summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/doc/latex/rbt-mathnotes/examples/cheat-sheet.tex
blob: 11ff82a5a908e028a1f5e61b906fa24aa353a1bd (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
%% cheat-sheet.tex
%% Copyright 2021 Rebecca B. Turner.
%
% This work may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.3
% of this license or (at your option) any later version.
% The latest version of this license is in
%   http://www.latex-project.org/lppl.txt
% and version 1.3 or later is part of all distributions of LaTeX
% version 2005/12/01 or later.
%
% This work has the LPPL maintenance status `maintained'.
% 
% The Current Maintainer of this work is Rebecca B. Turner.
%
% This work consists of the files:
%     README.md
%     rbt-mathnotes.tex
%     rbt-mathnotes.sty
%     rbt-mathnotes.cls
%     rbt-mathnotes-util.sty
%     rbt-mathnotes-messages.sty
%     rbt-mathnotes-hw.cls
%     rbt-mathnotes-formula-sheet.cls
%     examples/cheat-sheet.tex
%     examples/multivar.tex
%     examples/topology-hw-1.tex
% and the derived files:
%     rbt-mathnotes.pdf
%     examples/cheat-sheet.pdf
%     examples/multivar.pdf
%     examples/topology-hw-1.pdf

\documentclass{rbt-mathnotes-formula-sheet}
\usepackage{nicefrac}
\ExplSyntaxOn
\NewDocumentCommand \normalized { m }
  { \frac { #1 } { \| #1 \| } }
\let \gr \grad
\def \ddx { \frac{d}{dx} }
% VL = vector literal
\NewDocumentCommand \vl { m } { \left\langle #1 \right\rangle }
\ExplSyntaxOff

