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Diffstat (limited to 'FuVar.tex')
| -rw-r--r-- | FuVar.tex | 115 |
1 files changed, 69 insertions, 46 deletions
@@ -90,8 +90,7 @@ \section{Preface} These are just my personal notes of the \themodule{} course, and definitively -not a rigorously constructed mathematical text. The good looking \LaTeX{} -typesetting may trick you into thinking it is rigorous, but really, it is not. +not a rigorously constructed mathematical text. \section{Derivatives of vector valued scalar functions} @@ -185,11 +184,11 @@ typesetting may trick you into thinking it is rigorous, but really, it is not. \begin{definition}[Directional derivative] A function \(f(\vec{x})\) has a directional derivative in direction - \(\vec{r}\) (with \(|\vec{r}|=1\)) of + \(\vec{v}\) (with \(|\vec{v}|=1\)) of \[ - \frac{\partial f}{\partial\vec{r}} - = \nabla_\vec{r} f = \vec{r} \dotp \grad f - = \sum_{i=1}^m r_i \partial_{x_i} f + \frac{\partial f}{\partial\vec{v}} + = \nabla_\vec{v} f = \vec{v} \dotp \grad f + = \sum_{i=1}^m v_i \partial_{x_i} f \] \end{definition} @@ -274,8 +273,8 @@ typesetting may trick you into thinking it is rigorous, but really, it is not. \begin{remark} The previous method is obtained by studying the second directional derivative - \(\nabla_\vec{r}\nabla_\vec{r} f\) at the stationary point in direction of a - vector \(\vec{r} = \vec{e}_1\cos(\alpha) + \vec{e}_2\sin(\alpha)\). + \(\nabla_\vec{v}\nabla_\vec{v} f\) at the stationary point in direction of a + vector \(\vec{v} = \vec{e}_1\cos(\alpha) + \vec{e}_2\sin(\alpha)\). \end{remark} \begin{method}[Determine the type of stationary point in higher dimensions] @@ -447,51 +446,29 @@ typesetting may trick you into thinking it is rigorous, but really, it is not. \label{thm:transform-coords} Given two ``nice'' functions \(x(u,v)\) and \(y(u,v)\), that means are a bijection from \(S\) to \(S'\) with continuous partial derivatives and - nonzero Jacobian determinant \(|\mx{J}_f| = \partial_u x \partial_v y - + nonzero Jacobian determinant \(|\mx{J}| = \partial_u x \partial_v y - \partial_v x \partial_u y\), which transform the coordinate system. Then \[ - \iint_S f(x,y) \,ds = \iint_{S'} f(x(u,v), y(u,v)) |\mx{J}_f| \,ds . + \iint_S f(x,y) \,ds = \iint_{S'} f(x(u,v), y(u,v)) |\mx{J}| \,ds . \] \end{theorem} \begin{theorem}[Transformation of coordinates] The generalization of theorem \ref{thm:transform-coords} is quite simple. For an \(m\)-integral of a function \(f:\mathbb{R}^m\to\mathbb{R}\) over a - region \(B\), we let \(\vec{x}(\vec{u})\) be ``nice'' functions that + region \(B\), we let \(\vec{g}(\vec{u})\) be ``nice'' functions that transform the coordinate system. Then as before \[ - \int_B f(\vec{x}) \,ds = \int_{B'} f(\vec{x}(\vec{u})) |\mx{J}_f| \,ds . + \int_B f(\vec{r}) \,ds = \int_{B'} f(\vec{g}(\vec{u})) |\mx{J}_\vec{g}| \,ds . \] \end{theorem} -\begin{table} - \centering - \begin{tabular}{l >{\(}l<{\)} >{\(}l<{\)}} - \toprule - & \text{Volume } dv & \text{Surface } d\vec{s}\\ - \midrule - Cartesian & - & dx\,dy \\ - Polar & - & rd\,rd\phi \\ - Curvilinear & - & |\mx{J}_f|\,du\,dv \\ - \midrule - Cartesian & dx\,dy\,dz & \uvec{z}\,dx\,dy \\ - Cylindrical & r\,dr\,d\phi\,dz & \uvec{z}r\,dr\,d\phi \\ - & & \uvec{\phi}\,dr\,dz \\ - & & \uvec{r}r\,d\phi\,dz \\ - Spherical & r^2\sin\theta\, dr\,d\theta\,d\phi & - \uvec{r}r^2\sin\theta\,d\theta\,d\phi \\ - Curvilinear & |\mx{J}_f|\,du\,dv\,dw & - \\ - \bottomrule - \end{tabular} - \caption{Differential elements for integration.} -\end{table} - \begin{application}[Physics] Given the mass \(m\) and density function \(\rho\) of an object, its \emph{center of mass} is calculated with \[ - \vec{x}_c = \frac{1}{m}\int_V \vec{x}\rho(\vec{x}) \,dv - \stackrel{\rho\text{ const.}}{=} \frac{1}{V} \int_V \vec{x}\,dv . + \vec{x}_c = \frac{1}{m}\int_V \rho(\vec{r})\vec{r} \,dv + \stackrel{\rho\text{ const.