<1-1> (1.2.1)
[\mbox {v}]=\mbox{L}\mbox{T}^{-1}
<1-2> (1.3.1)
y=ax+b
<1-3> (1.3.2)
y=b\sin(ax)+c
<1-4> (1.3.3)
y=f(x)
<1-5> (1.3.4)
y=f(x_1,x_2,x_3,\cdots)
<1-6> (1.3.5)
y=x^2
<1-7>
4=x^2
<1-8> (1.3.6)
y^2=x
<1-9> (1.3.7)
x=f^{-1}(y)
<1-10> (1.3.8)
y=f(x)=ax+b
<1-11> (1.3.9)
x=f^{-1}(y)=\frac{y-b}{a}
<1-12> (1.3.10)
\begin{array}{l}
y=\left\{
\begin{array}{l}
e^x \\ \\ \sin x \\ \\ \sin(x^3) \\ \\
(\sin x)^{1/3}
\end{array} \right.
<1-13> (1.3.11)
\begin{array}{l}
y=\left\{
\begin{array}{l}
\ln y \\ \\ \sin^{-1}y \\ \\
\sqrt[3]{\sin^{-1}y} \\ \\
\sin^{-1}(y^3)
\end{array}\right.
<1-13> (1.3.11)
\begin{array}{rl}
x&=f^{-1}(y) \\ \\
&=\left\{
\begin{array}{l}
\ln y \\ \\
\sin^{-1}y \\ \\
\sqrt[3]{\sin^{-1}y} \\ \\
\sin^{-1}(y^3)
\end{array} \right. \\ \\
\end{array}
\end{array}
<2-1> (2.1.1)
\lim_{\Delta\rightarrow0}
\left[\frac{\Delta f(x)}{\Delta x}\right]
=\frac{df(x)}{dx}\equiv f^\prime(x)
<2-2> (2.1.2)
\begin{array}{l}
\displaystyle{\Delta f(x)=\frac{df(x)}{dx}\Delta x} \\ \\
\displaystyle{\mbox{or}\quad df(x)
=\frac{df(x)}{dx}dx}
\end{array}
<2-3>
(f\pm g)^\prime=f^\prime\pm g^\prime
<2-4>
(kf)^\prime=kf^\prime
<2-5>
(fg)^\prime=f^\prime g+fg^\prime
<2-6>
\frac{dy}{dx}=\frac{dy}{dz}\frac{dz}{dx}=
\frac{df(z)}{dz}\frac{dg(x)}{dx}
<2-7> (2.1.3)
\frac{d^nf(x)}{dx^n}\equiv f^{(n)}(x),
\quad(n=0,1,2,\cdots)
<2-8> (2.1.4)
\frac{df(x)}{dx}=\frac{d}{dx}f(x)
<2-9> (2.1.5)
\begin{array}{rl}
\displaystyle{\frac{df(x)}{dx}}
&=\displaystyle{\lim_{\Delta x\rightarrow0}
\left[\frac{f(x+\Delta x)-f(x)}{\Delta x}\right]}}
\\ \\
&=\displaystyle{\lim_{\Delta x\rightarrow0}
\displaystyle{
\left[\frac{\Delta f(x)}{\Delta x}\right]
}
\end{array}
<2-10> (2.1.6)
\frac{df(x)}{dx}\quad\Rightarrow\quad
\frac{d}{dx}f(x)\quad\mbox{or}\quad Df(x)
<2-11> (2.2.1)
\begin{array}{rl}
f(x)=
&f(a) \\ \\
&+\displaystyle{\frac{f^{(1)}(a)}{1!}(x-a)} \\ \\
&+\displaystyle{\frac{f^{(2)}(a)}{2!}(x-a)^2} \\ \\
&+\cdots \\ \\
&+\displaystyle{\frac{f^{(n)}(a)}{n!}(x-a)^n} \\ \\
&+\cdots
\end{array}
<2-12>
n!=1\times2\times3\cdots\times(n-1)\times n
<2-13> (2.2.2)
\begin{array}{rl}
f(x)=&f(0) \\ \\
&+\displaystyle{\frac{f^{(1)}(0)}{1!}x} \\ \\
&+\displaystyle{\frac{f^{(2)}(0)}{2!}x^2}} \\ \\
&+\cdots \\ \\
&+\displaystyle{\frac{f^{(n)}(0)}{n!}x^n} \\ \\
&+\cdots
\end{array}
<2-14> (2.2.3)
f(x)=f(0)+ f^{(1)}(0)x
<2-15> (2.2.4)
(1+x)^n=1+ nx
<2-16>
e^x
<2-17>
1+x+\frac{x^2}{2}
<2-18>
\sin x
<2-19>
x-\frac{x^3}{6}+\frac{x^5}{120}
<2-20>
\cos x
<2-21>
1-\frac{x^2}{2}+\frac{x^4}{24}
<2-22>
\ln(1+x)
<2-23>
x-\frac{x^2}{2}+\frac{x^3}{3}
<2-24>
(1+x)^\alpha
<2-25>
1+\alpha x+\frac{\alpha(\alpha-1)}{2}x^2
<2-26> (2.2.5)
\frac{d^2\theta}{dt^2}=-\frac{g}{\ell}\sin\theta
<2-27> (2.2.6)
\frac{d^2\theta}{dt^2}=-\frac{g}{\ell}\theta
<2-28> (2.2.7)
\theta=A\sin(\omega t),\quad
\left(\omega=\sqrt{g/\ell}\right)
<2-29> (2.3.1)
\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=
\frac{f^{\prime\prime}(a)}{ g^{\prime\prime}(a)}
<2-30>
\left\{\begin{array}{rl}
f(x)
&=\displaystyle{
\left[\frac{df(x)}{dx}\right]_{x=a}(x-a)} \\ \\
&+\displaystyle{\frac{1}{2}
\left[\frac{d^2f(x)}{dx^2}\right]_{x=a}(x-a)^2}\\ \\
g(x)
&=\displaystyle{
\left[\frac{dg(x)}{dx}\right]_{x=a}(x-a)} \\ \\
&+\displaystyle{\frac{1}{2}
\left[\frac{d^2g(x)}{dx^2}\right]_{x=a}(x-a)^2}
\end{array}\right.
<2-31>
\left\{\begin{array}{rl}
f(x)
&=\displaystyle{
\left[\frac{df(x)}{dx}\right]_{x=a}(x-a)} \\ \\
&=\displaystyle{f^\prime(a)(x-a)}}\\ \\
g(x)
&=\displaystyle{\left[
\frac{dg(x)}{dx}\right]_{x=a}(x-a)} \\ \\
&=\displaystyle{g^\prime(a)(x-a)}}
\end{array}\right.
<2-32>
\lim_{x\rightarrow a}\frac{f(x)}{g(x)}
=\lim_{x\rightarrow }
\frac{f^\prime(a)(x-a)}{g^\prime(a)(x-a)}
=\frac{f^\prime(a) }{g^\prime(a)}
<2-33>
A=\lim_{x\rightarrow 0}\frac{\sin(x)}{x}
<2-34>
A=\lim_{x\rightarrow 0}\frac{\cos(x)}{1}=1
<2-35>
f(x)=\frac{x}{e^{ax}-1}
<2-36>
\left\{\begin{array}{l}
\displaystyle{\frac{dx}{dt}=1}\\ \\
\displaystyle{\frac{
d(e^{ax}-1)}{dx}=\frac{de^{ax}}{dx}=ae^{ax}
\end{array}\right.