\title{Formula Sheet}
\author{Rebecca Turner}
\date{2019-11-12}

% "The most common size for index cards in North America and UK is 3 by 5
% inches (76.2 by 127.0 mm), hence the common name 3-by-5 card. Other sizes
% widely available include 4 by 6 inches (101.6 by 152.4 mm), 5 by 8 inches
% (127.0 by 203.2 mm) and ISO-size A7 (74 by 105 mm or 2.9 by 4.1 in)."
\mathnotes{
  height = 4in ,
  width = 6in ,
}
\begin{document}
\maketitle
\begin{gather*}
% 12.2: Vectors
% 12.3: Dot product
  \textstyle\vec a \cdot \vec b = \sum_i a_i b_i = |\vec a| |\vec b| \cos \theta. \\
% 12.4: Cross product
  \vec a \times \vec b
  % = \left| \begin{array}{rrr}
    % \hat{i} & \hat{j} & \hat{k} \\
    % a_1 & a_2 & a_3 \\
    % b_1 & b_2 & b_3 \\
  % \end{array} \right| \\
  = \langle a_2 b_3 - a_3 b_2,
    \quad a_3b_1 - a_1b_3, \\
    a_1b_2 - a_2b_1 \rangle.\quad
  |\vec a \times \vec b| = |\vec a| |\vec b| \sin \theta.
% 12.5: Equations of lines and planes.
\shortintertext{Param.\ eqns.\ of line through $\langle x_0,y_0,z_0 \rangle$
par.\ to $\langle a,b,c \rangle$:}
  x = x_0 + at,
  \quad y = y_0 + bt,
  \quad z = z_0 + ct. \\
\text{Symm.\ eqns.: }
  \frac{x-x_0}{a}
  = \frac{y-y_0}{b}
  = \frac{z-z_0}{c}. \\
\shortintertext{Vec.\ eqn.\ of plane through $\vec r$ with $\vec n$ normal:}
  \vec n \cdot (\vec r - \vec r_0) = 0,
  \quad \vec n \cdot \vec r = \vec n \cdot \vec r_0. \\
% 13.1: Vector functions
% 13.2: Derivatives/integrals of vector functions
% 13.3: Arc length and curvature
\shortintertext{Length along a vec.\ fn.\ $\vec r(t)$:}
  \textstyle\int_a^b \left|\vec r'(t)\right|\,dt = \int_a^b \sqrt{\sum_i
  r_i'(t)^2}\,dt, \\
\shortintertext{Unit tang.\ $\vec T(t) = \vec r'(t)/\left|\vec
r'(t)\right|$, so curvature of $\vec r(t)$ w/r/t the arc len.\ fn. $s$:}
  \kappa = \left|\frac{d\vec T}{ds}\right|
  = \frac{\left| \vec T'(t) \right|}{\left| \vec r'(t) \right|}
  = \frac{\left| \vec r'(t) \times \vec r''(t) \right|}{\left| \vec r'(t)
  \right|^3}. \\
\text{Unit normal:}\quad
  \vec N(t) = \vec T'(t)\,/\,\left| \vec T'(t) \right| \\
% 14.1: Functions of several variables
% 14.2: Limits and continuity
% 14.3: Partial derivatives
\text{Clairaut's thm.:}\quad
  f_{xy}(a,b) = f_{yx}(a,b) \\
% 14.4: Tangent planes & linear approximations
\shortintertext{Tan.\ plane to $z = f(x,y)$ at $\langle x_0, y_0,
z_0\rangle$:}
  z - z_0 = f_x(x_0, y_0) (x-x_0) \\
    + f_y(x_0, y_0) (y-y_0). \\
% Partial derivatives of f for each variable exist near a point and are
% continuous => f is differentiable at the po\int.
% 14.6: Directional derivatives and the gradient vector
\text{Grad.:}\quad
  \grad f(x,y) = \pd[f]x \hat{i} + \pd[f]y \hat{j}. \\
\shortintertext{Dir.\ deriv.\ towards $\vec u$ at $\langle x_0, y_0 \rangle$:}
  D_{\langle a,b\rangle} f(x_0, y_0) = f_x(x,y) a + f_y(x,y) b \\
  = \grad f(x,y) \cdot \vec u. \\
\shortintertext{Max of $D_{\vec u} f(\vec x) = \left|\grad f(\vec
x)\right|$. Tan.\ plane of $f$ at $\vec p$:}
  0 =
  f_x(\vec p)(x-\vec p_x)
  + f_y(\vec p)(y-\vec p_y) \\
  + f_z(\vec p)(z-\vec p_z).
% 14.7: Maximum and minimum values
\shortintertext{If $f$ has loc.\ extrem.\ at $\vec p$, then $f_x(\vec p) =
0$ (\& $f_y$, etc). If so, let}
  D = \left| \begin{array}{ll}
    f_{xx} & f_{xy} \\
    f_{yx} & f_{yy}
  \end{array}\right|
  = f_{xx} f_{yy} - (f_{xy})^2.
\shortintertext{%
  $D = 0$: no information.
  $D < 0$: saddle pt.
  $D > 0$: $f_{xx}(\vec p) > 0 \implies$ loc.\ min;
  $f_{xx}(\vec p) < 0 \implies$ loc.\ max.
  ($D$ is the \textbf{Hessian mat.})
\endgraf
  Set of possible abs. min and max vals of $f$ in reg.\ $D$: $f$ at critical
  pts.\ and extreme vals.\ on the boundary of $D$.
% 14.8: Lagrange multipliers
\endgraf
  Lagrange mults.: extreme vals of $f(\vec p)$ when $g(\vec p) = k$.
  Find all $\vec x, \lambda$ s.t.
}
  \grad f(\vec x) = \lambda \grad g(\vec x),\quad g(\vec x) = k.
\shortintertext{i.e.\ $f_x = \lambda g_x$, etc.}
% 15.1: Double integrals over rectangles
% 15.2: Iterated integrals
% 15.3: Double integrals over general regions
  \iint f(r\cos\theta, r\sin\theta)r\,dr\,d\theta. \\
  A = \iint_D \left(\sqrt{f_x(x,y)^2 + f_y(x,y)^2 + 1}\right) \,dA. \\
\shortintertext{Line int.s}
  \int_C f(x,y)\,ds = \\
    \int_a^b f(x(t), y(t))\sqrt{\left(\pd[x]t\right)^2 + \left(\pd[y]t\right)^2}\,dt \\
\shortintertext{If $C$ is a smooth curve given by $\vec r(t)$ from $a \le t
\le b$,}
  \int_C \grad f \cdot d\vec r = f(\vec r(b)) - f(\vec r(a)) \\
\text{Spherical coords:}\quad
  x = \rho \sin \phi \cos \theta \\
  y = \rho \sin \phi \sin \theta, z = \rho \cos \phi \\
  \curl \vec F = \\ \left< \pd[R]y - \pd[Q]z, \pd[P]z - \pd[R]x, \pd[Q]x -
  \pd[P]y\right>. \\
  \vec F = \langle P,Q,R \rangle,\quad
  \curl \vec F = \grad \times \vec F \\
  \vec F \text{ ``conservative''} \implies \exists f, \vec F = \grad f. \\
  \dive \vec F = \grad \cdot \vec F = \pd[P]x + \pd[Q]y + \pd[R]z. \\
  \curl(\grad f) = \vec 0,\quad \dive \curl \vec F = 0 \\
\shortintertext{If $C$ is a positively-oriented (ccw) closed curve, $D$
is bounded by $C$, and $\vec n$ represents the normal,}
  % \int_C P\,dx + Q\,dy = \iint_D\left( \pd[Q]{x} - \pd[P]{y} \right). \\
  \oint_C \vec F \cdot \vec n\,ds = \iint_D \dive \vec F(x,y)\,dA.
\end{gather*}

