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@@ -3,7 +3,7 @@ % !TeX root = FuVar.tex %% TODO: publish to CTAN -\documentclass[twocolumn, margin=normal]{tex/hsrzf} +\documentclass[twocolumn, margin=small]{tex/hsrzf} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Packages @@ -498,7 +498,7 @@ typesetting may trick you into thinking it is rigorous, but really, it is not. % and similarly the \emph{area moment of inertia} \(I\) \end{application} -\section{Parametric curves and line integrals} +\section{Parametric curves, line and surface integrals} \begin{definition}[Parametric curve] A parametric curve is a vector function \(\mathcal{C} : \mathbb{R} \to W @@ -620,10 +620,182 @@ typesetting may trick you into thinking it is rigorous, but really, it is not. \end{align*} \end{theorem} -\section{Surface integrals} +\begin{definition}[Parametric surface] + A parametric surface is a vector function \(\mathcal{S}: W \subseteq \mathbb{R}^2 \to + \mathbb{R}^3\). +\end{definition} + +\begin{theorem}[Area of a parametric surface] + The area spanned by a parametric surface \(\vec{s}(u,v)\), with continuous + partial derivatives and that satisfy \(\partial_u \vec{s} \crossp \partial_v + \vec{s} \neq \vec{0}\), is given by + \[ + A = \int_\mathcal{S} ds + = \iint |\partial_u \vec{s} \crossp \partial_v \vec{s}| \,dudv + \] +\end{theorem} + +\begin{definition}[Scalar surface integral] + Let \(f: \mathbb{R}^3 \to \mathbb{R}\) be a function on a parametric surface + \(\vec{s}: W \subseteq \mathbb{R}^2 \to \mathbb{R}^3\). The surface integral + of \(f\) over \(\mathcal{S}\) is + \[ + \int_\mathcal{S} f \,ds = + \iint_W f(\vec{s}(u,v)) \cdot + |\partial_u \vec{s} \crossp \partial_v \vec{s}| \,dudv + \] +\end{definition} \section{Vector analysis} +\begin{definition}[Flux] + In a vector field \(\vec{F}: \mathbb{R}^m \to \mathbb{R}^n\) we define the + \emph{flux} through a parametric surface \(\mathcal{S}\) as + \[ + \Phi = \int_\mathcal{S} \vec{F} \dotp d\vec{s} + = \int_\mathcal{S} \vec{F} \dotp \uvec{n} \,ds . + \] + If \(\mathcal{S}\) is a closed surface we write + \( + \mathring{\Phi} = \oint_\mathcal{S} \vec{F} \dotp d\vec{s} + \). +\end{definition} + +If we now take the normalized flux on the surface of an arbitrarily small +(limit) volume \(V\) we get the \emph{divergence} +\[ + \div \vec{F} = \lim_{V\to 0} \frac{1}{V} \oint_{\partial V} \vec{F}\dotp d\vec{s} . +\] + +\begin{theorem}[Formula for divergence] + Let \(\vec{F}: \mathbb{R}^m \to \mathbb{R}^m\) be a vector field. + The divergence of \(\vec{F} = (F_{x_1},\ldots, F_{x_m})^t\) is + \[ + \div\vec{F} = \sum_{i = 1}^m \partial_{x_i} F_{x_i} , + \] + as suggested by the (ab)use of the dot product notation. +\end{theorem} + +\begin{theorem}[Divergence theorem, Gauss's theorem] + Because the flux on the boundary \(\partial V\) of a volume \(V\) contains + information of the field inside of \(V\), it is possible relate the two with + \[ + \int_V \div \vec{F} \,dv = \oint_{\partial V} \vec{F} \dotp d\vec{s} . + \] +\end{theorem} + +\begin{definition}[Circulation, Vorticity] + The result of a closed line integral can be interpreted as a macroscopic + measure how much the field rotates around a given point, and is thus + sometimes called \emph{circulation} or \emph{vorticity}. +\end{definition} + +As before, if we now make the area \(A\) enclosed by the parametric curve for +the circulation arbitrarily small, normalize it, and use Gauss's theorem we get +a local