<2-37>
\lim_{x\rightarrow 0}\frac{x}{e^{ax}-1}=
\lim_{x\rightarrow 0}\frac{1}{ae^{ax}}=\frac{1}{a}
<2-38> (2.4.1)
x(t)=x_0+\int_{t_0}^{t}v(t)dt
<2-39> (2.4.2)
v(t)=v_0+\int_{t_0}^{t}a(t)dt
<3-1> (3.1.1)
z=f(x,y)
<3-2> (3.1.2)
P=\frac{nRT}{V}
<3-3> (3.1.3)
\lim_{h\rightarrow0}\frac{f(x+h,y)-f(x,y)}{h}
\equiv\frac{\partial f}{\partial x}
<3-4> (3.1.4)
\lim_{h\rightarrow0}\frac{f(x,y+h)-f(x,y)}{h}
\equiv\frac{\partial f}{\partial y}
<3-5> (3.1.5)
\frac{\partial}{\partial x}
\left(\frac{\partial f(x,y)}{\partial x}\right)
=\frac{\partial^2f(x,y)}{\partial x^2}=f_{xx}(x,y)
<3-6> (3.1.6)
\frac{\partial}{\partial y}
\left(\frac{\partial f(x,y)}{\partial x}\right)
=\frac{\partial^2f(x,y)}{ \partial y\partial x}
=f_{xy}(x,y)
<3-7> (3.1.7)
f(x,y) =x^2y+xy^2+y^3
<3-8> (3.1.8)
\begin{array}{l}
\left\{\begin{array}{l}
f_x(x,y)=2xy+y^2 \\ \\
f_y(x,y)=x^2+2xy+3y^2
\end{array}\right. \\ \\
\left\{\begin{array}{l}
f_{xx}(x,y)=2y \\ \\
f_{xy}(x,y)=2x+2y \\ \\
f_{yx}(x,y)=2x+2y \\ \\
f_{yy}(x,y)=2x+6y
\end{array}\right.
\end{array}
<3-9> (3.1.9)
\Delta f(x,y) =f(x+\Delta x, y+\Delta y)-f(x,y)
<3-10>
\Delta f(x,y)=
\left\{\frac{f(x+\Delta x, y+\Delta y)
-f(x, y+\Delta y)}
{\Delta x}\right\}\Delta x
+\left\{\frac{f(x, y+\Delta y)
-f(x, y)}{\Delta y}\right\}\Delta y
<3-11>
\begin{array}{rl}
\Delta f(x,y)
&\rightarrow
\displaystyle{\left\{\frac{f(x+\Delta x, y)-
f(x, y)}{\Delta x}\right\}\Delta x}
+\displaystyle{\left\{\frac{f(x, y+\Delta y)-
f(x, y)}{\Delta y}\right\}\Delta y} \\ \\
&\rightarrow
\displaystyle{
\frac{\partial f(x, y)}{\partial x}\Delta x+
\frac{\partial f(x, y)}{\partial y}\Delta y} \\ \\
&=f_x(x, y)\Delta x+ f_y(x, y)\Delta y
\end{array}
<3-12> (3.1.10)
df(x,y)= \frac{\partial f(x, y)}{\partial x}dx+
\frac{\partial f(x, y)}{\partial y}dy
<3-13> (3.1.11)
\begin{array}{l}
\displaystyle{\frac{\partial P(x, y)}{\partial y}=
\frac{\partial Q(x, y)}{\partial x} \\ \\
\quad\mbox{namely}\quad P_y(x, y)=Q_x(x, y)
\end{array}
<3-14> (3.1.12)
\left\{\begin{array}{l}
\displaystyle{
\frac{\partial f(x, y)}{\partial x}=P(x,y)} \\ \\
\displaystyle{
\frac{\partial f(x, y)}{\partial y}=Q(x,y)}
\end{array}\right.
<3-15> (3.1.13)
df(x, y)=P(x,y)dx+Q(x,y)dy
<3-16>
\frac{\partial P}{\partial y}
=\frac{\partial\left(\displaystyle{
\frac{\partial f}{\partial x}}\right)}{\partial y}=
\frac{\partial^2f}{\partial y\partial x}=
\frac{\partial\left(\displaystyle{
\frac{\partial f}{\partial y}}\right)}{\partial x}=
\frac{\partial Q}{\partial x}
<3-17>
\mbox{(a)}\quad(3x^2+2xy-2y^2)dx+(x^2-4xy)dy
<3-18>
\left\{\begin{array}{l}
\displaystyle{
\frac{\partial P}{\partial y}=2x-4y} \\ \\
\displaystyle{\frac{
\partial Q}{\partial x}=2x-4y}
\end{array}\right.,\quad\mbox{therefore}\quad
\frac{\partial P}{\partial y}
=\frac{\partial Q}{\partial x}
<3-19>
df(x, y)=P(x,y)dx+Q(x,y)dy
<3-20>
\begin{array}{l}
\displaystyle{\frac{\partial f}{\partial x}}=
P(x,y)=3x^2+2xy-2y^2} \\ \\
\displaystyle{\frac{\partial f}{\partial y}}=
Q(x,y)=x^2-4xy}
\end{array}
<3-21>
f(x, y)=x^3+x^2y-2xy^2+C
<3-22> (3.1.14)
\left\{\begin{array}{rl}
d\left[f(x)g(x)\right]&=g(x)df(x)+f(x)dg(x) \\ \\
&=\displaystyle{
\left(g\frac{df}{dx}+f\frac{dg}{dx}\right)dx
\end{array}
<3-23>
\lim_{\Delta x\rightarrow0}\left[
\frac{z(x+\Delta x)-z(x)}{\Delta x}\right]
=\frac{dz}{dx}
<3-24>
\lim_{\Delta t\rightarrow0}\left[
\frac{x(t+\Delta t)-x(t)}{\Delta t}\right]
=\frac{dx}{dt}
<3-25> (3.1.15)
\frac{dz}{dt}=\frac{dz}{dx}\frac{dx}{dt}
<3-26> (3.1.16)
dz=\frac{df(x)}{dx}dx=\frac{df}{dx}\frac{dg}{dt}dt
<3-27>
dz=\left(\frac{\partial z}{\partial x}\frac{dx}{dt}
+\frac{\partial z}{\partial y}\frac{dy}{dt}\right)dt
<3-28> (3.1.17)
\frac{dz}{dt}=\frac{\partial z}{\partial x}
\frac{dx}{dt}+\frac{\partial z}{\partial y
\frac{dy}{dt}
<3-29> df(x)=\frac{df(x)}{dx}dx
<3-30> dg(x,y)=\frac{\partial g(x,y)}{\partial x}dx
+\frac{\partial g(x,y)}{\partial y}dy
<3-31>
dz=\left(\frac{\partial z}{\partial x}
\frac{\partial x}{\partial r}
+\frac{\partial z}{\partial y}
\frac{\partial y}{\partial r}\right)dr
+\left(\frac{\partial z}{\partial x}
\frac{\partial x}{\partial s}
+\frac{\partial z}{\partial y}
\frac{\partial y}{\partial s}\right)ds
<3-32> (3.1.18)
\left\{\begin{array}{l}
\displaystyle{\frac{\partial z}{\partial r}
=\frac{\partial z}{\partial x}
\frac{\partial x}{\partial r}
+\frac{\partial z}{\partial y}
\frac{\partial y}{\partial r}} \\ \\
\displaystyle{\frac{\partial z}{\partial s}
=\frac{\partial z}{\partial x}
\frac{\partial x}{\partial s}
+\frac{\partial z}{\partial y}
\frac{\partial y}{\partial s}}
\end{array}\right.
<3-33>
\left\{\begin{array}{l}
x+ct\equiv p(x,t) \\ \\
x-ct\equiv q(x,t)
\end{array}\right.
<3-34> (3.1.19)
u(x,t)=f(p)+g(q)
<3-35>
\left\{\begin{array}{l}
p(x,t)\equiv x+ct \\ \\
q(x,t)\equiv x-ct
\end{array}\right.
