From 30ef4e9ca8b3c236fb98c420f844a74f83ffedee Mon Sep 17 00:00:00 2001
From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com>
Date: Mon, 13 Jan 2025 10:46:58 -0600
Subject: [PATCH] new course

---
 .../LNG_v1.0.py}                              |  31 ++++--
 pages/Math4121/Math4121_L1.md                 | 105 ++++++++++++++++++
 pages/Math4121/Math4121_L10.md                |   1 +
 pages/Math4121/Math4121_L11.md                |   1 +
 pages/Math4121/Math4121_L12.md                |   1 +
 pages/Math4121/Math4121_L13.md                |   1 +
 pages/Math4121/Math4121_L14.md                |   1 +
 pages/Math4121/Math4121_L15.md                |   1 +
 pages/Math4121/Math4121_L16.md                |   1 +
 pages/Math4121/Math4121_L17.md                |   1 +
 pages/Math4121/Math4121_L18.md                |   1 +
 pages/Math4121/Math4121_L19.md                |   1 +
 pages/Math4121/Math4121_L2.md                 |   1 +
 pages/Math4121/Math4121_L20.md                |   1 +
 pages/Math4121/Math4121_L21.md                |   1 +
 pages/Math4121/Math4121_L22.md                |   1 +
 pages/Math4121/Math4121_L23.md                |   1 +
 pages/Math4121/Math4121_L24.md                |   1 +
 pages/Math4121/Math4121_L25.md                |   1 +
 pages/Math4121/Math4121_L26.md                |   1 +
 pages/Math4121/Math4121_L27.md                |   1 +
 pages/Math4121/Math4121_L28.md                |   1 +
 pages/Math4121/Math4121_L29.md                |   1 +
 pages/Math4121/Math4121_L3.md                 |   1 +
 pages/Math4121/Math4121_L4.md                 |   1 +
 pages/Math4121/Math4121_L5.md                 |   1 +
 pages/Math4121/Math4121_L6.md                 |   1 +
 pages/Math4121/Math4121_L7.md                 |   1 +
 pages/Math4121/Math4121_L8.md                 |   1 +
 pages/Math4121/Math4121_L9.md                 |   1 +
 pages/Math4121/_meta.js                       |   6 +
 pages/Math4121/index.md                       |  23 ++++
 pages/_meta.js                                |   3 +
 33 files changed, 184 insertions(+), 12 deletions(-)
 rename pages/{CSE347/lecture_note_generator.py => Math4121/LNG_v1.0.py} (68%)
 create mode 100644 pages/Math4121/Math4121_L1.md
 create mode 100644 pages/Math4121/Math4121_L10.md
 create mode 100644 pages/Math4121/Math4121_L11.md
 create mode 100644 pages/Math4121/Math4121_L12.md
 create mode 100644 pages/Math4121/Math4121_L13.md
 create mode 100644 pages/Math4121/Math4121_L14.md
 create mode 100644 pages/Math4121/Math4121_L15.md
 create mode 100644 pages/Math4121/Math4121_L16.md
 create mode 100644 pages/Math4121/Math4121_L17.md
 create mode 100644 pages/Math4121/Math4121_L18.md
 create mode 100644 pages/Math4121/Math4121_L19.md
 create mode 100644 pages/Math4121/Math4121_L2.md
 create mode 100644 pages/Math4121/Math4121_L20.md
 create mode 100644 pages/Math4121/Math4121_L21.md
 create mode 100644 pages/Math4121/Math4121_L22.md
 create mode 100644 pages/Math4121/Math4121_L23.md
 create mode 100644 pages/Math4121/Math4121_L24.md
 create mode 100644 pages/Math4121/Math4121_L25.md
 create mode 100644 pages/Math4121/Math4121_L26.md
 create mode 100644 pages/Math4121/Math4121_L27.md
 create mode 100644 pages/Math4121/Math4121_L28.md
