diff --git a/chapters/measures_integrals.tex b/chapters/measures_integrals.tex index 1f7f583..6ec57eb 100644 --- a/chapters/measures_integrals.tex +++ b/chapters/measures_integrals.tex @@ -10,4 +10,5 @@ \subfile{sections/integrals.tex} \subfile{sections/int_real_nums.tex} \subfile{sections/product_measures.tex} + \subfile{sections/trans_thm.tex} \end{document} \ No newline at end of file diff --git a/chapters/sections/trans_thm.tex b/chapters/sections/trans_thm.tex new file mode 100644 index 0000000..874ce12 --- /dev/null +++ b/chapters/sections/trans_thm.tex @@ -0,0 +1,182 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section{The Transformation Theorem} + +\begin{defi} + Let $U, V \subset \realn^n$ be open. A mapping $T: U \rightarrow V$ is said to be a diffeomorphism if it is bijective and if + $T$ and $\inv{T}$ are continuously differentiable. Analogously we define + \begin{gather*} + C^r \text{-diffeomorphism if it is } r \text{-times differentiable} \\ + C^{\infty} \text{-diffeomorphism if it is infinitely differentiable} + \end{gather*} +\end{defi} + +\begin{rem} + \begin{enumerate}[(i)] + \item In physics, $f$ and $f \circ T$ are often denoted with the same symbol + \item We can apply the chain rule to $T \circ \inv{T} = \idf_V$ + \[ + DT(\inv{T}(y)) \cdot D\inv{T}(y) = I_V + \] + Since $\inv{T}$ is surjective, $DT(x)$ is invertible $\forall x \in U$. According to the theorem about inverse functions, + the inverse $\inv{T}$ of a bijective mapping is continuously differentiable if $DT(x)$ is invertible + \item If $T$ is a diffeomorphism, then $\inv{T}$ is one too. + \end{enumerate} +\end{rem} + +\begin{eg} + \begin{enumerate}[(i)] + \item Polar coordinates: + \begin{align*} + T: (0, \infty) \times (0, 2\pi) &\longrightarrow \realn^2 \setminus \set{[0, \infty] \times \set{0}} \\ + (r, \phi) &\longmapsto (r \cos\phi, r \sin\phi) + \end{align*} + + \item Another diffeomorphism would be + \begin{align*} + T: \oball[1](0) &\longrightarrow \realn^n \\ + x &\longmapsto \frac{x}{\sqrt{1 - \norm{x}}} + \end{align*} + + \item An example for a mapping that is no diffeomorphism would be + \begin{align*} + T: \realn &\longrightarrow \realn \\ + x &\longmapsto x^3 + \end{align*} + The Jacobian "matrix" $T'(x) = 3x^2$ is not invertible. + + \item Another counter example would be + \begin{align*} + T: (0, \infty) \times \realn &\longrightarrow \realn^2 \setminus \set{0} \\ + (r, \phi) &\longmapsto (r\cos\phi, r\sin\phi) + \end{align*} + This function is not injective, so it's not a diffeomorphism. + \end{enumerate} +\end{eg} + +\begin{thm}[Transformation Theorem] + Let $U, V \subset \realn^n$ and $T: U \rightarrow V$ a diffeomorphism. + Then $f: V \rightarrow \realn$ is integrable over $V$ if and only if $f \circ T \cdot \abs{\det DT}$ is integrable over $U$. + In this case + \[ + \int_V d \dd{\lambda^n} = \int_U f \circ T \cdot \abs{\det DT} \dd{\lambda^n} + \] +\end{thm} +\begin{proof} + Without proof. +\end{proof} + +\begin{eg}[Area of the unit circle] + The area is defined as + \[ + \lambda^2(K_1(0)) = \int_{\realn^2} \charfun_{K_1(0)} \dd{\lambda^2} + \] + We transform into polar coordinates: + \begin{align*} + U &= (0, \infty) \times (0, 2\pi) \\ + V &= \realn^2 \setminus \underbrace{([0, \infty] \times \set{0})}_{\lambda^2-null set} + \end{align*} + We define the transformation + \[ + T: (r, \phi) \longmapsto (r \cos\phi, r \sin\phi) + \] + Which results in + \begin{align*} + \det DT(r, \phi) &= r \\ + \charfun_{K_1(0)} \circ T(r, \phi) &= \charfun_{(0, 1]}(r) + \end{align*} + So we can calculate + \begin{align*} + \lambda^2(K_1(0)) &= \int_B \charfun_{(0, 1]}(x, y) \dd{\lambda^2}(x, y) \\ + &= \int_U \charfun_{(0, 1]}(r) \cdot r \cdot \dd{\lambda^2}(r, \phi) \\ + &= \int_0^{\infty} \int_0^{2\pi} \charfun_{(0, 1]} r \dd{\phi} \dd{r} \\ + &= 2\pi \int_0^{\infty} \charfun_{(0, 1]}(r) \dd{r} = 2\pi \int_0^1 r \dd{r} \\ + &= \pi r^2 = \pi + \end{align*} +\end{eg} + +\begin{rem} + \begin{enumerate}[(i)] + \item Consider + \begin{align*} + T: \realn^n &\longrightarrow \realn^n \\ + x &\longmapsto Ax ~~A \in \realn^{n \times n} + \end{align*} + If $\exists \inv{A}$, then $T$ is a diffeomorphism with $DT = A$ + \[ + \implies \int f \dd{\lambda^2} = \abs{\det A} \int f \circ T \dd{\lambda^2} + \] + + \item Let $A$ be an orthogonal matrix (so a rotation/mirroring). + \[ + \det A = \pm 1 \implies \abs{\det A} = 1 + \] + Thus, rotations and mirrorings do not change the volume. + + \item Let $A = \diag(a, a, \cdots, a) ~~a \in (0, \infty)$ (this is a scaling matrix). Then + \[ + \det A = a^n + \] + which means that continuous scaling of a factor $a$ scales the $\lambda^n$-volume by $a^n$. + + \item This is a "generalization" of the substitution rule + \[ + \int_{\realn} f(g(x)) g'(x) \dd{x} = \int_{\realn} f(y) \dd{y} + \] + \end{enumerate} +\end{rem} + +\begin{eg} + We want to compute + \[ + K = \int_{\realn} e^{-x^2} \dd{x} + \] + Consider + \[ + K^2 = \int_{\realn} e^{-x^2} \dd{x} \int_{\realn} e^{-y^2} \dd{y} = \int_{\realn^2} e^{-(x^2 + y^2)} \dd{\lambda^2(x, y)} + \] + By transforming $f = e^{-(x^2 + y^2)}$ into polar coordinates + \begin{align*} + K^2 &= \int_U f \circ T \abs{\det DT} \dd{\lambda^2} \\ + &= \int_V e^{-r^2} \cdot r \dd{\lambda^2(r, \phi)} \\ + &= \int_0^{\infty} \int_0^{2\pi} r e^{-r^2} \dd{r}\dd{\phi} \\ + &= 2\pi \int_0^{\infty} r e^{-r^2} \dd{r} \\ + &= 2\pi \limn\left(-\frac{1}{2} e^{-n^2} + \frac{1}{2}\right) = \pi + \end{align*} + Thus $K = \sqrt{\pi}$. +\end{eg} + +\begin{eg}[Integrability of radial functions] + Let $f: [0, \infty] \rightarrow \realn$ be measureable and set + \begin{align*} + F: \realn^n &\longrightarrow \realn \\ + x &\longmapsto f(\norm{x}) + \end{align*} + $\dnorm$ is the Euclidian norm. Under which conditions is $F$ $\lambda^n$-integrable? + Let $D := (0, \infty) \times \underbrace{(0, \pi)^{n-2} \times (0, 2\pi)}_{D_{\phi}}$. And define + \begin{align*} + T: D &\longrightarrow \realn^n \setminus A \\ + (r, \phi) &\longmapsto \begin{pmatrix} + r \cos\phi_1 \\ + r \sin\phi_1 \cos\phi_2 \\ + r \sin\phi_1 \sin\phi_2 \cos\phi_3 \\ + \vdots \\ + r \sin\phi_1 \cdots \sin\phi_{n-2} \cos\phi_{n-1} \\ + r \sin\phi_1 \cdots \sin\phi_{n-2} \sin\phi_n + \end{pmatrix}^T + \end{align*} + Then $\norm{T(r, \phi)} = r$ and + \[ + \abs{\det DT(r, \phi)} = r^{n-1} \sin^{n-2}\phi_1 \sin^{n-3} \phi_2 \cdots \sin\phi_{n-2} = r^{n-1} A_n(\phi) + \] + Thus + \begin{align*} + \int_{\realn^n} \abs{F(x)} \dd{\lambda^n}(x) &= \int_D \underbrace{\abs{F \circ T(r, \phi)}}_{f(r)} \abs{\det DT(r, \phi)} \dd{\lambda^n}(r, \phi) \\ + &= \int_{D_{\phi}} \int_0^{\infty} r^{n-1} \abs{f(r)} A_n(\phi) \dd{r} \dd{\lambda^{n-1}}(\phi) \\ + &= \int_0^{\infty} r^{n-1} \abs{f(x)} \dd{r} \underbrace{\int_{D_{\phi}} \abs{A_n(\phi)} \dd{\lambda^{n-1}}(\phi)}_{< \infty} + \end{align*} + So $F$ is $\lambda^n$-integrable if $r^{n-1} f(x)$ is integrable over $[0, \infty)$. +\end{eg} +\end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index ce95a2c..284e110 100644 Binary files a/script.pdf and b/script.pdf differ diff --git a/script.tex b/script.tex index 9f90fd8..021a393 100644 --- a/script.tex +++ b/script.tex @@ -34,6 +34,7 @@ \DeclareMathOperator{\spn}{span} %\DeclareMathOperator{\dim}{dim} \DeclareMathOperator{\sgn}{sgn} +\DeclareMathOperator{\diag}{diag} \newcommand{\natn}{\mathbb{N}} \newcommand{\intn}{\mathbb{Z}}