\pagebreak
\raggedright Common derivs:
$f(g(x)) \to g'(x) f'(g(x))$,
$b^x \to b^x \ln b$,
$f^{-1}(x) \to 1/f'(f^{-1}(x))$,
$\ln x \to 1/x$,
$\sin x \to \cos x$, $\cos x \to -\sin x$,
$\tan x \to \sec^2 x$,
$\sin^{-1} x \to 1/\sqrt{1-x^2}$,
$\cos^{-1} x \to -(\sin^{-1}x)'$ (etc.),
$\tan^{-1} x \to 1/(1+x^2)$,
$\sec^{-1} x \to 1/(|x|\sqrt{x^2-1})$.

Common ints (don't forget $+C$):
\begin{gather*}
  x^n \to \frac{x^{n + 1}}{n + 1} + C \quad \text{when } n \ne -1 \\
  1/x \to \ln |x| \\
  \tan x \to -\ln(\cos x) \\
  \int uv'\,dx = uv - \int u'v\,dx \quad\text{(Int.\ by parts)} \\
  \int u\,dv = uv-\int v\,du \\
  \int_{g(a)}^{g(b)} f(u)\,du = \int_a^b f(g(x))g'(x)\,dx
  \quad\text{$u$-substitution.}
\intertext{E.x.\ in $\int 2x \cos x^2\,dx$, let $u=x^2$, find $du/dx=2x
\implies du = 2x\,dx$, subs.\ $\int \cos u\,du = \sin u + C = \sin x^2 +
C$.}
  \iint_R f(x,y)\,dA = \int_\alpha^\beta \int_a^b f(r\cos\theta,
  r\sin\theta)r\,dr\,d\theta
\end{gather*}
\begin{itemize}
  \item Integrand contains $a^2-x^2$, let $x = a\sin\theta$ and use $1 -
  \sin^2 \theta = \cos^2 \theta$.
  \item $a^2 + x^2$, let $x = a\tan\theta$, use $1 + \tan^2 \theta = \sec^2
  \theta$.
  \item $x^2 - a^2$, let $x = a\sec\theta$, use $\sec^2\theta - 1 = \tan^2
  \theta$.
\end{itemize}

\begin{gather*}
  \lim_{x \to 0} \sin x/x = 1 \\
  \lim_{x \to 0} (1-\cos x)/x = 0 \\
  \lim_{x \to \infty} x \sin(1/x) = 1 \\
  \lim_{x \to 0} (1+x)^{1/x} = e \\
  \lim_{x \to 0} (e^{ax}-1)/(bx) = a/b \\
  \lim_{x \to 0^+} x^x = 1 \\
  \lim_{x \to 0^+} x^{-n} = \infty \\
  \text{For $0/0$ or $\pm\infty/\infty$,}\quad
  \lim_{x \to c} f(x)/g(x) = \lim_{x \to c} f'(x)/g'(x) \\
  \text{For $g(x)$ cont.\ at $L$,}
  \lim_{x \to c} f(x) = L \implies \lim_{x \to c} g(L)
\end{gather*}

\end{document}