 create mode 100644 pages/Math4121/Math4121_L29.md
 create mode 100644 pages/Math4121/Math4121_L3.md
 create mode 100644 pages/Math4121/Math4121_L4.md
 create mode 100644 pages/Math4121/Math4121_L5.md
 create mode 100644 pages/Math4121/Math4121_L6.md
 create mode 100644 pages/Math4121/Math4121_L7.md
 create mode 100644 pages/Math4121/Math4121_L8.md
 create mode 100644 pages/Math4121/Math4121_L9.md
 create mode 100644 pages/Math4121/_meta.js
 create mode 100644 pages/Math4121/index.md

diff --git a/pages/CSE347/lecture_note_generator.py b/pages/Math4121/LNG_v1.0.py
similarity index 68%
rename from pages/CSE347/lecture_note_generator.py
rename to pages/Math4121/LNG_v1.0.py
index f2461b7..a448e3c 100644
--- a/pages/CSE347/lecture_note_generator.py
+++ b/pages/Math4121/LNG_v1.0.py
@@ -1,13 +1,20 @@
-course_code=input('We will follow the naming pattern of {class}_L{lecture number}.md, enter the course code to start.\n')
-start=input('enter the number of lecture that you are going to start.\n')
-end=input('Enter the end of lecture (exclusive).\n')
-start=int(start)
-end=int(end)
-while start<end:
-    # create a empty text file
-    fp = open(f'{course_code}_L{start}.md', 'w')
-    fp.write(f'# Lecture {start}')
-    fp.close()
-    start+=1
-
+import os
+from pathlib import Path
+
+course_code=input('We will follow the naming pattern of {class}_L{lecture number}.md, enter the course code to start.\n')
+start=input('enter the number of lecture that you are going to start.\n')
+end=input('Enter the end of lecture (exclusive).\n')
+start=int(start)
+end=int(end)
+
+cur_dir = os.path.dirname(os.path.abspath(__file__))
+
+while start<end:
+    # create a empty text file
+    file_name = Path.joinpath(cur_dir, f'{course_code}_L{start}.md')
+    fp = open(file_name, 'w')
+    fp.write(f'# Lecture {start}')
+    fp.close()
+    start+=1
+
 print("Complete")
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L1.md b/pages/Math4121/Math4121_L1.md
new file mode 100644
index 0000000..75f1071
--- /dev/null
+++ b/pages/Math4121/Math4121_L1.md
@@ -0,0 +1,105 @@
+# Lecture 1
+
+## Chapter 5: Differentiation
+
+### The derivative of a real function
+
+#### Definition 5.1
+
+Let $f$ be a real-valued function on an interval $[a,b]$ ($f: [a,b] \to \mathbb{R}$). 
+
+We say that $f$ is _differentiable_ at a point $x\in [a,b]$ if the limit
+
+$$
+\lim_{t\to x} \frac{f(t)-f(x)}{t-x}
+$$
+
+exists.
+
+Then we defined the derivative of $f$, $f'$, a function whose domain is the set of all $x\in [a,b]$ at which $f$ is differentiable, by
+
+$$
+f'(x) = \lim_{t\to x} \frac{f(t)-f(x)}{t-x}
+$$
+
+#### Theorem 5.2
+
+Let $f:[a,b]\to \mathbb{R}$. If $f$ is differentiable at $x\in [a,b]$, then $f$ is continuous at $x$.
+
+Proof:
+
+We need to show that $\lim_{t\to x} f(t) = f(x)$.
+
+Equivalently, we need to show that
+
+$$
+\lim_{t\to x} (f(t)-f(x)) = 0
+$$
+
+So for $t\ne x$, since $f$ is differentiable at $x$, we have
+
+$$
+\begin{aligned}
+\lim_{t\to x} (f(t)-f(x)) &= \lim_{t\to x} \left(\frac{f(t)-f(x)}{t-x}\right)(t-x) \\
+&= \lim_{t\to x} \left(\frac{f(t)-f(x)}{t-x}\right) \lim_{t\to x} (t-x) \\
+&= f'(x) \cdot 0 \\
+&= 0
+\end{aligned}
+$$
+
+Therefore, differentiable is a stronger condition than continuous.