<3-36>
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial p}{\partial x}}=1,
& \quad\displaystyle{\frac{\partial p}{\partial t}}=c
\\ \\
\displaystyle{\frac{\partial q}{\partial x}}=1,
& \quad\displaystyle{\frac{\partial q}{\partial t}}=-c
\end{array}\right.
<3-37>
\left\{\begin{array}{l}
\displaystyle{\frac{\partial u}{\partial x}
=\frac{df}{dp}\frac{dp}{dx}+
\frac{dg}{dq}\frac{\partial q}{\partial x}
=\frac{df}{dp}+\frac{dg}{dq}} \\ \\
\displaystyle{\frac{\partial u}{\partial t}
=\frac{df}{dp}\frac{dp}{dt}+
\frac{dg}{dq}\frac{\partial q}{\partial t}
=c\left(\frac{df}{dp}-\frac{dg}{dq}\right)}
\end{array}\right.
<3-38>
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial^2u}{\partial x^2}}
&=\displaystyle{\frac{\partial}{\partial x}
\left(\frac{df}{dp}+\frac{dg}{dq}\right)} \\ \\
&=\displaystyle{
\left[\frac{d}{dp}\left(\frac{df}{dp}\right)\right]
\frac{\partial p}{\partial x}+\left[\frac{d}{dq}
\left(\frac{dg}{dq}\right)\right]
\frac{\partial q}{\partial x} \\ \\
&=\displaystyle{\frac{d^2f}{dp^2}
+\frac{d^2g}{dq^2}} \\ \\
\displaystyle{\frac{\partial^2u}{\partial t^2}}
&=\displaystyle{\frac{\partial}{\partial t}
\left[c\left(\frac{df}{dp}
-\frac{dg}{dq}\right)\right]} \\ \\
&=\displaystyle{c\left(\left[\frac{d}{dp}
\left(\frac{df}{dp}\right)\right]
\frac{\partial p}{\partial t}+
\left[\frac{d}{dq}\left(\frac{dg}{dq}\right)\right]
\frac{\partial q}{\partial t} \right)\\ \\
&=c^2\displaystyle{\left(\frac{d^2f}{dp^2}
+\frac{d^2g}{dq^2}\right)}
\end{array}\right.
<3-39>
\frac{\partial^2u}{\partial t^2}-
c^2\frac{\partial^2u}{\partial x^2}=0
<3-40>
F(x+dx,y+dy)=F(x,y)
+\frac{\partial F(x,y)}{\partial x}dx
+\frac{\partial F(x,y)}{\partial y}dy
<3-41> (3.1.19)
\begin{array}{rl}
dF&\equiv F(x+dx,y+dy)-F(x,y) \\ \\
&=Adx+Bdy
\end{array}
<3-42> (3.1.20)
\left\{\begin{array}{l}
\displaystyle{
\frac{\partial F(x,y)}{\partial x}=A} \\ \\
\displaystyle{\frac{
\partial F(x,y)}{\partial y}=B}
\end{array}\right.
<3-43> (3.1.21)
G=F(x,y)-xA
<3-44> (3.1.22)
\left\{\begin{array}{l}
G\rightarrow G+dG \\ \\
F\rightarrow F+dF \\ \\
x\rightarrow x+dx \\ \\
A\rightarrow A+dA
\end{array}\right.
<3-45>
\begin{array}{rl}
G+dG&=F+dF-(x+dx)(A+dA) \\ \\
&=F+dF-(xA+xdA+Adx+dxdA)
\end{array}\right.
<3-46>
\begin{array}{rl}
G+dG&=(F-xA)+(dF-xdA-Adx)-dxdA \\ \\
&=(F-xA)+(Ddy-xdA)-dxdA
\end{array}\right
<3-47> (3.1.23)
dG=Bdy-xdA
.
<3-48>
\mbox{(i)}\quad f(x,v)=\frac{m}{2}v^2-V(x)
<3-49>
\mbox{(ii)}\quad\frac{d}{dt}
\left(\frac{\partial f}{\partial v}
\right)-\frac{\partial f}{\partial x}=0
<3-50>
\mbox{(iii)}\quad\frac{d}{dt}(mv)=-\frac{dV}{dx}
<3-51>
\mbox{(iv)}\quad
\frac{\partial f(x,v)}{\partial v}=p
<3-52>
\mbox{(v)}\quad g=vp-f(x,v)
<3-53>
d(vp)=vdp+pdv
<3-54>
\begin{array}{rl}
df&=\displaystyle{\frac{\partial f}{\partial v}dv+
\frac{\partial f}{\partial x}dx} \\ \\
&=\displaystyle{pdv-\frac{dV}{dx}dx}
\end{array}
<3-55>
\begin{array}{rl}
dg&=d(vp)-df \\ \\
&=vdp+pdv-\displaystyle{\left(pdv
-\frac{dV}{dx}dx\right) \\ \\
&=vdp+\displaystyle{\frac{dV}{dx}dx}
\end{array}
<3-56>
dg=\frac{\partial g}{\partial x}dx
+\frac{\partial g}{\partial p}dp
<3-57>
\mbox{(vi)}\qqad\left\{\begin{array}{l}
\displaystyle{\frac{\partial g}{\partial x}
=\frac{dV}{dx}} \\ \\
\displaystyle{\frac{\partial g}{\partial p}
=v=\frac{p}{m}}
\end{array}\right.
<3-58>
\mbox{(vii)}\quad\begin{array}{rl}
g&=\displaystyle{\frac{p}{m}p-\left[\frac{m}{2}
\left(\frac{p}{m}\right)^2-V(x)\right] \\ \\
&=\displaystyle{\frac{p^2}{2m}+V(x)
\end{array}
<3-59>
\frac{dg}{dt}=\frac{\partial g}{\partial x}
\frac{dx}{dt}
+\frac{\partial g}{\partial p}\frac{dp}{dt}
<3-60>
\mbox{(ix)}\qquad\begin{array}{rl}
\displaystyle{\frac{dg}{dt}}
&=\displaystyle{\frac{\partial g}{\partial x}
\frac{\partial g}{\partial p}
+\frac{\partial g}{\partial p}
\left(-\frac{\partial g}{\partial x}\right)} \\ \\
&=0
\end{array}
<3-61>
dU=\left(\frac{\partial U}{\partial S}\right)_VdS
+\left(\frac{\partial U}{\partial V}\right)_SdV
<3-62>
\left\{\begin{array}{l}
\displaystyle{T=
\left(\frac{\partial U}{\partial S}\right)_V} \\ \\
\displaystyle{
p=-\left(\frac{\partial U}{\partial V}\right)_S
\end{array}\right.
<3-63>
\mbox{(a)}\quad dU=TdS-pdV
<3-64>
\mbox{(b)}\quad\left\{\begin{array}{ll}
U &
(\mbox{Internal Energy with the variables}
(S,V))\\ \\
F=U-TS &
(\mbox{Helmholtz Free Energy with the variables}
(T,V))\\ \\
H=U+pV &
(\mbox{Enthalpy with the variables}
(S,p))\\ \\
G=H-TS &
(\mbox{Gibbs Free Energy with the variables}
(T,p))
\end{array}\right.
<3-65>
\mbox{(c)}\quad
dF=\frac{\partial F}{\partial T}dT
+\frac{\partial F}{\partial V}dV
<3-66>
\begin{array}{rl}
dF&=dU-d(TS) \\ \\
&=TdS-pdV-(TdS+SdT) \\ \\
&=-SdT-pdV
\end{array}
<3-67>
\mbox{(d)}\quad\left\{\begin{array}{l}
\displaystyle{S=
-\left(\frac{\partial F}{\partial T}\right)_V} \\ \\
\displaystyle{
p=-\left(\frac{\partial F}{\partial V}\right)_T}
\end{array}\right.
