University of Tokyo 1S1 Mathematical Science Foundation： Differential and Integral Calculus

The other part of 数理科学基礎 (with 東大1S1数理科学基礎：線形代数) seems to be looking at Common Materials for Mathematical Science Foundations as well.
It seems that the evaluation is based only on exams, and attendance is not particularly taken into account.
Exams
- There is hardly any discussion about epsilon-delta.
- The main focus is on computational problems.
Continued: University of Tokyo 1S2 Differential and Integral Calculus

Chapter 11. Partial Derivatives and Tangent Planes

The definition of Partial Differentiation is as we know.
It seems that $\frac{\partial f}{\partial x}$ can also be written as $f_{x}$ .
- Oh, they don’t use dashes or anything (blu3mo).
- It can be confusing, right? (takker)
There are four patterns for Second Partial Derivatives.
- $f_{xx}, f_{x y}, f_{y x}, f_{yy}$ .
- Well, that’s how it is (blu3mo).
The Gradient Vector is represented as $(f_{x} (x, y), f_{y} (x, y))^{T}$ .
- This represents the direction of steepest ascent on the tangent plane.
  - Well, that’s what the Gradient Descent Method uses (blu3mo).
Critical Points
- Points where $f_{x} = f_{y} = 0$ .
- In other words, even if we take the total derivative, it will be 0. Chapter 10. Graphs of Functions of Two Variables
Functions of Two Variables
- Functions like $f (x, y)$ .
- It can also be said to be a mapping from $R^{2}$ to $R$ .
Different functions have different 3D graph shapes.
- $f (x, y) = a x + b y + c$ represents a plane.
  - This can be defined by vectors, like in high school.
- $f (x, y) = g (x^{2} + y^{2})$ represents a rotational surface.
  - A rotational surface can be imagined as a solid formed by rotating a graph.
  - The input to function g is the distance from the origin to (x,y).
  - Therefore, if the coordinates have the same distance from the origin, they will have the same output.
- $f (x, y) = g (x)$ represents a cylindrical surface.
  - This one is easier to imagine.
- Rotational Paraboloid/Hyperbolic Paraboloid
  - $f (x) = x^{2} + y^{2}$ : Rotational Paraboloid
  - $f (x) = x^{2} - y^{2}$ : Hyperbolic Paraboloid
  - Well, that’s the shape it is (blu3mo).
  - TODO: This seems to correspond to Trigonometric Functions and Hyperbolic Functions, but how do they correspond? (blu3mo)
    - I was going to give a hint, but it would give away the answer, so I won’t (takker).
Let’s consider $f (x, y) = p x^{2} + q x y + r y^{2}$ .
- If we complete the square, it becomes a form like $k (x + y)^{2} + l y^{2}$ .
  - Here, if we consider a new coordinate system $X = x + y, Y = y$ , it can be expressed as $k X^{2} + l Y^{2}$ .
    - I see~ (blu3mo).
    - They said that when studying linear algebra, even more complex equations can be expressed in the above form.
  - In this case, if both $k$ and $l$ are positive, it becomes a Rotational Paraboloid, and if one of them is negative, it becomes a Hyperbolic Paraboloid (6278c2ba79e11300007fcc0a).
    - It is a distorted shape because we are changing the coordinate system.
Methods for understanding the shape of a graph (without using differentiation)
- Cut the graph with a plane perpendicular to the x and y axes to obtain a cross-section of the graph.
  - The curve $x = x_{0} + t cos θ_{0}, y = y_{0} + t sin θ_{0}, z = f (x_{0} + t cos θ_{0}, y_{0} + t sin θ_{0})$ represents the cross-section.
- Draw Contour Lines.
  - We need to find the relationship between x and y when z is constant.
- Determine the sign of the graph and draw it.
  - This can be done using Completing the Square or Factoring.

Chapter 6.

Differential equations

Chapter 5.

