From 6e95b496bb256be384350cc3083d14d04b741839 Mon Sep 17 00:00:00 2001 From: Robert Date: Sun, 4 Apr 2021 19:33:59 +0200 Subject: [PATCH] Finished Line Integrals --- chapters/int_submanifold.tex | 10 + chapters/sections/line_integrals.tex | 409 +++++++++++++++++++++++++ chapters/sections/solution_methods.tex | 2 +- script.pdf | Bin 1198007 -> 1226065 bytes script.tex | 7 +- 5 files changed, 425 insertions(+), 3 deletions(-) create mode 100644 chapters/int_submanifold.tex create mode 100644 chapters/sections/line_integrals.tex diff --git a/chapters/int_submanifold.tex b/chapters/int_submanifold.tex new file mode 100644 index 0000000..b72fe3d --- /dev/null +++ b/chapters/int_submanifold.tex @@ -0,0 +1,10 @@ +% !TeX root = ../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} + \chapter{Integration over Submanifolds} + \vspace*{\fill}\par + \pagebreak + + \subfile{sections/line_integrals.tex} +\end{document} \ No newline at end of file diff --git a/chapters/sections/line_integrals.tex b/chapters/sections/line_integrals.tex new file mode 100644 index 0000000..928bde5 --- /dev/null +++ b/chapters/sections/line_integrals.tex @@ -0,0 +1,409 @@ +% !TeX root = ../../script.tex +\documentclass[../../script.tex]{subfiles} + +\begin{document} +\section{Line Integrals} + +\begin{defi} + Let $I$ be an interval and $n \in \natn$. A parametrized curve (or path) in $\realn^n$ is a continuous mapping + \[ + \gamma: I \longrightarrow \realn^n + \] + A parametrized curve is said to be regular if it is $C^1$ and $\gamma'(t) \ne 0 ~~\forall t \in I$. + It is said to be piecewise regular if there is a disjoint decomposition + \[ + I = I_1 \cup I_2 \cup \cdots \cup I_n + \] + into partial intervals such that $\gamma$ is regular on each partial interval. + + A curve is a subset of $\realn^n$ that is the image of a parametrized curve. If $\curve$ is a curve, then + \[ + \gamma: I \longrightarrow \realn^n + \] + is said to be the parametrization of $\curve$, if $\gamma(I) = \curve$ and if $\gamma$ is injective on $\interior{I}$. + The curves in this chapter will always be regular. +\end{defi} + +\begin{eg} + \begin{enumerate}[(i)] + \item $\alpha, \kappa > 0$: + \begin{align*} + \gamma: \realn &\longrightarrow \realn^3 \\ + t &\longmapsto (\cos(\alpha t), \sin(\alpha t), \kappa t) + \end{align*} + This is the parametrization of a screw curve. + + \item The unit circle + \[ + \set[x^2 + y^2 = 1]{(x, y) \in \realn^2} + \] + is a curve with the parametrization + \begin{align*} + \gamma: [0, 2\pi] &\longrightarrow \realn^2 \\ + t &\longmapsto (\cos t, \sin t) + \end{align*} + + \item A square + \[ + \set[\max\set{\abs{x_1}, \abs{x_2}} = 1]{(x, y) \in \realn^2} + \] + is a piecewise regular curve. + \end{enumerate} +\end{eg} + +\begin{rem} + Let $\gamma: I \rightarrow \realn^n$ be regular, $f: \gamma(I) \rightarrow \realn$ be continuous and $a, b \in \interior{I}$. + A decomposition $Z$ is given by the grid points + \[ + a = t_0 < t_1 < \cdots < t_n = b + \] + The fineness of $Z$ is given by + \[ + m(Z) := \max_{t \in \set{0, 1, \cdots, n-1}} (t_{i+1} - t_i) + \] + We can represent $I$ in terms of $Z$ via + \[ + I(Z) := \sum_{i=0}^{n-1} f(\gamma(t_i)) \norm{\gamma(t_{i+1}) - \gamma(t_i)} + \] + Or