<3-68>
\begin{array}{rl}
dH&=dU+d(pV) \\ \\
&=TdS-pdV+(pdV+Vdp) \\ \\
&=TdS+Vdp
\end{array}
<3-69>
\mbox{(e)}\quad\left\{\begin{array}{l}
\displaystyle{T=
\left(\frac{\partial H}{\partial S}\right)_p}\\ \\
\displaystyle{V=
\left(\frac{\partial H}{\partial p}\right)_S
\end{array}\right.
<3-70>
\begin{array}{rl}
dG&=dH-d(TS) \\ \\
&=TdS+Vdp-(SdT+TdS) \\ \\
&=-SdT+Vdp
\end{array}
<3-71>
\mbox{(f)}\quad\left\{\begin{array}{l}
\displaystyle{
S=-\left(\frac{\partial G}{\partial T}\right)_p}
\\ \\
\displaystyle{
V=\left(\frac{\partial G}{\partial p}\right)_T
\end{array}\right.
<3-72> (3.2.1)
1\times1=1
<3-73> (3.2.2)
i\times i=-1
<3-74> (3.2.3)
1\times a+i\times b=a+ib\equiv z
<3-75> (3.2.4)
z=x+iy
<3-76> (3.2.5)
\left\{\begin{array}{l}
x=r\cos\theta \\ \\
y=r\sin\theta
\end{array}\right.
<3-77>
\begin{array}{rl}
z&=r(\cos\theta +i\sin\theta) \\ \\
&=\displaystyle{\left[\left(1-\frac{\theta^2}{2!}
+\frac{\theta^4}{4!}-\cdots\right)
+i\left(\theta-\frac{\theta^3}{3!}
+ \frac{\theta^5}{5!}-\cdots\right)\right]
\end{array}
<3-78>
\begin{array}{rl}
z&=\displaystyle{r\left[\left(1+\frac{(i\theta)^2}{2!}
+\frac{(i\theta)^4}{4!}+\cdots\right)
+\left(i\theta-\frac{(i\theta)^3}{3!}
+\frac{(i\theta)^5}{5!}+
\cdots\right)\right] \\ \\
&=\displaystyle{r\sum_{k=0}^{\infty}
\frac{(i\theta)^k}{k!}}
\end{array}
<3-79> (3.2.6)
\begin{array}{rl}
z&=x+iy \\ \\
&=r(\cos\theta+i\sin\theta) \\ \\
&=re^{i\theta}
\end{array}
<3-80> (3.2.7)
e^{i\theta}=\cos\theta+i\sin\theta
<3-81> (3.2.8)
\left\{\begin{array}{l}
r=|z|=\sqrt{x^2+y^2} \\ \\
\displaystyle{\tan\theta=\frac{y}{x}\quad
\mbox{or}\quad\theta=\mbox{arg}(z)
=\tan^{-1}\left(\frac{y}{x}\right)
\end{array}\right.
<3-82> (3.2.9)
\begin{array}{l}
360^\circ=2\pi\;\mbox{[radian]} \\ \\
\mbox{or}\\ \\
180^\circ=\pi\;\mbox{[radian]}\simeq3.14\;
\mbox{[radian]}
\end{array}
<3-83>
\pi/6\simeq0.52
<3-84>
e^{i\pi/6}=\sqrt{3}/2+i(1/2)
<3-85>
\pi/4\simeq0.79
<3-86>
e^{i\pi/4}=1/\sqrt{2}+i\left(1/\sqrt{2}\right)
<3-87>
\pi/3\simeq1.05
<3-88>
e^{i\pi/3}=1/2+i\left(\sqrt{3}/2\right)
<3-89>
\pi/2\simeq1.57
<3-90>
e^{i\pi/2}=i
<3-91>
3\pi/4\simeq2.36
<3-92>
e^{i3\pi/4}=-1/\sqrt{2}+i\left(1/\sqrt{2}\right)
<3-93>
\pi\simeq3.14
<3-94>
e^{i\pi}=-1
<3-95>
5\pi/4\simeq3.93
<3-96>
e^{i5\pi/4}=-1/\sqrt{2}-i\left(1/\sqrt{2}\right)
<3-97>
3\pi/2\simeq4.71
<3-98>
e^{i3\pi/2}=-i
<3-99>
7\pi/4\simeq5.50
<3-100>
e^{i7\pi/4}=1/\sqrt{2}-i\left(1/\sqrt{2}\right)
<3-101>
2\pi\simeq6.28
<3-102>
e^{i2\pi}=1
<3-103> (3.2.10)
\left\{\begin{array}{l}
\displaystyle{\cos\theta
=\frac{1}{2}\left(e^{i\theta}
+e^{-i\theta}\right)} \\ \\
\displaystyle{\sin\theta
=\frac{1}{2i}\left(e^{i\theta}
-e^{-i\theta}\right)}
\end{array}\right.
<3-104> (3.2.11)
\left(\cos\theta+i\sin\theta\right)^n
=\cos(n\theta)+i\sin(n\theta)
<3-105>
r=r_1r_2
<3-106>
\theta=\theta_1+\theta_2\quad\Rightarrow
\quad\mbox{arg}(z)
=\mbox{arg}(z_1)+\mbox{arg}(z_2)
<3-107> (3.2.12)
z^*=x-iy
<3-108> (3.2.13)
z^*=re^{-i\theta}
<3-109> (3.2.14)
\sqrt{zz^*}=|z|=r
<3-110>
\begin{array}{rl}
\cos z&=\displaystyle{
\frac{1}{2}\left[\left(\cos x+i\sin x\right)e^{-y}
+\left(\cos x-i\sin x\right)e^y\right]} \\ \\
&=\displaystyle{
\frac{1}{2}\left[\left(e^y+e^{-y}\right)\cos x
-i\left(e^y-e^{-y}\right)\sin x\right]}
\end{array}
<3-111>
\left\{\begin{array}{l}
u=\displaystyle{
\frac{1}{2}\left(e^{y}+e^{-y}\right)\cos x} \\ \\
v=\displaystyle{
-\frac{1}{2}\left(e^{y}-e^{-y}\right)\sin x}
\end{array}\right.
<3-112>
\left\{\begin{array}{l}
\displaystyle{
\frac{1}{2}\left(e^{y}+e^{-y}\right)
=\cosh y} \\ \\
\displaystyle{
-\frac{1}{2}\left(e^{y}-e^{-y}\right)
=\sinh x}
\end{array}\right.
<3-113>
\left\{\begin{array}{l}
u=\cosh y\cos x \\ \\
v=-\sinh y\sin x
\end{array}\right.
<3-114>
\sin z=\cosh y\sin x+i\sinh y\cos x
<3-115>
z=-1\quad\mbox{or}\quad z^2-z+1=0
<3-116>
z=\frac{1\pm\sqrt{-3}}{2}
=\frac{1\pm i\sqrt{3}}{2}
<3-117>
z=-1,\quad\frac{1+i\sqrt{3}}{2},\quad
\frac{1-i\sqrt{3}}{2}
<3-118>
\left\{\begin{array}{l}
\displaystyle{e^{i(\pi/4)}
=\cos\left(\frac{\pi}{4}\right)
+i\sin\left(\frac{\pi}{4}\right)
=\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}} \\ \\
\displaystyle{e^{i(3\pi/4)}
=\cos\left(\frac{3\pi}{4}\right)
+i\sin\left(\frac{3\pi}{4}\right)
=-\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}} \\ \\
\displaystyle{e^{i(5\pi/4)}
=\cos\left(\frac{5\pi}{4}\right)+
i\sin\left(\frac{5\pi}{4}\right)=
-\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}} \\ \\
\displaystyle{e^{i(7\pi/4)}
=\cos\left(\frac{7\pi}{4}\right)+
i\sin\left(\frac{7\pi}{4}\right)=
\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}}}
\end{array}\right.
