new course

pages/Math4121/LNG_v1.0.py

@@ -1,11 +1,18 @@
+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
-    fp = open(f'{course_code}_L{start}.md', 'w')
+    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

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)
+
+$$  
+\begin{aligned}
+\left(\frac{f}{g}\right)'(x) &= \lim_{t\to x}\left(\frac{f(t)g(x)}{g(t)g(x)} - \frac{f(x)g(x)}{g(t)g(x)}\right) \\
+&= \frac{1}{g(t)g(x)}\left(\lim_{t\to x} (f(t)g(x)-f(x)g(t))\right) \\
+\end{aligned}
+$$
+

pages/Math4121/Math4121_L10.md

@@ -0,0 +1 @@
+# Lecture 10

pages/Math4121/Math4121_L11.md

@@ -0,0 +1 @@
+# Lecture 11

pages/Math4121/Math4121_L12.md

@@ -0,0 +1 @@
+# Lecture 12

pages/Math4121/Math4121_L13.md

@@ -0,0 +1 @@
+# Lecture 13

pages/Math4121/Math4121_L14.md

@@ -0,0 +1 @@
+# Lecture 14

pages/Math4121/Math4121_L15.md

@@ -0,0 +1 @@
+# Lecture 15

pages/Math4121/Math4121_L16.md

@@ -0,0 +1 @@
+# Lecture 16

pages/Math4121/Math4121_L17.md

@@ -0,0 +1 @@
+# Lecture 17

pages/Math4121/Math4121_L18.md

@@ -0,0 +1 @@
+# Lecture 18

pages/Math4121/Math4121_L19.md

@@ -0,0 +1 @@
+# Lecture 19

pages/Math4121/Math4121_L2.md

@@ -0,0 +1 @@
+# Lecture 2

pages/Math4121/Math4121_L20.md

@@ -0,0 +1 @@
+# Lecture 20

pages/Math4121/Math4121_L21.md

@@ -0,0 +1 @@
+# Lecture 21

pages/Math4121/Math4121_L22.md

@@ -0,0 +1 @@
+# Lecture 22

pages/Math4121/Math4121_L23.md

@@ -0,0 +1 @@
+# Lecture 23

pages/Math4121/Math4121_L24.md

@@ -0,0 +1 @@
+# Lecture 24

pages/Math4121/Math4121_L25.md

@@ -0,0 +1 @@
+# Lecture 25

pages/Math4121/Math4121_L26.md

@@ -0,0 +1 @@
+# Lecture 26

pages/Math4121/Math4121_L27.md

@@ -0,0 +1 @@
+# Lecture 27

pages/Math4121/Math4121_L28.md

@@ -0,0 +1 @@
+# Lecture 28

pages/Math4121/Math4121_L29.md

@@ -0,0 +1 @@
+# Lecture 29

pages/Math4121/Math4121_L3.md

@@ -0,0 +1 @@
+# Lecture 3

pages/Math4121/Math4121_L4.md

@@ -0,0 +1 @@
+# Lecture 4

pages/Math4121/Math4121_L5.md

@@ -0,0 +1 @@
+# Lecture 5

pages/Math4121/Math4121_L6.md

@@ -0,0 +1 @@
+# Lecture 6

pages/Math4121/Math4121_L7.md

@@ -0,0 +1 @@
+# Lecture 7

pages/Math4121/Math4121_L8.md

@@ -0,0 +1 @@
+# Lecture 8

pages/Math4121/Math4121_L9.md

@@ -0,0 +1 @@
+# Lecture 9

pages/Math4121/_meta.js

@@ -0,0 +1,6 @@
+export default {
+    index: "Course Description",
+    "---":{
+        type: 'separator'
+    }
+}

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.
+

pages/_meta.js

@@ -32,6 +32,9 @@ export default {
     CSE347: {
       type: 'page',
     },
+    Math4121: {
+      type: 'page',
+    },
     about: {
         display: 'hidden'
     },