+
+> There exists some function that is continuous but not differentiable.
+> 
+> For example, $f(x) = |x|$ is continuous at $x=0$, but not differentiable at $x=0$.
+>
+> We can see that the left-hand limit and the right-hand limit are not the same.
+>
+> $$ \lim_{t\to 0^-} \frac{|t|-|0|}{t-0} = -1 \quad \text{and} \quad \lim_{t\to 0^+} \frac{|t|-|0|}{t-0} = 1 $$
+>
+> Therefore, the limit does not exist. for $f(x) = |x|$ at $x=0$.
+
+#### Theorem 5.3
+
+Suppose $f$ is differentiable at $x\in [a,b]$ and $g$ is differentiable at a point $x\in [a,b]$. Then $f+g$, $fg$ and $f/g$ are differentiable at $x$, and
+
+(a) $(f+g)'(x) = f'(x) + g'(x)$  
+(b) $(fg)'(x) = f'(x)g(x) + f(x)g'(x)$  
+(c) $\left(\frac{f}{g}\right)'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$, provided $g(x)\ne 0$
+
+Proof:
+
+Since the limit of product is the product of the limits, we can use the definition of the derivative to prove the theorem.
+
+(a)
+
+$$
+\begin{aligned}
+(f+g)'(x) &= \lim_{t\to x} \frac{(f+g)(t)-(f+g)(x)}{t-x} \\
+&= \lim_{t\to x} \frac{f(t)-f(x)}{t-x} + \lim_{t\to x} \frac{g(t)-g(x)}{t-x} \\
+&= f'(x) + g'(x)
+\end{aligned}
+$$
+
+(b)
+
+Since $f$ is differentiable at $x$, we have $\lim_{t\to x} f(t) = f(x)$.
+
+$$
+\begin{aligned}
+(fg)'(x) &= \lim_{t\to x} \left(\frac{f(t)g(t)-f(x)g(x)}{t-x}\right) \\
+&= \lim_{t\to x} \left(f(t)\frac{g(t)-g(x)}{t-x} + g(x)\frac{f(t)-f(x)}{t-x}\right) \\
+&= f(t) \lim_{t\to x} \frac{g(t)-g(x)}{t-x} + g(x) \lim_{t\to x} \frac{f(t)-f(x)}{t-x} \\
+&= f(x)g'(x) + g(x)f'(x)
+\end{aligned}
+$$
+
+(c)
+
+$$  
+\begin{aligned}
+\left(\frac{f}{g}\right)'(x) &= \lim_{t\to x}\left(\frac{f(t)g(x)}{g(t)g(x)} - \frac{f(x)g(x)}{g(t)g(x)}\right) \\
+&= \frac{1}{g(t)g(x)}\left(\lim_{t\to x} (f(t)g(x)-f(x)g(t))\right) \\
+\end{aligned}
+$$
+
diff --git a/pages/Math4121/Math4121_L10.md b/pages/Math4121/Math4121_L10.md
new file mode 100644
index 0000000..72b00ef
--- /dev/null
+++ b/pages/Math4121/Math4121_L10.md
@@ -0,0 +1 @@
+# Lecture 10
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L11.md b/pages/Math4121/Math4121_L11.md
new file mode 100644
index 0000000..9cdf10f
--- /dev/null
+++ b/pages/Math4121/Math4121_L11.md
@@ -0,0 +1 @@
+# Lecture 11
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L12.md b/pages/Math4121/Math4121_L12.md
new file mode 100644
index 0000000..6c6b1b2
--- /dev/null
+++ b/pages/Math4121/Math4121_L12.md
@@ -0,0 +1 @@
+# Lecture 12
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L13.md b/pages/Math4121/Math4121_L13.md
new file mode 100644
index 0000000..1282ee9
--- /dev/null
+++ b/pages/Math4121/Math4121_L13.md
@@ -0,0 +1 @@
+# Lecture 13
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L14.md b/pages/Math4121/Math4121_L14.md
new file mode 100644
index 0000000..873d41d
--- /dev/null
+++ b/pages/Math4121/Math4121_L14.md
@@ -0,0 +1 @@
+# Lecture 14
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L15.md b/pages/Math4121/Math4121_L15.md
new file mode 100644
index 0000000..56bee61