<3-119>
\begin{array}{rl}
|w|^2&=\displaystyle{
\frac{i+1}{\cos\theta-i\sin\theta}
\left(\frac{i+1}{\cos\theta-i\sin\theta }\right)^*} \\ \\
&=\displaystyle{
\frac{(i+1)(-i+1)}{( \cos\theta-i\sin\theta)
(\cos\theta+i\sin\theta)}} \\ \\
&=\displaystyle{
\frac{2}{\cos^2\theta+\sin^2\theta}} \\ \\
&=2
\end{array}
<4-1> (4.1.1)
\frac{dF(x)}{dx}=f(x)
<4-2> (4.1.2)
\frac{dF(x)}{dx}=f(x)
<4-3> (4.1.3)
\int f(x)dx
<4-4>
f(x)
<4-5>
F(x)=\int f(x)dxdA
<4-6>
x^n
<4-7>
\frac{x^{n+1}}{n+1}
<4-8>
\frac{1}{x}
<4-9>
\ln|x|
<4-10>
\frac{1}{x^{n+1}}\quad
\mbox{($n$ is a natural number.)}\quad
\mbox{(*)}
<4-11>
-\frac{1}{nx^n}
<4-12>
e^x
<4-13>
e^x
<4-14>
\sin x
<4-15>
-\cos x
<4-16>
\cos x
<4-17>
\sin x
<4-18>
\sin^2x \quad \mbox{(*)}
<4-19>
-\frac{1}{4}\sin 2x+\frac {x}{2}
<4-20>
\cos^2x \quad \mbox{(*)}
<4-21>
\frac{1}{4}\sin 2x+\frac {x}{2}
<4-22>
\frac{1}{\sqrt{a+x}}\quad\mbox{(a is a constant.)}
\quad\mbox{(*)}
<4-23>
2\sqrt{a+x}
<4-24>
\frac{1}{(a+x)^{3/2}}\quad\mbox{(a is a constant.)}
\quad\mbox{(*)}
<4-25>
-\frac{2}{\sqrt{a+x}}
<4-26>
\sqrt{a^2+x^2}\quad\mbox{(a is a constant.)} \quad
\mbox{(*)}
<4-27>
\frac{x\sqrt{a^2+x^2}}{2}+\frac{a^2}{2}\ln\left|x
+\sqrt{a^2+x^2}\right|
<4-28>
\sqrt{a^2-x^2}\quad\mbox{(a is a constant.)}
\quad \mbox{(*)}
<4-29>
\frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\sin^{-1}
\left(\frac{x}{a}\right)
<4-30> (4.2.1)
F(b)-F(a)\equiv\int_a^b f(x)dx
<4-31> (4.2.2)
<5-1> (5.1.1)
\frac{dN}{dt}=-kN
<5-2> (5.1.2)
L\frac{d^2I}{dt^2}+R\frac{dI}{dt}+\frac{I}{C}=0
<5-3> (5.1.3)
\alpha\frac{d^2\psi}{dx^2}+V(x)\psi=E\psi
<5-4> (5.1.4)
\frac{\partial^2\Phi}{\partial x^2}+
\frac{\partial^2\Phi}{\partial y^2}
+\frac{\partial^2\Phi}{\partial z^2}=0
<5-5> (5.1.5)
\frac{\partial\phi}{\partial t}=
D\frac{\partial^2\phi}{\partial x^2
<5-6>
\left[\mbox{A term proportional to }
\frac{d^2y}{dx^2}\right]
+\left[\mbox{A term proportional to }
\frac{dy}{dx}\right]
+\left[\mbox{A term proportional to }y\right]
=\left[
\mbox{A funcion $Q(x)$ independent of $y$}
\right]
<5-7> (5.1.6)
\frac{dy}{dx}=p(x)q(y)
<5-8> (5.1.7)
\frac{1}{q(y)}\frac{dy}{dx}=p(x)
<5-9> (5.1.8)
\int\frac{1}{q(y)}\frac{dy}{dx}dx=\int p(x)dx+C
<5-10> (5.1.9)
\int\frac{1}{q(y)}dy=\int p(x)dx+C
<5-11>
\int\frac{1}{y+1}dy=\int\frac{1}{x+1}dx+C
<5-12> (5.1.10)
\int\frac{1}{x+a}dx=\ln|x+a|
<5-13>
\ln|y+1|=\ln|x+1|+C
<5-14>
\ln|y+1|-\ln|x+1|=\ln\left|\frac{y+1}{x+1}\right|=C
<5-15>
\left|\frac{y+1}{x+1}\right|=e^C
<5-16>
\frac{y+1}{x+1}=C
<5-17>
y=C(x+1)-1
<5-18>
\left\{\begin{array}{l}
\displaystyle{\frac{dy}{dx}=C} \\ \\
\displaystyle{\frac{y+1}{x+1}=C}
\end{array}\right.
<5-19> (5.1.11)
\frac{dy}{dx}+p(x)y=q(x)
<5-20> (5.1.12)
\frac{dy}{dx}+p(x)y=0
<5-21>
\int\frac{1}{y}dy=
-\int p(x)dx+C\quad
\mbox{(C is an integration constant.)}
<5-22> (5.1.13)
y=Ce^{\displaystyle{-\int p(x)dx}}
<5-23> (5.1.14)
y=C(x)e^{\displaystyle{-\int p(x)dx}}
<5-24> (5.1.15)
\frac{dC(x)}{dx}=
q(x)e^{\displaystyle{\int p(x)dx}}
<5-25> (5.1.16)
\begin{array}{rl}
C(x)&=\int X(x)dx+C \\ \\
&=\int\left\{q(x)e^{\displaystyle{
\int p(x)dx}}\right\}dx+C
\end{array}
<5-26> (5.1.17)
y=e^{-\displaystyle{\int p(x)dx}}
\left[\int\left\{q(x)e^{\displaystyle{
\int p(x)dx}}\right\}dx+C\right]
<5-27> (5.1.18)
L\frac{dI}{dt}+RI=V(t)
<5-28> I(t)=
\frac{1}{L}e^{-(R/L)t}
\left[\int\left\{e^{(R/L)t}V(t)\right\}dt
+C_1\right]
<5-29>
I(t)=\frac{V_0}{R}+C_2e^{-(R/L)t}
<5-30>
I(t)=\frac{V_0}{R}\left\{1-e^{-(R/L)t}\right\}
<5-31>
e^{-(R/L)t}\simeq 1-\frac{R}{L}t
<5-32>
I(t)\simeq
\frac{V_0}{R}\left[1-\left\{1-
\frac{R}{L}t\right\}\right]=\frac{V_0}{L}t
<5-33> (5.1.19)
\frac{dy}{dx}=f\left(\frac{y}{x}\right)
<5-34>
\frac{y(x)}{x}=u(x)
<5-35>
\frac{dy}{dx}=\frac{d(xu)}{dx}=u+x\frac{du}{dx}
<5-36>
u+x\frac{du}{dx}=f(u)
<5-37> \quad \mbox{(*)}
\frac{du}{dx}=\frac{f(u)-u}{x}
<5-38>
\frac{dy}{dx}=\frac{x^2+y^2}{xy}
<5-39>
\begin{array}{rl}
\frac{x^2+y^2}{xy}&=\frac{x}{y}+\frac{x}{y} \\ \\
&=\left(\frac{y}{x}\right)^{-1}+\frac{x}{y}
\end{array}
<5-40>
\frac{du}{dx}=\frac{1/u}{x}
<5-41>
u=\sqrt{\ln(x^2)+C}
<5-43> (5.1.20)
\frac{dy}{dx}=-\frac{p(x,y)}{q(x,y)}
<5-44> (5.1.21)
\frac{\partial p}{\partial y}
=\frac{\partial q}{\partial x}
<5-45>
\mbox{(I)}\quad\frac{\partial P}{\partial y}
=\frac{\partial Q}{\partial x}
<5-46>
\mbox{(II)}\quad\left\{\begin{array}{l}
\displaystyle{\frac{\partial f}{\partial x}}
=P(x,y) \\ \\
\displaystyle{\frac{\partial f}{\partial y}}
=Q(x,y)
\end{array}\right.rtial f}{\partial y}}
=Q(x,y)
\end{array}\right.