Covered until high school level.
- Logarithmic Functions <-> Exponential Functions
  - (Strictly speaking, they are not inverse functions unless the function is defined to be a bijection)
    - For continuous functions, if it is a strictly Monotonic Functions within the domain, it is a bijection.
      - There are various types of monotonic functions, right? (takker)
        
        I only know about $f (a) < f (x)$ for all $x \in A$ and $a < x$ (this is the case of strictly increasing monotonicity).- The inequality becomes problematic when it includes an equal sign (nishio).
  - - When the right side is used, it becomes unclear whether to choose a or b when creating the inverse image because f(a) and f(b) are equal.
  - General and strict | Monotonic function - Wikipedia Maybe this? (takker)
  - Oh, it’s just a matter of whether the equal sign is included or not.
    - Of course, that’s an important difference.
    - I see (blu3mo).
  - It was distinguished as monotone non-decreasing and monotone increasing in the text (blu3mo).
    - I prefer this way of saying it (nishio)(erniogi)(takker).
      - “Strict” is a relative expression and even just by looking at the name, it’s not clear what it means.
  - I thought there were various types like uniformly continuous functions.
In the case of real numbers, it’s okay because the domain is defined (takker).
- $exp : R \to R_{+}$
- $ln : R_{+} \to R$
  - Note: $R_{+} := {x \in R ∣ x > 0}$
- So, within the range of the domain, it’s a strict monotonic function, right? (blu3mo)
- When extended to complex numbers, it doesn’t work.
  - This was a point I struggled with in [/blu3mo/Generalization of Euler’s Form for Complex Numbers](https://scrapbox.io/blu3mo/複素数のEuler Formの一般化) (blu3mo).
  - Nostalgic (takker).
Trigonometric functions <-> Inverse trigonometric functions (arcsin, etc.).
- The domain is also restricted for the same reason.
- So, arcsin, etc. have a limited domain.
  - I learned this in IB (blu3mo).
- The calculation of derivatives can also be derived from formulas.
  - When y=arcsinx, siny=x, so we find dx/dy and then rearrange it.
Hyperbolic Functions

Lec 4.

Differentiable Functions
Continuity and Differentiability
Intermediate Value Theorem
Mean Value Theorem
- $\exists c \frac{f ( b ) - f ( a )}{( b - a )} = f^{'} (c), a < c < b$
  - It means that there is a linear function from a to b that has the same value (slope at c) as $f^{'} (c)$ .
  - It’s more convenient not to use fractions (takker).
    - $f (b) = f^{'} (c) (b - a) + f (a) for \exists c \in (a, b)$
    - And what does this form do…?
We’ll learn about monotone increasing later (blu3mo)(blu3mo)❓
Derivative of Inverse Functions
- Theorem 4: If there is an inverse function g(y) of f(x), the derivative is given by $g^{'} (y) = \frac{1}{f ^{'} ( g ( y ))}$ .
  - This is quite non-trivial and useful (blu3mo).
  - Condition: $f (x)$ is monotonically increasing and differentiable on an open interval I, and $f^{'} (x) \neq = 0$ .
    - I want to understand why this condition is necessary ❓
    - (/icons/cheering)(takker)
Primitive Functions
- I’ll study this on my own (blu3mo)❓
Thoughts
- Lately, I’ve been learning quite a few obvious theorems, but how much should I understand them? (blu3mo)
  - Should I just know the conditions of the theorems? Should I understand the proofs once? Should I be able to recall the proofs at any time? Should I have the ability to prove them from scratch?
  - (takker)
    - Should I just know the conditions of the theorems?
      - If you’re doing the last-minute cramming for an exam, it’s necessary.
      - Otherwise, it’s honestly not necessary.
        
        Even if the conditions of the theorem become vague, you can simply write a proof on the spot or check if it’s correct by concretizing the symbolic constants.
        
        Of course, if you do this for every theorem during the exam, you’ll run out of time, so it’s more like doing problem exercises in advance and repeatedly writing proofs on the spot whenever you forget.
        
