Home Calculus III - Lecture 7

Calculus III - Lecture 7

Lecture highlights:

  1. $T$ is a functor
  2. A big picture for what is to come.


Few meth today, mostly the cold hard truth. Well is mathematics truth? I don’t know. (For the few of you who want to say something, I don’t care either. (This might suggest that I absolutely think mathematics is truth and am very firm on this, or I don’t think mathematics is truth. Whatever.))

Throughout early education, we are too used to think maths in terms of formulas in coordinates. At first, it is convenient, but it blocks us from understanding higher-level maths.

(Other meth like this in the future will likely be excluded from these posts. Too repetitive.)

Jacobean is a great mathematician. His hometown was in Germany. Unfortunately, it is no longer in Germany

Lecture notes

Again for any vector $\vec{u}$, we have $f_{\vec{u}}(p) := \left.\dfrac{d}{dt}\right|_{t=0} f(p+t\vec{u})$. By the chain rule, this is equal to $\dfrac{\partial f}{\partial x^i} (p) u^i$. Note once again that we use the Einstein notation here.

But since these stuff are linear, we have that $\dfrac{\partial f}{\partial x^i}$ and $f_{\vec{u}}(p)$ for any $\vec{u} \in \mathbb{R}^n$ determine each other.

For every $p$, we have a linear map $df_p: T_p U \to \mathbb{R}$. This corresponds to the mapping $(p, \vec{u})\mapsto f_{\vec{u}} (p)$. This map is linear on $\vec{u}$.

We define smooth on an open set $U$ since that for any $p \in U$, $\vec{u}\in \mathbb{R}^n$, there exists some $t$ such that $p+t\vec{u}\in U$ by a basic property of open sets.

In summary, what is most important, is that partial differentiation of $f$ at $p$ corresponds to $df_p \in T^\ast_p U$, which is the cotangent space of $U$ at $p$

For a smooth map $F:U \to V$, where $U$ is open in $\mathbb{E}^n$ and $V$ is in $\mathbb{E}^m$

The Taylor expansion of $F$ at $p \in U$ is, for any $q \in p+\vec{u}$

\[F(q) \sim F(p) + F_{\vec{u}} (p) + \frac{1}{2!} F_{\vec{u}\vec{u}} (p) + \cdots.\]

In particular $F(q) \sim F(p) + F_{\vec{u}} (p)$ is the linear approximation. More formally, it is the affine map $\mathbb{E}^n \to \mathbb{E}^m$ where $q \mapsto F(p) + F_{\vec{u}} (p)$.

Applying the tangent functor, we get the map $T_p F: T_p U \to T_{F(p)} V$ which $(p, \vec{u})\mapsto (F(p), F_{\vec{u}} (p))$. This is called the tangent, in geometric terms, or linearisation, in algebraic terms, of $F$ at $p$. This and the linear approximation determine each other.

For my personal interpretation of the interesting terminology, tangent space is a space where tangents can exist in, and tangents are actually tangents from an intuitive understanding. Don’t be confused.

As a realisation, the map could be thought of as between

\[T_p U = T_p \mathbb{E}^n \equiv (\mathbb{E}^n, p) \to (\mathbb{E}^m, F(p)) \equiv T_{F(p)} \mathbb{E}^m = T_{F(p)} V.\]

POV: You went out to touch grass and stumbled upon a blue moon in the sky! He actually gave an actually concrete example with literal numbers! Consider the tangent of $y = \sin x$ at $(\frac{\pi}{2} , \frac{1}{2})$. Its linearisation is $y = \sin \frac{\pi}{6} + \left(\cos \frac{\pi}{2} \right)x$.

A smooth map from an open set of a Euclidean space to an open set of another Euclidean space is called a calculus-smooth map. Note that this is not a standard terminology. Fun fact: calculus-smooth maps form a category under composition.