<5-47>
\mbox{(III)}\quad df(x,y)=P(x,y)dx+Q(x,y)dy
<5-48> (5.1.22)
p(x,y)dx+q(x,y)dy=0
<5-49> (5.1.23)
du(x,y)=p(x,y)dx+q(x,y)dy
<5-50> (5.1.24)
\left\{\begin{array}{l}
\displaystyle{\frac{\partial u}{\partial x}}
=p(x,y) \\ \\
\displaystyle{\frac{\partial u}{\partial y}}
=q(x,y)
\end{array}\right.
<5-51>
\frac{dy}{dx}=-\frac{x+y+1}{x-y^2+3}
<5-52>
\frac{\partial p(x,y)}{\partial y}=
\frac{\partial q(x,y)}{\partial x}=1
<5-53>
\left\{\begin{array}{rl}
\displaystyle{\frac{\partial u(x,y)}{\partial x}}
&=p(x,y) \\ \\
&=x+y+1 \\ \\
\displaystyle{\frac{\partial u(x,y)}{\partial y}}
&=q(x,y) \\ \\
&=x-y^2+3
\end{array}\right.
<5-54>
u=\frac{x^2}{2}+xy+x+g(y)
<5-55>
\frac{\partial u}{\partial y}=x+\frac{dg}{dy}
<5-56>
\frac{dg}{dy}=-y^2+3
<5-57>
g=-\frac{y^3}{3}+3y+A
<5-58>
u=\frac{x^2}{2}+xy-\frac{y^3}{3}+x+3y+A
<5-59>
\frac{x^2}{2}+xy-\frac{y^3}{3}+x+3y=C
<5-60> (5.1.25)
\frac{dy}{dx}+p(x)y=q(x)y^n \quad (n\ne0,1)
<5-61> (5.1.26)
\frac{d^2y}{dx^2}+p(x)y^2+q(x)y+r(x)=0
<5-62>
\frac{dx}{dt}=
k\left(a-\frac{x}{2}\right)
\left(b-\frac{x}{2}\right)
<5-63>
\mbox{(1)}\quad
\int\frac{1}{\left(a-
\displaystyle{\frac{x}{2}}\right)
\left(b-\displaystyle{\frac{x}{2}}\right)}dx
=k\int dt+C
<5-64>
\frac{1}{\left(a-\displaystyle{\frac{x}{2}}\right)
\left(b-\displaystyle{\frac{x}{2}}\right)}
=\frac{1}{a-b}\left[\frac{1}{b-
\displaystyle{\frac{x}{2}}}-
\frac{1}{a-\displaystyle{\frac{x}{2}}}\right]
<5-65>
\mbox{(2)}\quad\frac{1}{a-b}
\int\frac{1}{b-\displaystyle{
\frac{x}{2}}}dx-\frac{1}{a-b}\int\frac{1}{a-
\displaystyle{\frac{x}{2}}}dx=kt+C
<5-66>
\int\frac{1}{a-\displaystyle{\frac{x}{2}}}dx
=-2\ln(x-2a)
<5-67>
\ln\frac{x-2a}{x-2b}=-\frac{b-a}{2}(kt+C)
<5-68>
\mbox{(3)}\quad
x=2ab\left[
\frac{1-e^{(b-a)kt/2}}{a-be^{(b-a)kt/2}}
\right]
<5-69>
e^y\simeq 1+y
<5-70>
A(\mbox{radio nucleide})\rightarrow
B(\mbox{radio nucleide})\rightarrow
C(\mbox{stable nucleide})
<5-71>
\Delta N_{A}(t)=-\lambda_{A}N_{A}(t)\Delta t
<5-72>
N_{A}(t)-\lambda_{A}N_{A}(t)\Delta t
<5-73>
N_{B}(t)-\lambda_{B}N_{B}(t)\Delta t
+\lambda_{A}N_{A}(t)\Delta t
<5-74>
N_{C}(t)+\lambda_{B}N_{B}(t)\Delta t
<5-75>
\left\{\begin{array}{l}
\mbox{(1)}\quad N_{A}(t+\Delta t)=
N_{A}(t)-\lambda_{A}N_{A}(t)\Delta t \\ \\
\mbox{(2)}\quad N_{B}(t+\Delta t)=
N_{B}(t)-\lambda_{B}N_{B}(t)\Delta t+
\lambda_{A}N_{A}(t)\Delta t \\ \\
\mbox{(3)}\quad N_{C}(t+\Delta t)=
N_{C}(t)+\lambda_{B}N_{B}(t)\Delta t
\end{array}\right.
<5-76>
\frac{N_A(t+\Delta t)-N_A(t)}{\Delta t}
<5-77>
\frac{N_B(t+\Delta t)-N_B(t)}{\Delta t}
<5-78>
\frac{N_C(t+\Delta t)-N_C(t)}{\Delta t}
<5-79>
\left\{\begin{array}{l}
\mbox{(4)}\quad \displaystyle{
\frac{N_{A}(t)}{dt}}=-\lambda_{A}N_{A}(t) \\ \\
\mbox{(5)}\quad \displaystyle{
\frac{N_{B}(t)}{dt}}=-\lambda_{B}N_{B}(t)
+\lambda_{A}N_{A}(t) \\ \\
\mbox{(6)}\quad \displaystyle{
\frac{N_{C}(t)}{dt}}=\lambda_{B}N_{B}(t)
\end{array}\right.