--- /dev/null
+++ b/pages/Math4121/Math4121_L15.md
@@ -0,0 +1 @@
+# Lecture 15
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L16.md b/pages/Math4121/Math4121_L16.md
new file mode 100644
index 0000000..543dc8f
--- /dev/null
+++ b/pages/Math4121/Math4121_L16.md
@@ -0,0 +1 @@
+# Lecture 16
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L17.md b/pages/Math4121/Math4121_L17.md
new file mode 100644
index 0000000..eeda5de
--- /dev/null
+++ b/pages/Math4121/Math4121_L17.md
@@ -0,0 +1 @@
+# Lecture 17
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L18.md b/pages/Math4121/Math4121_L18.md
new file mode 100644
index 0000000..078e0d2
--- /dev/null
+++ b/pages/Math4121/Math4121_L18.md
@@ -0,0 +1 @@
+# Lecture 18
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L19.md b/pages/Math4121/Math4121_L19.md
new file mode 100644
index 0000000..ca72bc8
--- /dev/null
+++ b/pages/Math4121/Math4121_L19.md
@@ -0,0 +1 @@
+# Lecture 19
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L2.md b/pages/Math4121/Math4121_L2.md
new file mode 100644
index 0000000..2707d3e
--- /dev/null
+++ b/pages/Math4121/Math4121_L2.md
@@ -0,0 +1 @@
+# Lecture 2
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L20.md b/pages/Math4121/Math4121_L20.md
new file mode 100644
index 0000000..ff21618
--- /dev/null
+++ b/pages/Math4121/Math4121_L20.md
@@ -0,0 +1 @@
+# Lecture 20
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L21.md b/pages/Math4121/Math4121_L21.md
new file mode 100644
index 0000000..9ce48b9
--- /dev/null
+++ b/pages/Math4121/Math4121_L21.md
@@ -0,0 +1 @@
+# Lecture 21
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L22.md b/pages/Math4121/Math4121_L22.md
new file mode 100644
index 0000000..e0a3957
--- /dev/null
+++ b/pages/Math4121/Math4121_L22.md
@@ -0,0 +1 @@
+# Lecture 22
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L23.md b/pages/Math4121/Math4121_L23.md
new file mode 100644
index 0000000..9ef05c6
--- /dev/null
+++ b/pages/Math4121/Math4121_L23.md
@@ -0,0 +1 @@
+# Lecture 23
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L24.md b/pages/Math4121/Math4121_L24.md
new file mode 100644
index 0000000..f92a2b5
--- /dev/null
+++ b/pages/Math4121/Math4121_L24.md
@@ -0,0 +1 @@
+# Lecture 24
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L25.md b/pages/Math4121/Math4121_L25.md
new file mode 100644
index 0000000..391d6ac
--- /dev/null
+++ b/pages/Math4121/Math4121_L25.md
@@ -0,0 +1 @@
+# Lecture 25
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L26.md b/pages/Math4121/Math4121_L26.md
new file mode 100644
index 0000000..9ba5345
--- /dev/null
+++ b/pages/Math4121/Math4121_L26.md
@@ -0,0 +1 @@
+# Lecture 26
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L27.md b/pages/Math4121/Math4121_L27.md
new file mode 100644
index 0000000..cd8bdf8
--- /dev/null
+++ b/pages/Math4121/Math4121_L27.md
@@ -0,0 +1 @@
+# Lecture 27
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L28.md b/pages/Math4121/Math4121_L28.md
new file mode 100644
