\documentclass[11pt,a4paper,twocolumn,catalan]{article}
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\newcommand{\nablax}{\vec\nabla\times\vec}
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\title{Càlcul vectorial}
\author{Manel Bosch}
\date{}
\begin{document}
\maketitle
\noindent
\textbf{Operadors vectorials}\newline
1.\underline{Coordenades cartesianes}\newline
Siguin: $$f(x,y,z) \qquad \mathrm{i}\qquad \vec f(\vec r) = f_x(\vec r)\hat\imath + f_y(\vec r)\hat\jmath + f_z(\vec r)\hat k$$
\begin{itemize}
\item \textbf{Gradient}: $$\nablac f = \left({\frac{\partial}{\partial x}f,\frac{\partial}{\partial y}f,\frac{\partial}{\partial z}f}\right)$$
\item \textbf{Divergència}: $$\nablac\vec f = \frac{\partial}{\partial x}f+\frac{\partial}{\partial y}f+\frac{\partial}{\partial z}f $$
\item \textbf{Rotacional}: $$\displaystyle \nablax f = \begin{vmatrix}\vi& \vj& \vk \\\partial/\partial x & \partial /\partial y & \partial/\partial z \\  f_x& f_y & f_z \\ \end{vmatrix}$$
\item \textbf{Laplacià}: $$\nabla^2 f = \frac{\partial ^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{\partial^2 f}{\partial z^2}$$
\end{itemize}
2.\underline{Coordenades Polars}\newline
Siguin: $$(x,y)\to (\rho,\theta);\quad f(\rho,\phi); \quad\vec{f}=f_\rho\hat\rho + f_\phi\hat\phi$$
\begin{itemize}
\item \textbf{Gradient}: $$\nablac f = \frac{\partial f}{\partial\rho}\hat\rho + \frac{\partial f}{\partial\phi}\hat\phi $$
\item \textbf{Divergència}: $$\nablac\vec f = \frac 1\rho\frac{\partial f_\rho}{\partial \rho}+\frac{\partial f_\phi}{\partial \phi} $$
\item \textbf{Laplacià}: $$\nabla^2 f =\frac{1}{\rho}\frac{\partial}{\partial \rho}\left({\rho\frac{\partial f}{\partial\rho}}\right)+\frac{1}{\rho^2}\frac{\partial^2 f}{\partial \phi^2}$$
\end{itemize}
3.\underline{Coordenades cilíndriques}\newline
$$f(x,y,z) \to f(\rho,\phi,z); \qquad \vec f(\vec r) = (f_\rho,f_\phi,f_z)$$
\begin{itemize}
\item \textbf{Gradient}: $$\nablac f = \frac{\partial f}{\partial \rho}\hat\rho + \frac 1\rho \frac{\partial f}{\partial \phi}\hat\phi +\frac{\partial f}{\partial z}\hat z$$
\item \textbf{Divergència}: $$\nablac\vec f = \frac 1\rho \frac{\partial (\rho\cdot f_\rho)}{\partial \rho}+\frac 1\rho \frac{\partial f_\phi}{\partial \phi}+\frac{\partial f_ z}{\partial z} $$
\item \textbf{Rotacional}: $$\displaystyle \nablax f = \frac 1\rho \begin{vmatrix}\hat\rho& \rho\hat\phi& \hat z \\\partial/\partial \rho & \partial /\partial \phi & \partial/\partial z \\  f_\rho& \rho f_\phi & f_z \\ \end{vmatrix}$$
\item \textbf{Laplacià}: $$\nabla^2 f =\frac 1\rho\frac{\partial}{\partial\rho}\left({\rho\frac{\partial f}{\partial\rho}}\right) +\frac {1}{\rho^2} \frac{\partial^2 f}{\partial \phi^2}+\frac{\partial^2 f}{\partial z}  $$
\end{itemize}

4.\underline{Coordenades esfèriques}\newline
$$f(x,y,z) \to f(\rho,\theta,\phi); \qquad \vec f(\vec r) = (f_\rho,f_\theta,f_\phi)$$
\begin{itemize}
\item \textbf{Gradient}: $$\nablac f = \frac{\partial f}{\partial r}\hat r +\frac 1r \frac{\partial f}{\partial \theta}\hat\theta +\frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\hat \phi$$
\item \textbf{Divergència}: $$\nablac\vec f = \frac {1}{r^2} \frac{\partial}{\partial r}\left({r^2f_r}\right)+\frac{1}{r\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta\cdot f_\theta)+\frac{1}{r\sin\theta} \frac{\partial f_\phi}{\partial \phi} $$
\item \textbf{Rotacional}: $$\nablax f=\frac{1}{r^2\sin\theta}\begin{vmatrix}\hat{r} &r\hat{\theta} & r\sin\theta\hat{\phi} \\{\partial}/{\partial r} & {\partial}/{\partial \theta} & {\partial}/{\partial \phi}\\f_r & rf_\theta & r\sin\theta f_\phi\end{vmatrix}$$
\item \textbf{Laplacià}: $$\nabla^2 f = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial f}{\partial r}\right)+ \frac{1}{r^2\,{\rm \sin}\,\theta}\frac{\partial}{\partial \theta}\left({\rm \sin}\,\theta\frac{\partial f}{\partial \theta}\right) +$$ $$+ \frac{1}{r^2{\rm \sin}^2\theta}\frac{\partial^2 f}{\partial \phi^2}  $$
\end{itemize}\newpage