<5-80>
\mbox{(7)}\quad N_{A}(t)=C_Ae^{-\lambda_{A}t}
<5-81>
\mbox{(8)}\quad N_{A}(t)=N_0e^{-\lambda_{A}t}
<5-82>
\mbox{(9)}\quad \frac{N_{B}(t)}{dt}
+\lambda_{B}N_{B}(t)
=\lambda_{A}N_0e^{-\lambda_{A}t}
<5-83>
\mbox{(10)}\quad \frac{N_{B}(t)}{dt}
+\lambda_{B}N_{B}(t)=0
<5-84>
\mbox{(11)}\quad N_{B}(t)
=C_Be^{-\lambda_{B}t}
<5-85>
\mbox{(12)}\quad N_{B}(t)
=C_B(t)e^{-\lambda_{B}t}
<5-86>
\frac{dN_{B}(t)}{dt}=\frac{dC_{B}(t)}{dt}
e^{-\lambda_{B}t}
-\lambda_{B}C_{B}(t)e^{-\lambda_{B}t}
<5-87>
\frac{dC_{B}(t)}{dt}=
\lambda_{A}N_{0}e^{(\lambda_{B}-\lambda_{A})t}
<5-88>
C_{B}(t)=\frac{\lambda_{A}N_{0}}
{\lambda_{B}-\lambda_{A}}
\left[e^{(\lambda_{B}-\lambda_{A})t}-1\right]
<5-89>
N_{B}(t)=\frac{\lambda_{A}N_{0}}
{\lambda_{B}-\lambda_{A}}
\left[e^{-\lambda_{A}t}}-e^{-\lambda_{B}t}}\right]
<5-90>
\frac{dN_{C}(t)}{dt}=
\frac{\lambda_{A}\lambda_{B}N_{0}}
{\lambda_{B}-\lambda_{A}}
\left[e^{-\lambda_{A}t}}-
e^{-\lambda_{B}t}}\right]
<5-91>
\int e^{-at}dt=-\frac{e^{-at}}{a}
<5-92>
N_{C}(t)=
\frac{\lambda_{A}\lambda_{B}N_{0}}
{\lambda_{B}-\lambda_{A}}
\left[-\frac{e^{-\lambda_{A}t}}{\lambda_{A}}
+\frac{e^{-\lambda_{B}t}}{\lambda_{B}}\right]
+C_C
<5-93>
N_{C}(t)=\frac{N_{0}}{\lambda_{A}-\lambda_{B}}
\left[\lambda_{A}(1-e^{-\lambda_{B}t})
-\lambda_{B}(1-e^{-\lambda_{A}t})\right]
<5-94>
N_{A}(t)=N_{0}e^{-\lambda_{A}T}=\frac{N_{0}}{2}
<5-95>
T=\frac{\ln 2}{\lambda_{A}}\simeq
\frac{0.69}{\lambda_{A}}
<5-96> (5.1.27)
\frac{d^2y}{dx^2}+p(x)\frac{dy}{dx}+q(x)y=r(x)
<5-97> (5.1.28)
\frac{d^2y}{dx^2}+p(x)\frac{dy}{dx}+q(x)y=0
<5-98> (5.1.29)
\left[\frac{d^2}{dx^2}+p(x)\frac{d}{dx}
+q(x)\right]y\equiv L(y)=0
<5-99> (5.1.30)
L(y_{1})=0\quad\mbox{and}\quad L(y_{2})=0
<5-100> (5.1.31)
L(C_{1}y_{1}+C_{2}y_{2})
=\frac{d^2(C_{1}y_{1}+C_{2}y_{2})}{dx^2}+
p(x)\frac{d(C_{1}y_{1}+C_{2}y_{2})}{dx}
+q(x)(C_{1}y_{1}+C_{2}y_{2})
=C_{1}\left[\frac{d^2y_{1}}{dx^2}
+p(x)\frac{dy_{1}}{dx}+q(x)y_{1}\right]
+C_{2}\left[\frac{d^2y_{2}}{dx^2}
+p(x)\frac{dy_{2}}{dx}+q(x)y_{2}\right]
=C_{1}L(y_{1})+C_{2}L(y_{2})
=0
<5-101> (6.1.32)
C_{1}y_{1}+C_{2}y_{2}=0
<5-102> (5.1.33)
W(x)=\left|\begin{array}{cc}
y_{1}(x) & y_{2}(x) \\ \\
y'_{1}(x) & y'_{2}(x)
\end{array}\right|
=y_{1}(x)y'_{2}(x)-y'_{1}(x)y_{2}(x)
<5-103> (5.1.34)
\left\{\begin{array}{l}
\mbox{if }W(x)\ne0,
\mbox{ $y_1$ and $y_2$ are linear independent.}
\\ \\
\mbox{if }W(x)=0,
\mbox{ $y_1$ and $y_2$ are linear dependent.}
\end{array}\right.
<5-104> (5.1.35)
\frac{d^2y}{dx^2}+p\frac{dy}{dx}+qy=r(x)
<5-105> (5.1.36)
\frac{d^2y}{dx^2}+p\frac{dy}{dx}+qy=0
<5-106> (5.1.37)
\lambda^2+p\lambda+q=0
<5-107>
D^2y+pDy+qy=(D^2+pD+q)y(x)=0
<5-108>
(\lambda^2+p\lambda+q)y(x)=0
<5-109>
\left\{\begin{array}{l}
\displaystyle{\lambda_1=
\frac{1}{2}\left(-p+\sqrt{p^2-4q}\right)} \\ \\
\displaystyle{\lambda_2
=\frac{1}{2}\left(-p-\sqrt{p^2-4q}\right)}
\end{array}\right.
<5-110>
y=C_1e^{\lambda_1x}+C_2e^{\lambda_2x}
<5-111>
\begin{array}{l}
y=C_1y_1(x)+C_2y_(2) \\ \\
\mbox{where}\quad \left\{\begin{array}{l}
y_1(x)=e^{\lambda_1x} \\ \\
y_2(x)=e^{\lambda_2x}
\end{array}\right.
\end{array}right.
\end{array}
<5-112>
\begin{array}{l}
\left\{\begin{array}{l}
\displaystyle{\lambda_1=
-\frac{1}{2}+i\gamma} \\ \\
\displaystyle{\lambda_2=-\frac{1}{2}-i\gamma}
\end{array}\right. \\ \\
\mbox{where}\quad\gamma=
\displaystyle{\frac{\sqrt{4q-p^2}}{2}}
\end{array}\begin{array}{l}
\left\{\begin{array}{l}
\displaystyle{\lambda_1=
-\frac{1}{2}+i\gamma} \\ \\
\displaystyle{\lambda_2=
-\frac{1}{2}-i\gamma}
\end{array}\right. \\ \\
\mbox{where}\quad\gamma=
\displaystyle{\frac{\sqrt{4q-p^2}}{2}}
\end{array}
<5-113>
\begin{array}{rl}
y&=C_1e^{(-p/2+i\gamma)x}+
C_2e^{(-p/2-i\gamma)x} \\ \\
&=e^{(-p/2)x}\left[C'_1\cos(\gamma x)+
C'_2\sin(\gamma x)\right]
\end{array}
<5-114>
\begin{array}{l}
y=C_1y_1(x)+C_2y_(2) \\ \\
\mbox{where}\quad \left\{\begin{array}{l}
y_1(x)=e^{(-p/2)x}\cos(\gamma x) \\ \\
y_2(x)=e^{(-p/2)x}\sin(\gamma x)
\end{array}\right.
\end{array}
<5-115>
y_1=e^{(-p/2)x}
<5-116>
y_2(x)=C(x)e^{(-p/2)x}
<5-117>
\frac{d^2C(X)}{dx^2}=0
<5-118>
C(x)=C'_1+C'_2x
<5-119>
y_2=C'2xe^{-(p/2)x}
<5-120>
y=(C_1+C_2x)e^{-(p/2)x}
<5-121>
\begin{array}{l}
y=C_1y_1(x)+C_2y_(2) \\ \\
\mbox{where}\quad \left\{\begin{array}{l}
y_1(x)=e^{(-p/2)x} \\ \\
y_2(x)=xe^{(-p/2)x}
\end{array}\right.
\end{array}
<5-122> (5.1.38)
y(x)=C_1y_1(x)+C_2y_2(x)
<5-123> (5.1.39)
y(x)=C_1(x)y_1(x)+C_2(x)y_2(x)
<5-124> (5.1.40)
\frac{dC_1(x)}{dx}y_1(x)+
\frac{dC_2(x)}{dx}y_2(x)=0
<5-125> (5.1.41)
\frac{dC_1(x)}{dx}\frac{dy_1(x)}{dx}+
\frac{dC_2(x)}{dx}\frac{dy_2(x)}{dx}=r(x)
<5-126> (5.1.52)
\left\{\begin{array}{l}
\displaystyle{\frac{dC_1(x)}{dx}=
-\frac{y_2(x)r(x)}{W(x)}} \\ \\
\displaystyle{\frac{dC_2(x)}{dx}=
\frac{y_1(x)r(x)}{W(x)}}
\end{array}\right. \\ \\
\mbox{where}\quad W(x)=
y_1(x)y'_2(x)-y'_1(x)y_2(x)
<5-127> (5.1.43)
\left\{\begin{array}{l}
C_1(x)=C''_1-\displaystyle{
\int\frac{r(x)y_2(x)}{W(x)}dx} \\ \\
C_2(x)=C''_2+\displaystyle{
\int\frac{r(x)y_1(x)}{W(x)}dx}
\end{array}\right.