index 0000000..3f89dbe
--- /dev/null
+++ b/pages/Math4121/Math4121_L28.md
@@ -0,0 +1 @@
+# Lecture 28
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L29.md b/pages/Math4121/Math4121_L29.md
new file mode 100644
index 0000000..c21aece
--- /dev/null
+++ b/pages/Math4121/Math4121_L29.md
@@ -0,0 +1 @@
+# Lecture 29
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L3.md b/pages/Math4121/Math4121_L3.md
new file mode 100644
index 0000000..58e3216
--- /dev/null
+++ b/pages/Math4121/Math4121_L3.md
@@ -0,0 +1 @@
+# Lecture 3
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L4.md b/pages/Math4121/Math4121_L4.md
new file mode 100644
index 0000000..b90a43d
--- /dev/null
+++ b/pages/Math4121/Math4121_L4.md
@@ -0,0 +1 @@
+# Lecture 4
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L5.md b/pages/Math4121/Math4121_L5.md
new file mode 100644
index 0000000..23c2f01
--- /dev/null
+++ b/pages/Math4121/Math4121_L5.md
@@ -0,0 +1 @@
+# Lecture 5
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L6.md b/pages/Math4121/Math4121_L6.md
new file mode 100644
index 0000000..7ef0413
--- /dev/null
+++ b/pages/Math4121/Math4121_L6.md
@@ -0,0 +1 @@
+# Lecture 6
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L7.md b/pages/Math4121/Math4121_L7.md
new file mode 100644
index 0000000..9b385d4
--- /dev/null
+++ b/pages/Math4121/Math4121_L7.md
@@ -0,0 +1 @@
+# Lecture 7
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L8.md b/pages/Math4121/Math4121_L8.md
new file mode 100644
index 0000000..ea02e1a
--- /dev/null
+++ b/pages/Math4121/Math4121_L8.md
@@ -0,0 +1 @@
+# Lecture 8
\ No newline at end of file
diff --git a/pages/Math4121/Math4121_L9.md b/pages/Math4121/Math4121_L9.md
new file mode 100644
index 0000000..5b73bd5
--- /dev/null
+++ b/pages/Math4121/Math4121_L9.md
@@ -0,0 +1 @@
+# Lecture 9
\ No newline at end of file
diff --git a/pages/Math4121/_meta.js b/pages/Math4121/_meta.js
new file mode 100644
index 0000000..f9b7f54
--- /dev/null
+++ b/pages/Math4121/_meta.js
@@ -0,0 +1,6 @@
+export default {
+    index: "Course Description",
+    "---":{
+        type: 'separator'
+    }
+}
diff --git a/pages/Math4121/index.md b/pages/Math4121/index.md
new file mode 100644
index 0000000..b398fd8
--- /dev/null
+++ b/pages/Math4121/index.md
@@ -0,0 +1,23 @@
+# Math 4121
+
+Riemann integration; measurable functions; measures; the Lebesgue integral; integrable functions; $L^p$ spaces.
+
+## Textbook
+
+Principles of Mathematical Analysis by Walter Rudin
+
+A radical Approach to Lebesgue's Theory of Integration by David
+
+## Grade
+
+| item | percentage |
+| --- | --- |
+| Homework | 40% |
+| Midterm 1 | 15% |
+| Midterm 2 | 15% |
+| Final | 30% |
+
+## Homework
+
+Due every Monday.
+
diff --git a/pages/_meta.js b/pages/_meta.js
index a48c129..a9217b2 100644
--- a/pages/_meta.js
+++ b/pages/_meta.js
@@ -32,6 +32,9 @@ export default {
     CSE347: {
       type: 'page',
     },
+    Math4121: {
+      type: 'page',
+    },
     about: {
         display: 'hidden'
     },