measure called \emph{curl} +\[ + \curl \vec{F} = + \lim_{A\to 0} \frac{\uvec{n}}{A} \oint_{\partial A} \vec{F} \dotp d\vec{s} +\] +Notice that the curl is a vector, normal to the enclosed surface \(A\). + +\begin{theorem}[Formula for curl] + Let \(\vec{F}\) be a vector field. In 2 dimensions + \[ + \curl \vec{F} = \left(\partial_x F_y - \partial_y F_x\right)\uvec{z}. + \] + And in 3D + \[ + \curl \vec{F} = \begin{pmatrix} + \partial_y F_z - \partial_z F_y \\ + \partial_z F_x - \partial_x F_z \\ + \partial_x F_y - \partial_y F_x + \end{pmatrix} + = \begin{vmatrix} + \uvec{x} & \uvec{y} & \uvec{z} \\ + \partial_x & \partial_y & \partial_z \\ + F_x & F_y & F_z + \end{vmatrix} . + \] +\end{theorem} + +\begin{theorem}[Stokes' theorem] + \[ + \int_\mathcal{S} \curl \vec{F} \dotp d\vec{s} + = \oint_{\partial\mathcal{S}} \vec{F} \dotp d\vec{r} + \] +\end{theorem} + +\begin{theorem}[Green's theorem] + The special case of Stokes' theorem in 2D is knowns as Green's theorem. + \[ + \int_\mathcal{S} \partial_x F_y - \partial_y F_x \,ds + = \oint_{\partial\mathcal{S}} \vec{F} \dotp d\vec{r} + \] +\end{theorem} + +\begin{definition}[Laplacian operator] + A second vector derivative is so important that it has a special name. For a + scalar function \(f: \mathbb{R}^m \to \mathbb{R}\) the divergence of the + gradient + \[ + \laplacian = \div (\grad f) = \sum_{i=1}^m \partial_{x_i}^2 f_{x_i} + \] + is called the \emph{Laplacian operator}. +\end{definition} + +\begin{definition}[Vector Laplacian] + The Laplacian operator can be extended on a vector field \(\vec{F}\) to the + \emph{Laplacian vector} by applying the Laplacian to each component: + \[ + \vlaplacian \vec{F} + = (\laplacian F_x)\uvec{x} + + (\laplacian F_y)\uvec{y} + + (\laplacian F_z)\uvec{z} . + \] + The vector laplacian can also be defined as + \[ + \vlaplacian \vec{F} = \grad (\div \vec{F}) - \curl (\curl \vec{F}). + \] +\end{definition} + +\begin{theorem}[Product rules and second derivatives] + Let \(f,g\) be sufficiently differentiable scalar functions \(D + \subseteq\mathbb{R}^m \to \mathbb{R}\) and \(\vec{A}, \vec{B}\) be + sufficiently differentiable vector fields in \(\mathbb{R}^m\) (with \(m = 2\) + or 3 for equations with the curl). + \begin{itemize} + \item Rules with the gradient + \begin{align*} + \grad (\div \vec{A}) &= \curl \curl \vec{A} + \vlaplacian \vec{A} \\ + \grad (f\cdot g) &= (\grad f)\cdot g + f\cdot \grad g \\ + \grad (\vec{A} \dotp \vec{B}) &= + (\vec{A} \dotp \grad) \vec{B} + + (\vec{B} \dotp \grad) \vec{A} \\ + & + \vec{A} \crossp (\curl \vec{B}) + + \vec{B} \crossp (\curl \vec{A}) + \end{align*} + \item Rules with the divergence + \begin{align*} + \div (\grad f) &= \laplacian f \\ + \div (\curl \vec{A}) &= 0 \\ + \div (f\cdot \vec{A}) &= (\grad f) \dotp \vec{A} + f\cdot (\div \vec{A}) \\ + \div (\vec{A}\crossp\vec{B}) &= (\curl \vec{A})\dotp \vec{B} + - \vec{A} \cdot (\curl\vec{B}) + \end{align*} + \item Rules with the curl + \begin{align*} + \curl (\grad f) &= \vec{0} \\ + \curl (\curl \vec{A}) &= \grad (\div \vec{A}) - \vlaplacian \vec{A} \\ + \curl (\vlaplacian \vec{A}) &= \vlaplacian (\curl \vec{A}) \\ + \curl (f\cdot \vec{A}) &= (\grad f)\crossp \vec{A} + f\cdot \curl \vec{A} \\ + \curl (\vec{A}\crossp\vec{B}) &= + (\vec{B} \dotp \grad) \vec{A} - (\vec{A} \dotp \grad) \vec{B} \\ + &+ \vec{A} \dotp (\div \vec{B}) - \vec{B} \dotp (\div \vec{A}) + \end{align*} + \end{itemize} +\end{theorem} + \section*{License} \doclicenseText |