        By writing it repeatedly every time you forget, you’ll eventually end up memorizing it.
    - Well, I want at least this much.
      - Should I understand the proofs once?
    - With this, I won’t have to rely on memorization.- When should we be able to recall proofs?
A simple example is the various trigonometric identities.
- Even if you forget $(sin θ)^{2}$ , if you know how to expand it, you can reconstruct it on the spot.
It is often useful to apply the techniques used in proofs to solve various problems.
Should we have the ability to prove from scratch?
- The meaning of “from scratch” is a bit unclear.
- If it means being able to prove even if you forget the proof strategy, then you have already lost the understanding of the proof.
- To be more precise, the intention was whether we should acquire universal mathematical skills to discover these proofs on our own as we learn proofs.
- But I don’t think that is really required.
- Oh, that would require being a genius, a madman, or a god, so it’s okay.
- However, it would be good to try manipulating proofs or considering alternative approaches if you have the time.
  - With this, you can discover methods from scratch to some extent.

Lec 3.

Continuation of Epsilon-Delta Proof.

Lec 2.

Epsilon-Delta Proof.

Lec 1.

Predicate Logic.
What is Predicate Logic?
- Assertions about variables.
  - Assertions that hold for any value or for a certain value.
  - Indicated by \forall and \exists.
  - Types:
    - Assertions with \forall are Universal Propositions.
      - $\forall x Q (x)$
    - Assertions with \exists are Existential Propositions.
      - $\exists x Q (x)$
    - The names are easy to understand.
- Negation of propositions.
  - The negation of $\forall x Q (x)$ is $\exists x \neg Q (x)$ .
    - This negation basically says that there exists a counterexample.
  - The negation of $\exists x Q (x)$ is $\forall x \neg Q (x)$ .
  - I feel a bit of De Morgan’s Laws in this.
    - The principles are the same.
      - 60bb12c21280f000001426b6
      - I see.
- Concrete examples.
  - If you define a Subset properly, it should be $x \in X ⟹ y \in Y$ .
    - Ah, I see.
    - The language of sets is converted into Logical Formulas.
- The one that makes “if-then” into a logical formula.
  - When I saw it in Math Girl, I couldn’t understand it, but now I get it.
  - $P ⟹ Q$ is equivalent to $\neg P \lor Q$ .
    - I understand that “if P is false or Q is true, then ‘P implies Q’ is true.”
  - With this, for example, when defining an Injective Function,
    - “If” version: $\forall x_{1} \forall x_{2} x_{1} \neq = x_{2} ⟹ f (x_{1}) \neq = f (x_{2})$
    - Logical formula version: $\forall x_{1} \forall x_{2} x_{1} = x_{2} \lor f (x_{1}) \neq = f (x_{2})$
      - The important thing is that “if” becomes “or” in the logical formula.
      - I see, this will take some time to get used to.
    - Is “if” treated as something different from a logical symbol?
      - In logic, “if” $⟹$ is one of the logical symbols, just like $\land$ or $\lor$ , so I felt a discrepancy in this comparison.
      - Ah, that’s right.
        