in integral representation + \[ + I(Z) = \int_a^b \underbrace{\sum_{i=0}^{n-1} f(\gamma(t_i)) \frac{\norm{\gamma(t_{i+1}) - \gamma(t_i)}}{\norm{t_{i+1} - t_i}} \charfun_{[t_i, t_{i+1})}(t)}_{g_Z(t)} \dd{t} + \] + So let $(Z_j)$ be a sequence of decompositions with + \[ + m(Z_j) \conv{j \rightarrow \infty} 0 + \] + Let $t \in [a, b]$ not be a grid point of any $Z_j$. Then there exists a unique grid poiont $t_{j, i_j}$ such that $t \in [t_{j, i_j}, t_{j, i_{j+1}}]$. Then + \[ + \limes{j}{\infty} t_{j, i_j} = \limes{j}{\infty} t_{j, i_{j+1}} = t + \] + And thus + \[ + \limes{j}{\infty} g_{Z_j}(t) = f(\gamma(t)) \norm{\gamma'(t)} + \] + $\forall t$ that are not grid points of $Z_j$, this means tahat + \[ + g_{Z_j} \conv{j \rightarrow \infty} f \norm{\gamma'} + \] + almost everywhere. The dominated convergence theorem then tells us + \[ + I(Z_j) = \int_a^b g_{Z_j}(t) \dd{t} \conv{j \rightarrow \infty} \int_a^b f(\gamma(t)) \norm{\gamma'(t)} \dd{t} + \] + Special case: For $f \equiv 1$ one gets the arc length. +\end{rem} + +\begin{defi}[Line Integrals, Arc Length] + Let $I$ be an interval and $\gamma: I \rightarrow \realn^n$ a parametrized curve. Define the functions + \begin{align*} + f: \gamma(I) \longrightarrow \realn && E: \gamma(I) \longrightarrow \realn^n + \end{align*} + Then + \[ + \int_{\gamma} f \dd{s} := \int_I f(\gamma(t)) \norm{\gamma'(t)} \dd{t} + \] + is said to be a scalar line integral (line integral of first kind), and + \[ + \int_{\gamma} \innerproduct{E}{\dd{s}} := \int_I \innerproduct{E(\gamma(t))}{\gamma'(t)} \dd{t} + \] + is said to be a vector line integral (line integral of second kind). + The function $f$ or the vector field $E$ are integrable along $\gamma$ if the according integral exists. + The integral + \[ + \int_{\gamma} \dd{s} + \] + is the arc length of $\gamma$, and $\gamma$ is said to be rectifiable if this integral is finite. + + If the curve $\gamma$ is closed, i.e. if $I = [a, b] ~~a, b \in \realn$ and + \[ + \gamma(a) = \gamma(b) + \] + Then the above integrals are often notated as + \begin{align*} + \oint_{\gamma} \dd{s} && \oint_{\gamma} \innerproduct{E}{\dd{s}} + \end{align*} + to emphasize that the curve is closed. This changes nothing about the formulas, it is merely visual. + I will try to adhere to this style. +\end{defi} + +\begin{eg}[Circumference of the unit circle] + Define + \begin{align*} + \gamma: [0, 2\pi] &\longrightarrow \realn^2 \\ + t &\longrightarrow (\cos(t), \sin(t)) + \end{align*} + and derive this function + \[ + \gamma'(t) = (-\sin(t), \cos(t)) \implies \norm{\gamma'(t)} = 1 + \] + Then the circumference is + \[ + \oint_{\gamma} \dd{s} = \int_0^{2\pi} \dd{t} = 2\pi + \] +\end{eg} + +\begin{rem} + \begin{enumerate}[(i)] + \item If $\gamma$ is only piecewise regular then the integrands might not be defined for all $t$. + \item Line integrals don't depend on the chosen parametrization. This means if $\curve$ is a curve and + \begin{align*} + \gamma: I \rightarrow \curve && \rho: J \rightarrow \curve + \end{align*} + are parametrizations, then + \[ + \int_{\gamma} f \dd{s} = \int_{\rho} f \dd{s} + \] + We also write + \[ + \int_{\curve} f \dd{s} + \] + The same holds for vector integrals. + \item Both kinds of