$$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Self-defined math definitions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Math symbol commands \newcommand{\intd}{\,{\rm d}} % Symbol 'd' used in integration, such as 'dx' \newcommand{\diff}{{\rm d}} % Symbol 'd' used in differentiation \newcommand{\Diff}{{\rm D}} % Symbol 'D' used in differentiation \newcommand{\pdiff}{\partial} % Partial derivative \newcommand{DD}[2]{\frac{\diff}{\diff #2}\left( #1 \right)} \newcommand{Dd}[2]{\frac{\diff #1}{\diff #2}} \newcommand{PD}[2]{\frac{\pdiff}{\pdiff #2}\left( #1 \right)} \newcommand{Pd}[2]{\frac{\pdiff #1}{\pdiff #2}} \newcommand{\rme}{{\rm e}} % Exponential e \newcommand{\rmi}{{\rm i}} % Imaginary unit i \newcommand{\rmj}{{\rm j}} % Imaginary unit j \newcommand{\vect}[1]{\boldsymbol{#1}} % Vector typeset in bold and italic \newcommand{\phs}[1]{\dot{#1}} % Scalar phasor \newcommand{\phsvect}[1]{\boldsymbol{\dot{#1}}} % Vector phasor \newcommand{\normvect}{\vect{n}} % Normal vector: n \newcommand{\dform}[1]{\overset{\rightharpoonup}{\boldsymbol{#1}}} % Vector for differential form \newcommand{\cochain}[1]{\overset{\rightharpoonup}{#1}} % Vector for cochain \newcommand{\bigabs}[1]{\bigg\lvert#1\bigg\rvert} % Absolute value (single big vertical bar) \newcommand{\Abs}[1]{\big\lvert#1\big\rvert} % Absolute value (single big vertical bar) \newcommand{\abs}[1]{\lvert#1\rvert} % Absolute value (single vertical bar) \newcommand{\bignorm}[1]{\bigg\lVert#1\bigg\rVert} % Norm (double big vertical bar) \newcommand{\Norm}[1]{\big\lVert#1\big\rVert} % Norm (double big vertical bar) \newcommand{\norm}[1]{\lVert#1\rVert} % Norm (double vertical bar) \newcommand{\ouset}[3]{\overset{#3}{\underset{#2}{#1}}} % over and under set % Super/subscript for column index of a matrix, which is used in tensor analysis. \newcommand{\cscript}[1]{\;\; #1} % Star symbol used as prefix in front of a paragraph with no indent \newcommand{\prefstar}{\noindent$\ast$ } % Big vertical line restricting the function. % Example: $u(x)\restrict_{\Omega_0}$ \newcommand{\restrict}{\big\vert} % Math operators which are typeset in Roman font \DeclareMathOperator{\sgn}{sgn} % Sign function \DeclareMathOperator{\erf}{erf} % Error function \DeclareMathOperator{\Bd}{Bd} % Boundary of a set, used in topology \DeclareMathOperator{\Int}{Int} % Interior of a set, used in topology \DeclareMathOperator{\rank}{rank} % Rank of a matrix \DeclareMathOperator{\divergence}{div} % Curl \DeclareMathOperator{\curl}{curl} % Curl \DeclareMathOperator{\grad}{grad} % Gradient \DeclareMathOperator{\tr}{tr} % Trace \DeclareMathOperator{\span}{span} % Span $$

止于至善

As regards numerical analysis and mathematical electromagnetism

James Munkres《拓扑学》笔记前言

许久以前，我读到了侯捷先生于《深入浅出MFC》一书中所写的“勿在浮砂筑高台”这句话，颇受警醒与启发。如今在工科领域已摸索多年，亦逐渐真切而深刻地认识到，若没有坚实、完整、细致的数学理论作为基石，任何技术工作与所谓的“唯象”理论研究也都无异于浮沙筑台，经不起举一反三的推敲与寻根究底的质疑。而这也像是不系安全带的无照司机驾车在高速公路上疾驰，终究逃不掉翻到沟里的命运。拓扑学作为数学理论大厦的一块基石，既起到了稳固上层建筑的作用，又提供了不同于我们通常的眼光和感性直观的视角与思路，值得细心学习和反复体会。

James Munkres《拓扑学》笔记以叙述性语言呈现而非纯粹抽象符号则是基于这样的考虑：无论多么复杂的概念或定理，都一定能够用明晰和形象的语言予以透彻阐释并深入浅出。如果做不到这一点，便说明对它们的理解和把握还不到位。正所谓，思想需诉诸语言表达，而语言的形式与结构亦反作用于思想。同时，学习者若能够以这种方式进行总结和训练，就如同在虚拟的讲台上面向虚拟的听众讲授，从而将被动的输入型学习转化为积极主动的输出型学习，以达到最佳的学习效果。

笔记放于如下豆瓣相册中，后续逐步更新。

posted @ 2019-04-03 07:12  皮波迪博士  阅读(1413)  评论(0)  编辑  收藏  举报