}}{=} \frac{1}{V} \int_V \vec{r}\,dv . \] The (scalar) \emph{moment of inertia} \(J\) of an object is given by \[ @@ -517,6 +494,14 @@ typesetting may trick you into thinking it is rigorous, but really, it is not. \end{align*} \end{theorem} +% \begin{figure} +% \centering +% \includegraphics{img/multivariable-chain-rule} +% \caption{ +% Multivariable chain rule. +% } +% \end{figure} + \begin{theorem}[Multivariable chain rule] Let \(\vec{x}: \mathbb{R} \to \mathbb{R}^m\) and \(f: \mathbb{R}^m \to \mathbb{R}\), so that \(f\circ\vec{x}: \mathbb{R} \to \mathbb{R}\), then @@ -537,31 +522,39 @@ typesetting may trick you into thinking it is rigorous, but really, it is not. \end{theorem} \begin{definition}[Line integral in a scalar field] - Let \(\mathcal{C}:[a,b]\to\mathbb{R}^n, t \mapsto \vec{x}(t)\) be a - parametric curve. The \emph{line integral} in a field \(f(\vec{x})\) is the + Let \(\mathcal{C}:[a,b]\to\mathbb{R}^n, t \mapsto \vec{r}(t)\) be a + parametric curve. The \emph{line integral} in a field \(f(\vec{r})\) is the integral of the signed area under the curve traced in \(\mathbb{R}^n\), and is computed with \[ - \int_\mathcal{C} f(\vec{x}) \,d\ell - = \int_\mathcal{C} f(\vec{x}) \,|d\vec{x}| - = \int_a^b f(\vec{x}(t)) |\vec{x}'(t)| \, dt . + \int_\mathcal{C} f(\vec{r}) \,d\ell + = \int_\mathcal{C} f(\vec{r}) \,|d\vec{r}| + = \int_a^b f(\vec{r}(t)) |\vec{r}'(t)| \, dt . \] \end{definition} \begin{application}[Length of a parametric curve] - By computing the line integral of the function \(1(\vec{x})\) we get the + By computing the line integral of the function \(1(\vec{r})\) we get the length of the parametric curve \(\mathcal{C}:[a,b]\to\mathbb{R}^n\). \[ \int_\mathcal{C}d\ell - = \int_\mathcal{C} |d\vec{x}| + = \int_\mathcal{C} |d\vec{r}| = \int_a^b \sqrt{\sum_{i=1}^n x'_i(t)^2} \,dt \] The special case with the scalar function \(f(x)\) results in \(\int_a^b\sqrt{1+f'(x)^2}\,dx\). \end{application} +\begin{figure} + \centering + \includegraphics{img/line-integral} + \caption{ + Line integral in a vector field. + } +\end{figure} + \begin{definition}[Line integral in a vector field] - The line integral in a vector field \(\vec{F}(\vec{x})\) is the ``sum'' of + The line integral in a vector field \(\vec{F}(\vec{r})\) is the ``sum'' of the projections of the field's vectors on the tangent of the parametric curve \(\mathcal{C}\). \[ @@ -587,7 +580,7 @@ typesetting may trick you into thinking it is rigorous, but really, it is not. \end{definition} \begin{theorem} - For a twice partially differentiable vector field \(\vec{F}(\vec{x})\) in + For a twice partially differentiable vector field \(\vec{F}\) in \(n\) dimensions without ``holes'', i.e. in which each closed curve can be contracted to a point (simply connected open set), the following statements are equivalent: @@ -627,6 +620,14 @@ typesetting may trick you into thinking it is rigorous, but really, it is not. \mathbb{R}^3\). \end{definition} +\begin{figure} + \centering + \includegraphics{img/surface-integral} + \caption{ + Surface integral. + } +\end{figure} + \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 @@ -648,6 +649,28 @@ typesetting may trick you into thinking it is rigorous, but really, it is not. \] \end{definition} +\begin{table} + \centering + \begin{tabular}{l >{\(}l<{\)} >{\(}l<{\)}} + \toprule + & \text{Volume } dv & \text{Surface } d\vec{s}\\ + \midrule + Cartesian & - & dx\,dy \\ + Polar & - & rd\,rd\phi \\ + Curvilinear & - & |\mx{J}_f|\,du\,dv \\ + \midrule + Cartesian & dx\,dy\,dz & \uvec{z}\,dx\,dy \\ + Cylindrical & r\,dr\,d\phi\,dz & \uvec{z}r\,dr\,d\phi \\ + & & \uvec{\phi}\,dr\,dz \\ + & & \uvec{r}r\,d\phi\,dz \\ + Spherical & r^2\sin\theta\, dr\,d\theta\,d\phi & + \uvec{r}r^2\sin\theta\,d\theta\,d\phi \\ + Curvilinear & |\mx{J}_f|\,du\,dv\,dw & - \\ + \bottomrule + \end{tabular} + \caption{Differential elements for integration.} +\end{table} + \section{Vector analysis} \begin{definition}[Flux] @@ -729,7 +752,7 @@ Notice that the curl is a vector, normal to the enclosed surface \(A\). \end{theorem} \begin{theorem}[Green's theorem] - The special case of Stokes' theorem in 2D is knowns as Green's theorem. + The special case of Stokes' theorem in 2D is known 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} |