\textbf{CORBES I SUPERFÍCIES EN $\rr^3$}


Tenim $\vec r(t) = x(t)\vi + y(t)\vj + z(t)\vk$
\begin{itemize}
\item Equació d'una recta tangent a una corba en un punt $A(x_0,y_0,z_0)$: $$\frac{x-x_0}{\dd x(t)/\dd t} =\frac{y-y_0}{\dd y(t)/\dd t}=\frac{z-z_0}{\dd z(t)/\dd t} $$
\item Equació del pla normal a la corba en $A \to$ $\vec r-\vec r_0 \perp \dd\vec r/\dd t$:
$$\frac{\dd x(t)}{\dd t}(x-x_0)+\frac{\dd y(t)}{\dd t}(y-y_0)+\frac{\dd z(t)}{\dd t}(z-z_0)=0$$
\end{itemize}
Parametritzar en funció de la longitud d'arc:
$\vec r(t) = (x(t), y(t),z(t)) = r(x(z),y(z),z)$
\begin{itemize}
\item $\displaystyle \dd s = \left|\frac{\dd\vec r}{\dd t}\right|\dd t $
\item $\displaystyle \frac{\dd \vec r}{\dd s}\equiv\hat\sigma;\quad \lim_{\Delta S\to 0} \left|{\frac{\Delta\phi}{\Delta s}}\right|=K;\quad \frac{\dd \hat \sigma}{\dd S} = K\hat n$
\item $\displaystyle R=\frac 1K;\;\to\; \frac{1}{R^2}=\left|{\frac{\dd ^2\vec r}{\dd s}}\right|^2$
\item $\hat b = \hat \sigma\times\hat n;\qquad \displaystyle \frac{\dd \hat b}{\dd s} = \hat\sigma\times \displaystyle \frac{\dd\hat n}{\dd s};\qquad \displaystyle \frac{\dd \hat b}{\dd s}=\tau \hat n = \frac{1}{T}\hat n$
\item $\displaystyle \tau = \frac 1T = \left|{-R^2\cdot\frac{\dd \vec r}{\dd s}\left({\frac{\dd^2\vec r}{\dd s^2}\times\frac{\dd^3 \vec r}{\dd s^3}}\right)}\right|$
\end{itemize}
Fómules de Serret-Frenet: $\hat\sigma,\hat b$ i $\hat n$ formen un sistema de referència mòbil ortogonal, on
\begin{itemize}
\item $\displaystyle \hat\sigma =\frac{\dd\vec r}{\dd s} ;\; \hat n =\frac 1z\frac{\dd \hat b}{\dd s} = \frac 1K\frac{\dd^2\vec r}{\dd s^2} ;\; \hat b = \frac 1k\left({\frac{\dd\vec r}{\dd s}\times\frac{\dd^2\vec r}{\dd s^2}}\right)$
\end{itemize}
En funció d'un paràmetre $t$ qualsevol:
\begin{itemize}
\item $\displaystyle \hat\sigma(t)=\frac{\vec r'(t)}{|\vec r'(t)|} $
\item $\displaystyle \hat b(t) = \frac{\vec r'(t)\times \vec r''(t)}{|\vec r'(t)\times \vec r''(t)|} $
\item $\displaystyle \hat n(t) = \frac{(\vec r'(t)\times \vec r''(t))\times \vec r'(t)}{|(\vec r'(t)\times \vec r''(t))\times \vec r'(t)|} $
\item $\displaystyle \frac{1}{R} = \frac{\left|\displaystyle \frac{\dd \vec r}{\dd t}\times \frac{\dd^2\vec r}{\dd t^2}\right|}{\left|{\displaystyle \frac{\dd \vec r}{\dd t}}\right|^3}$
\item $\displaystyle\frac 1T = \left|{-\frac{\displaystyle \frac{\dd \vec r}{\dd t}\cdot\left({\frac{\dd^2 \vec r}{\dd t^2}\times \frac{\dd^3\vec r}{\dd t^3}}\right)}{\displaystyle \left|{\frac{\dd \vec r}{\dd t}\times \frac{\dd^2\vec r}{\dd t^2}}\right|^2}}\right| $
\end{itemize}
\sep
\textbf{Diferencials}

\underline{Polars}
\begin{itemize}
\item Posició: $\dd \vec r=\dd\rho\hat\rho + \rho\dd\phi\hat\phi$
\item Longitud: $\dd s^2= \dd\rho^2+\rho^2\dd\phi^2$
\item Superfície $\dd S= \rho\dd\rho\phi$
\end{itemize}
\underline{Cilíndriques}
\begin{itemize}
\item Posició: $\dd \vec r=\dd\rho\hat\rho + \rho\dd\phi\hat\phi + \dd z\hat z$
\item Longitud: $\dd s^2= \dd\rho^2 +\rho\dd\phi^2+\dd z^2$
\item Superfície $\dd S=\rho\dd\phi\dd z$ (per $\rho =$cte.)
\item Volum $\dd V=\rho\dd\rho\dd\phi\dd z$
\end{itemize}
\underline{Esfèriques}
\begin{itemize}
\item Posició: $\dd \vec r=\dd r\hat r + r\dd\theta\hat\theta + r\sin\theta\dd\phi\hat\phi$
\item Longitud: $\dd s^2=\dd r^2 + r^2\dd\theta^2 + r^2\sin^2\theta\dd\phi^2 $
\item Superfície $\dd S=r^2\sin\theta\dd\theta\dd\phi$ per a $r=$cte.
\item Volum $\dd V=r^2\sin\theta\dd r\dd\theta\dd\phi$
\end{itemize}

\end{document}