We define the total tangent space of $U$:

\[T U := \bigcup_{p\in U} T_p U = \bigcup_{p\in U}\{p\}\times \mathbb{R}^n = U \times \mathbb{R}^n \subset \mathbb{E}^n \times \mathbb{R}^n \equiv \mathbb{E}^{2n}.\]

The tangent functor $T$ maps $F: U\to V$ to $TF:TU\to TV$, which $(p, \vec{u}) \mapsto (F(p), F_{\vec{u}}(p))$. Note that $F_{\vec{u}}(p) = \overline{dF_p} \vec{u}$. This $\overline{dF_p}$ is a Jacobean matrix as below:

\[\overline{dF_p} := \begin{pmatrix} \dfrac{\partial F^1}{\partial x^1}(p) & \cdots & \dfrac{\partial F^1}{\partial x^n}(p) \\ \vdots & \dfrac{\partial F^i}{\partial x^j}(p) & \vdots \\ \dfrac{\partial F^m}{\partial x^1}(p) & \cdots & \dfrac{\partial F^m}{\partial x^n}(p) \end{pmatrix}.\]

Now we talk more about the tangent functor. This means that linearisation is a functor. It is an endofunctor on the category of calculus-smooth maps.

Note that $T$ is a functor means that $T$ satisfies the partial associativity law, where for compatible $F, G$, we have

\[T(F\circ G) = TF \circ TG.\]

This is in fact the chain rule.

We also have the preservation of identity, which says $ T 1 = 1$. So we have the Jacobean matrix $J 1_p = I$. Once again, we have a commutative diagram illustrating this.

The reason I sound like a broken record is that M sounds like a broken record, although I am grateful for this.

So we can reduce this abstract version of the chain rule into our ordinary chain rule, which is

\[\left[\dfrac{\partial (G\circ F)^i}{\partial x^j}\right] (p) = \sum_{l=1}^m \left[ \dfrac{\partial G^i (F(p))}{\partial y^l} \dfrac{\partial F^l}{\partial x^j}(p) \right].\]

Or in a simpler form with Jacobean matrices,

\[J(G\circ F)_p = (JG)_{F(p)} \cdot JF_p.\]

This can be obtained from the geometric perspective:

\[T_p(G\circ F) = T_{F(p)}G \circ T_p F.\]

This means that $T$ preserves the commutativity of diagrams in the category of calculus-smooth maps:

But this simply says that $T$ is a functor.

We have finished our review of multivariable calculus! Moving forward, we will be interested in exploring the details of calculus on manifolds! Here is a big picture.

In Calculus I and Calculus II, we have studied the category of calculus-smooth maps, which are between open sets of Euclidean spaces. In fact, real affine spaces are enough, but why not make our life easier by endowing it with a Euclidean structure?

Anyway, this is a full subcategory of the category of smooth maps between subsets of Euclidean spaces. A category is a full subcategory of another when the set of morphisms is the included in the target category. In general, the morphisms in this category are not linear, but they are almost linear. This will (hopefully) be explained in later lectures.

There is a category in between. It is the category of smooth maps between manifolds. All the inclusions here are full. This is Calculus III and the object of our study. Mathematicians care about maps between open subsets of Euclidean spaces and those between manifolds, but not subsets (without restriction) of Euclidean spaces. It is just too uninteresting. But it provides us with a langauge to understand Calculus III better.

In this category, the tangent functor is reinterpreted as the velocity functor $V$. Mathematically, this category is difficult to understand, but physically it is easier. Note that we have not defined smoothness on simply subsets of Euclidean spaces yet but anyway we continue.

Imagine a particle $p \in A \subset \mathbb{E}^m$. It has a velocity vector. Suppose a beam of light shines on it, the shadow in $B \subset \mathbb{E}^n$ also has a velocity vector. The velocity functor just sends velocity of the real particle to the velocity of the shadow. Next time, we will do this with more details.

Sorry I have no idea what he talked about there.

Unfortunately I don’t know how to handle $\LaTeX$ here. The evaluation vertical line cannot be displayed.

This post is licensed under CC BY 4.0 by the author.
Trending Tags

Calculus III - Lecture 6

Calculus III - Lecture 8

Trending Tags