        We call it a logical symbol when we express everything using and, or, not.
- \forall and \exists can be chained just by listing them.
  - $\forall a, b$ and $\forall a \forall b$ have the same meaning.
    - It’s a hassle to write $\forall$ each time.
    - In 6253d65579e1130000d36f06, I had the realization that $\forall a, b$ can be expressed without any problems by listing them as $\forall a \forall b$ .
    - Oh, I see.
      - That’s also important.
      - It’s obvious, but it would be good to try proving $\forall a, b; P (a, b) ⟺ \forall a \forall b; P (a, b)$ by hand to understand it.
      - Being able to question things and try to understand them on your own is the best thing about mathematics.- This is similar to programming where you can execute and see the results immediately.
Can you chain forall and exists together?
- Yes, you can do something like $\forall a, b \in Z \exists! q, r \geq 0; a = q b + r \land 0 \leq r < b$ (takker)
- You cannot reverse the order (blu3mo)
- It seems like something you have to try out to see (takker)
  - Does $\forall a \exists b; P (a, b) ⟺ \exists a \forall; b P (a, b)$ hold?
- If either forall or exists is chained, changing the order is not a problem, but you cannot rearrange forall and exists (blu3mo)
  - You can change $\forall a \forall b \exists c \exists d$ to $\forall b \forall a \exists d \exists c$
  - However, you cannot change $\forall a \forall b \exists c \exists d$ to $\forall a \exists c \forall b \exists d$
- You can create questions or problems like, does $\forall a \exists b; P (a, b) ⟸ \exists a \forall; b P (a, b)$ hold? (takker)
  - You can create various variations and explore them in depth to pass the time
Upper bound and lower bound
- b is an upper bound of set A if
- $\forall x \in A x ≦ b$ , in other words, b is greater than all elements of A in Japanese
  - It is important that it is defined as ≤ instead of <
Maximum element, minimum element: min/max within a set
- Basically, [].max(), [].min()
- There is no maximum element in [0, 1) because 1 is not an element of this set
  - Why not consider $lim_{x \to 1} x$ as the maximum element? (blu3mo)
    - Ah, but if we define limits using the epsilon-delta definition, then lim is not a function that returns a specific value
      - It is a concept separate from values
      - So we cannot call it the maximum element
    - Limits are not relevant (takker)
      - It simply does not fit the definition of a maximum element
      - $a is a maximum element of A : ⟺ a \in A \land \forall x \in A; x \leq a$
      - It is already written in 625d718779e11300008fbc5e. It was an unnecessary comment
Upper bound and lower bound
- Different definitions from maximum element
- The minimum element of the set of upper bounds of a set
- Here, 1 is an upper bound of [0, 1]
  - An upper bound b is defined as a≦b for any element a, not a<b
- The maximum element of [0, 1) is not 1, but the upper bound is 1
  - In fact, this is why the upper bound is defined
- The upper bound of set A is denoted as $sup A$ , and the lower bound as $in f A$
It is easier to understand by looking at the materials around here where the definitions are written clearly
- Well, it’s a tool to say that in a set like [0, 1), the highest value is 1
  - 1 itself is not included in the set, so it cannot be called the maximum element
Real numbers have continuity
- As an accurate representation of this,
- If a subset of the set of real numbers is bounded above, it has a least upper bound
- If it did not have continuity, then when there is a subset that is bounded above, there would be no least upper bound for the subset’s upper bounds
  - Then it would mean that it is not bounded above
  - No, that’s not right (blu3mo)
    - This is not a properly constructed contrapositive
- But it is indeed natural that “if there is a subset that is bounded above, there exists a least upper bound for the subset’s upper bounds”
  - Is that not right? Even if there are elements in the upper bounds, there may not be a least upper bound
    - For example, in (0, ∞], there are elements but no least upper bound
    - In edge cases, it is the empty set (takker)
      - That’s right (blu3mo)
  - This holds true because it is the essence of what it means to be a real number, so it does not hold in any arbitrary partially ordered set (takker)
    - The upper bound of $[0, 2] \cap Q$ is ${x \in Q ∣\forall y \in [0, 2] \cap Q; y \leq x}$ ( $= [2, \infty) \cap Q$ ), but there is no least upper bound in $[2, \infty) \cap Q$
Local maximum/minimum
- It gives the image of being defined by the second derivative, but the definition ” $∣ x - x_{0} ∣ < δ$ implies $f (x) \leq f (x_{0})$ ” is indeed a better definition- Flat parts are locally both maximum and minimum (as defined by ≤).
Local maximum/Local minimum
- Understanding (blu3mo)
- If we tweak the definition of local maximum a bit, it becomes a global maximum, right? (blu3mo)

Shutaro Aoyama

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University of Tokyo 1S1 Mathematical Science Foundation： Differential and Integral Calculus

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