lecture #datetime(day: 12, year: 2025, month: 2).display()
+
+== Chain rule for multivariate functions
+
+We find motivation for the chain rule.
+
+Consider a hiker whose path is given by
+
+$
+  arrow(c) (t) = <x(t), y(t)>
+$
+
+and
+
+$
+  f(x,y) = x dot y
+$
+
+What does $x'(t)$ represent? Speed in $x$-direction. Likewise for $y'(t)$.
+
+Say $x'(t) = 3$, $y'(t) = 4$. Then how far did we travel in $t$ seconds?
+
+Suppose our slope in the $x$ direction is given by $m_x = 2$. Suppose the slope
+in $y$ is $m_y = -2$. In fact $m_x = f_x (x,y)$ and $m_y = f_y (x,y)$ (here
+$f_k$ is the partial derivative with respect to $k$).
+
+So each change in $t$ of 1 leads to a change in elevation up 6 meters in
+$x$-axis and down 8 meters in $y$-axis.
+
+So the total change $Delta z$ is given by
+$
+  Delta z = m_x dot Delta x + m_y dot Delta y
+$
+and analogously in calculus land
+
+$
+  (dif z) / (dif t) = (diff f) / (diff x) dot x'(t) + (diff f) / (diff y) dot y'(t)
+$<chain-rule>
+
+In fact @chain-rule is the chain rule.
+
+#fact[
+  $
+    (dif f) / (dif t) = (diff f) / (diff x) dot (diff x) / (diff t) + (diff f) / (diff y) dot (diff y) / (diff t) + (diff f) / (diff z) dot (diff z) / (diff t)
+  $
+]
+
+#example[
+  Consider $f(x) = x^x$. What is $f'(x)$?
+
+  We can do this with logarithmic differentiation but we can also do this with the multivariable chain rule.
+
+  $
+    f(x,y) =
+  $
+]
+
+#example[
+  Find the derivative $dif/(dif t) (f(x,y))$, where $f(x,y) = x^y$, $x(t) = t$,
+  and $y(t) = 1$. Assume $t > 0$.
+]
+
+#example[
+  Find the partial derivative $diff/(diff s) f(x,y,z)$ where $f(x,y,z) = x^2 y^2 + z^3$, and
+
+  $
+    x(s,t) = s t \
+    y(s, t) = s^2 t \
+    z(s,t) = s t^2
+  $
+]
+
+== Implicit differentiation
+
+Review from single variable: given $f(x,y)$ we can differentiate each term with
+respect to $x$, then collect all $(dif y)/(dif x)$ terms together and solve for
+it as a variable to obtain $(dif y)/(dif x) = f'(x,y)$.
+
+We do something similar for more variables. Main idea: extraneous variables are
+held constant in practice.
+
+Example: consider the surface $3x^2 + 5y z + z^3 = 0$. We want $(diff y)/(diff
+z)$ at some point. Use implicit differentiation by viewing the surface as a
+level set of some larger function $F(x,y,z) = 3x^2 + 5y z + x^3$ (the level set
+part is when $F(x,y,z) = 0$).
+
+By applying the product rule (really the chain rule @chain-rule)
+$
+  (diff F) / (diff x) = diff / (diff z) (3x^2 + 5 y z + z^3) = diff / (diff z) z^3 = 0 + (5 (diff y) / (diff x) z + 5y) + 3z^2 \
+  (diff y) / (diff z) = - (5y + 3z^2) / (5z)
+$

documents/by-course/pstat-120a/course-notes/main.typ

@@ -1385,6 +1385,24 @@ A discrete example:
 
 == CDFs, PMFs, PDFs
 
+#definition[
+  Let $X$ be a random variable. If we have a function $f$ such that
+
+  $
+    P(X <= b) = integral^b_(-infinity) f(x) dif x
+  $
+  for all $b in RR$, then $f$ is the *probability density function* of $X$.
+]
+
+The probability that the value of $X$ lies in $(-infinity, b]$ equals the area
+under the curve of $f$ from $-infinity$ to $b$.
+
+If $f$ satisfies this definition, then for any $B subset RR$ for which integration makes sense,
+
+$
+  P(X in B) = integral_B f(x) dif x
+$
+
 Properties of a CDF:
 
 Any CDF $F(x) = P(X <= x)$ satisfies