integrals depend on the scalar product. + \item Both kinds of integrals are special cases of integrals over so called One-forms + \end{enumerate} +\end{rem} + +\begin{thm} + Let $\gamma: I \rightarrow \realn^n$ be a parametrized curve, and $\vartheta: J \rightarrow I$ a diffeomorphism + (so $\vartheta \in C^1$ and $\vartheta'(t) \ne 0 ~~\forall t \in J$). Let $f: \gamma(I) \rightarrow \realn$, then + \[ + \int_{\gamma} f \dd{s} = \int_{\gamma \circ \vartheta} f \dd{s} + \] +\end{thm} +\begin{proof} + We can assume $I, J$ to be open, since the endpoints of the integrals are a null set and thus don't matter. + W.l.o.g. let $\gamma$ be regular. Then + \begin{equation} + \begin{split} + \int_{\gamma \circ \vartheta} f \dd{s} &= \int_J f(\gamma \circ \vartheta)(t) \norm{(\gamma \circ \vartheta)'(t)} \dd{t} \\ + &= \int_J f(\gamma(\vartheta(t))) \norm{\gamma'(\vartheta(t)) \vartheta'(t)} \dd{t} \\ + &= \int_J f(\gamma(\vartheta(t))) \norm{\gamma'(\vartheta(t))} \abs{\vartheta'(t)} \dd{t} \\ + &= \int_I f(\gamma(\tau)) \norm{\gamma'(\tau)} \dd{\tau} \\ + &= \int_{\gamma} f \dd{s} + \end{split} + \end{equation} +\end{proof} + +\begin{rem} + \begin{enumerate}[(i)] + \item One can show that for a curve $\curve$ and parametrizations + \begin{align*} + \gamma: I \rightarrow \curve && \rho: J \rightarrow \curve + \end{align*} + there exists a diffeomorphism $\vartheta: J \rightarrow I$ such that + \[ + \rho = \gamma \circ \vartheta + \] + So the line integral of first degree doesn't depend on the parametrization. + + \item A line integral of second degree doesn't depend on the parametrization if the parametrizations run along the curve in the same direction. + So if $\vartheta' > 0$, $\vartheta$ is said to conserve orientation. If $\vartheta' < 0$ then the integral switches sign. + \end{enumerate} +\end{rem} + +\begin{eg} + Let $\gamma: I \rightarrow \realn^3$ be the trajectory of a point mass, and $F: \realn^3 \rightarrow \realn^3$ a time-independent forcefield. + The work done is then given by + \[ + W := \int_{\gamma} \innerproduct{F}{\dd{s}} + \] + The fact that the parametrization can be chosen arbitrarily means that the work done in a forcefield is independent from the velocity of the point mass. +\end{eg} + +\begin{rem} + \begin{enumerate}[(i)] + \item Line integrals are linear in $f$ or $E$, meaning for + \[ + f, g: \gamma(I) \rightarrow \realn, ~~\lambda \in \realn + \] + we have + \[ + \int_{\gamma}(g + \lambda g) \dd{s} = \int_{\gamma} f \dd{s} + \lambda \int_{\gamma} g \dd{s} + \] + + \item Parametrized curves over compact intervals can be reparametrized so that $I = [0, 1]$. + + \item Let + \begin{align*} + \gamma: [0, 1] \rightarrow \realn^n && \rho: [0, 1] \rightarrow \realn^n + \end{align*} + be curves with $\gamma(1) = \rho(0)$. Define + \begin{align*} + \inv{\gamma}: [0, 1] &\longrightarrow \realn^n & \gamma\rho: [0, 1] &\longrightarrow \realn^n \\ + t &\longrightarrow \gamma(1 - t) & t &\longrightarrow \begin{cases} + \gamma(2t), & t \le 0.5 \\ + \rho(2t + 1), & t > 0.5 + \end{cases} + \end{align*} + Then we have + \begin{align*} + &\int_{\inv{\gamma}} f \dd{s} = \int_{\gamma} f \dd{s} \\ + &\int_{\gamma\rho} f \dd{s} = \int_{\gamma} f \dd{s} + \int_{\rho} f \dd{s} \\ + &\int_{\inv{\gamma}} \innerproduct{E}{\dd{s}} = -\int_{\gamma} \innerproduct{E}{\dd{s}} \\ + &\int_{\gamma\rho} \innerproduct{E}{\dd{s}} = \int_{\gamma} \innerproduct{E}{\dd{s}} + \int_{\rho} \innerproduct{E}{\dd{s}} + \end{align*} + \end{enumerate} +\end{rem} + +\begin{defi} + Let $U \subset \realn^n$ be open and $f: U \rightarrow \realn$ a $C^1$-function. Define + \[ + \grad f = (\partial_1 f, \partial_2 f, \cdots, \partial_m f) + \] + The vector field $E: U \rightarrow \realn^n$ is said to be conservative if there is a function $g: U \rightarrow \realn$ such that + \[ + E = \grad{g} + \] + $g$ is the potential of $E$. +\end{defi} + +\begin{rem} + \begin{enumerate}[(i)] + \item In physics the sign is typically switched, so + \[ + E = -\grad{g} + \] + + \item The IDE + \[ + p(x, y) + q(x, y)y' = 0 + \] + is exact if and only if the vector field $(p, q)$ is conservative. + + \item If $E$ is conservative and $C^1$, then + \[ + \partial_i E_j = \partial_j E_i + \] + This condition is not sufficient in general. + + \item If $g$ is a potential for $E$, then the functions + \[ + g + c ~~c \in \realn + \] + are also potentials. + + \item If $E$ is conservative, $g$ a potential and $\gamma: [a, b] \rightarrow \realn^n$ a curve, then + \begin{align*} + \int_{\gamma} \innerproduct{E}{\dd{s}} &= \int_a^b \innerproduct{E(\gamma(t))}{\gamma'(t)} \dd{t} \\ + &= \int_a^b \left(\partial_1 g(\gamma(t)) \gamma_1'(t) + \cdots + \partial_n g(\gamma(t))\gamma_n'(t)\right) \dd{t} \\ + &= \int_a^b (g \circ \gamma)'(t) \dd{t} = g(\gamma(b)) - g(\gamma(a)) + \end{align*} + The vector line integral over conservative fields is independent from the chosen path (it only depends on the start and end points). + + \item Let $U$ be open, path-connected and $E: U \rightarrow \realn^n$ a conservative vector field. + Choose a fixed $x_0 \in U$, and for $x \in U$ choose a parametrized curve $\gamma_x$ from $x_0$ to $x$. + Then + \[ + x \longmapsto \int_{\gamma_x} \innerproduct{E}{\dd{s}} + \] + is a potential, because if $g$ is an arbitrary potential we have + \[ + \int_{\gamma_x} \innerproduct{E}{\dd{s}} = g(x) - g(x_0) \quad \forall x \in U + \] + \end{enumerate} +\end{rem} + +\begin{eg} + \begin{enumerate}[(i)] + \item Let + \begin{align*} + E: \realn^3 \setminus \set{0} &\longrightarrow \realn^3 \\ + x &\longmapsto -\frac{x}{\norm{x}^3} + \end{align*} + This field is conservative, with the potential + \begin{align*} + \phi: \realn^3 \setminus \set{0} &\longrightarrow \realn \\ + x &\longmapsto \rec{\norm{x}} + \end{align*} + + \item Let + \begin{align*} + E: \realn^2 \setminus \set{0} &\longrightarrow \realn^2 \\ + (x, y) &\longmapsto \left(-\frac{y}{x^2 + y^2}, \frac{x}{x^2 + y^2} \right) + \end{align*} + Then + \[ + \partial_1 E_2 = \frac{1}{x^2 + y^2} - \frac{2x^2}{(x^2 + y^2)} = \frac{y^2 - x^2}{(x^2 + y^2)^2} = \partial_2 E_1 + \] + We can calculate the line integral of $E$ along the unit circle + \begin{align*} + \gamma: [0, 2\pi] &\longrightarrow \realn^2 \\ + t &\longmapsto (\cos t, \sin t) + \end{align*} + Then + \[ + E(\gamma(t)) = (-\sin t, \cos t) = \gamma'(t) + \] + The integral is then + \[ + \int_{\gamma} \innerproduct{E}{\dd{s}} = \int_0^{2\pi} \norm{(-\sin t, \cos t)}^2 \dd{t} = 2\pi \ne 0 + \] + + \item In the chapter about differential equations we looked at an exact equation in \Cref{eg:817}: + \[ + (2x + y^2) + (2xy)y' = 0 + \] + We can now use curve integrals to calculate the potential function more easily. For that let $x_0 = (0, 0)$. + Then for $(\xi, \eta)$ we can define a curve connecting $x_0$ and $(\xi, \eta)$ for $t \in [0, 1]$: + \[ + t \longmapsto (\xi t, \eta t) + \] + Consider the vector field + \[ + E(x, y) = (2x + y^2, 2xy) + \] + Then + \begin{align*} + (\xi, \eta) \longmapsto \int_{\gamma} \innerproduct{E}{\dd{s}} &= \int_0^1 \innerproduct{E(\xi t, \eta t)}{(\xi, \eta)} \dd{t} \\ + &= \int_0^1 (2\xi^2 t + \eta^2\xi t^2 + 2\xi\eta^2 t^2) \dd{t} \\ + &= \xi^2 + \eta^2\xi + \end{align*} + \end{enumerate} +\end{eg} + +\begin{thm} + Let $U \subset \realn^n$ be an open subset. A continuous vector field $E: U \rightarrow \realn^n$ is conservative if and only if + for every closed curve $\gamma: [0, 1] \rightarrow U$ the following holds + \[ + \oint_{\gamma} \innerproduct{E}{\dd{s}} = 0 + \] +\end{thm} +\begin{proof} + Line integrals over $E$ are path independent. Let $\gamma, \rho: [0, 1] \rightarrow U$ be paths with + \begin{align} + \gamma(0) = \rho(0) && \gamma(1) = \rho(1) + \end{align} + Then $\gamma \inv{\rho}$ is closed, so + \begin{equation} + 0 = \int_{\gamma\inv{\rho}} \innerproduct{E}{\dd{s}} = \int_{\gamma} \innerproduct{E}{\dd{s}} - \int_{\rho} \innerproduct{E}{\dd{s}} + \end{equation} + Assume that $U$ is path continuous. Choose a fixed $x_0 \in U$ and let $g: U \rightarrow \realn$. Then + \begin{equation} + g(x) = \int_{x_0}^x \innerproduct{E}{\dd{s}} + \end{equation} + Performing a directional derivation in direction $h \in \realn^n$ yields + \begin{equation} + \begin{split} + g(x + ah) - g(x) &= \int_{x_0}^{x + ah} \innerproduct{E}{\dd{s}} - \int_{x_0}^x \innerproduct{E}{\dd{s}} \\ + &= \int_{x}^{x + ah} \innerproduct{E}{\dd{s}} \\ + &= \int_0^a \innerproduct{E(x + th)}{h} \dd{t} + \end{split} + \end{equation} + Here we have chosen a linear path of integration between $x_0$ and $x$, and between $x$ and $x + ah$. In other words, we're integrating along + \begin{equation} + t \longmapsto x + th + \end{equation} + Using the intermediate value theorem, we can find that $\exists \xi_a \in (0, a)$ such that + \begin{equation} + \int_0^a \innerproduct{E(x + th)}{h} \dd{t} = \innerproduct{E(x + \xi_a h)}{h} \cdot a + \end{equation} + Then we have + \begin{equation} + \partial_h g(x) = \limes{a}{0} \frac{g(x + ah) - g(x)}{a} = \limes{a}{0} \innerproduct{E(x + \xi_a h)}{h} = \innerproduct{E(x)}{h} + \end{equation} + So if $h$ is a standard basis $e_i$, then + \begin{equation} + \partial_i g(x) = E_i(x) + \end{equation} + Thus the partial derivative of $g$ is continuous, and therefore $g$ is continuously differentiable, and thus a potential. +\end{proof} +\end{document} \ No newline at end of file diff --git a/chapters/sections/solution_methods.tex b/chapters/sections/solution_methods.tex index bbeffcd..dbf5d04 100644 --- a/chapters/sections/solution_methods.tex +++ b/chapters/sections/solution_methods.tex @@ -306,7 +306,7 @@ The above condition is merely necessary! 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