<5-128> (5.1.44)
\begin{array}{l}
y_1(x)=\displaystyle{\left[C''_1-
\int\frac{r(x)y_2(x)}{W(x)}dx}\right]y_1(x)}
+\displaystyle{\left[C''_2+
\int\frac{r(x)y_1(x)}{W(x)}dx\right]y_2(x)}y_2(x)}
\\ \\
=\displaystyle{C''_1y_1(x)+C''_2y_2(x)
-y_1(x)\left(\int\frac{r(x)y_2(x)}{W(x)}dx}\right)
+y_2(x)\left(\int\frac{r(x)y_1(x)}{W(x)}dx}\right)}
\end{array}
<5-129> (5.2.1)
\frac{d^2x}{dt^2}=-\frac{g}{l}x
<5-130> (5.2.2)
\frac{d^2x}{dt^2}+\omega^2x=0
<5-131> (5.2.3)
\lambda^2+\omega^2=0
<5-132> (5.2.4)
x(t)=C_1e^{i\omega t}+C_2e^{-i\omega t}
<5-133>
\frac{dx}{dt}=i\omega(C_1e^{i\omega t}-
C_2e^{-i\omega t})
<5-134> (5.2.5)
\left\{\begin{array}{l}
C_1+C_2=0 \\ \\
i\omega(C_1-C_2)=\omega_0
\end{array}\right.
<5-135>
\left\{\begin{array}{l}
\displaystyle{C_1=\frac{\omega_0}{2i\omega}} \\ \\
\displaystyle{C_2=-\frac{\omega_0}{2i\omega}}
\end{array}\right.
<5-136> (5.2.6)
x(t)=\frac{\omega_0}{\omega}\sin(\omega t)
<5-137> (5.2.7)
\frac{d^2x}{dt^2}+2\gamma\frac{dx}{dt}+\omega_0^2x
=0
<5-138> (5.2.8)
\lambda^2+2\gamma\lambda+\omega_0^2=0
<5-139>
\left\{\begin{array}{l}
\lambda_1=-\gamma+\sqrt{\gamma^2-\omega_0^2} \\ \\
\lambda_2=-\gamma-\sqrt{\gamma^2-\omega_0^2}
\end{array}\right.
<5-140> (5.2.9)
x(t)=e^{-\gamma t}\left[C_1e^{\sqrt{\gamma^2-
\omega_0^2}}+C_2e^{-\sqrt{\gamma^2-\omega_0^2}}
\right]
<5-141> (5.2.10)
\begin{array}{l}
\displaystyle{\frac{d^2x}{dt^2}+2\gamma\frac{dx}{dt}
+\omega_0^2x=f\cos(\omega t)}, \\ \\
\mbox{(where$\quad\omega_0\ge\gamma$.)}
<5-142>
\frac{d^2x_0}{dt^2}+2\gamma\frac{dx_0}{dt}
+\omega_0^2x_0=f\cos(\omega t)
<5-143>
\frac{d^2F}{dt^2}+2\gamma\frac{dF}{dt}
+\omega_0^2F
<5-144>
\frac{d(f+g)}{dt}=\frac{df}{dt}+\frac{dg}{dt}
<5-145>
\begin{array}{l}
\displaystyle{\frac{d^2F}{dt^2}
+2\gamma\frac{dF}{dt}+\omega_0^2F} \\ \\
=\displaystyle{\left[\frac{d^2x}{dt^2}
+2\gamma\frac{dx}{dt}+\omega_0^2x\right]
-\left[\frac{d^2x_0}{dt^2}+
2\gamma\frac{dx_0}{dt}+\omega_0^2x_0\right]}=0
\end{array}
<5-146> (5.2.11)
F(t)=e^{-\gamma t}
\left[C_1e^{\sqrt{\gamma^2-\omega_0^2}t}
+C_2e^{-\sqrt{\gamma^2-\omega_0^2}t}\right]
<5-147> (5.2.12)
\frac{d^2z}{dt^2}+2\gamma\frac{dz}{dt}+
\omega_0^2z=fe^{i\omega t}
<5-148>
\left\{\begin{array}{l}
z=x+iy \\ \\
\displaystyle{\frac{dz}{dt}=\frac{dx}{dt}+
i\frac{dy}{dt}} \\ \\
\displaystyle{\frac{d^2z}{dt^2}=
\frac{d^2x}{dt^2}+i\frac{d^2y}{dt^2}} \\ \\
\displaystyle{e^{i\omega t}=\cos(\omega t)+
i\sin(\omega t)}
\end{array}
<5-149>
\frac{de^{i\omega t}}{dt}=i\omega e^{i\omega t}
<5-150>
\frac{d^2e^{i\omega t}}{dt^2}=
-\omega^2 e^{i\omega t}
<5-151> (5.2.13)
A=\frac{f}{\omega_0^2-\omega^2+i\omega\gamma}
<5-152> (5.2.14)
\left\{\begin{array}{l}
a=\displaystyle{
\frac{f}{\sqrt{(\omega_0^2-\omega^2)^2
+4\gamma^2\omega^2}}} \\ \\
\tan\phi=\displaystyle{
\frac{2\gamma\omega}{\omega_0^2
-\omega^2}}\quad\mbox{or}\quad
\displaystyle{\phi=\tan^{-1}
\left(
\frac{2\gamma\omega}{\omega_0^2-\omega^2}
\right)}
\end{array}\right.
<5-153> (5.2.15)
z(t)=ae^{i(\omega t-\phi)}
<5-154> (5.2.16)
x(t)=a\cos(\omega t-\phi)
<5-155> (5.2.17)
y(t)=e^{-\gamma t}\left[
C_1e^{\sqrt{\gamma^2-\omega_0^2}t}
+C_2e^{-\sqrt{\gamma^2-\omega_0^2}t}
\right]+
a\cos(\omega t-\phi)
<6-1> (6.1.1)
\left(\begin{array}{ccc}
1 & 5 & 2 \\ \\ 4 & 2 & 3
\end{array}\right)
<6-2> (6.1.2)
\left(\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\ \\
a_{21} & a_{22} & a_{23}
\end{array}\right)\pm\left(\begin{array}{ccc}
b_{11} & b_{12} & b_{13} \\ \\
b_{21} & b_{22} & b_{23}
\end{array}\right)=\left(\begin{array}{ccc}
a_{11}\pm b_{11} & a_{12}\pm b_{12}
& a_{13}\pm b_{13}
\\ \\
a_{21}\pm b_{21} & a_{22}\pm b_{22}
& a_{23}\pm b_{23}
\end{array}\right)
<6-3> (6.1.3)
k\left(\begin{array}{cc}
a_{11} & a_{12} \\ \\ a_{21} & a_{22}
\end{array}\right)=\left(\begin{array}{cc}
ka_{11} & ka_{12} \\ \\ ka_{21} & ka_{22} }
\end{array}\right)
<6-4>
A=\left(\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1L} \\ \\
a_{21} & a_{22} & \cdots & a_{2L} \\ \\
\cdots & \cdots & \cdots & \cdots \\ \\
a_{N1} & a_{N2} & \cdots & a_{NL}
\end{array}\right)
<6-5>
B=\left(\begin{array}{cccc}
b_{11} & b_{12} & \cdots & b_{1M} \\ \\
b_{21} & b_{22} & \cdots & b_{2M} \\ \\
\cdots & \cdots & \cdots & \cdots \\ \\
b_{L1} & b_{L2} & \cdots & b_{LM}
\end{array}\right)
<6-6>
C=\left(\begin{array}{cccc}
c_{11} & c_{12} & \cdots & c_{1M} \\ \\
c_{21} & c_{22} & \cdots & c_{2M} \\ \\
\cdots & \cdots & \cdots & \cdots \\ \\
c_{N1} & c_{N2} & \cdots & c_{NM}
\end{array}\right)
<6-7> (6.1.4)
\begin{array}{l}
\left\{\begin{array}{l}
c_{11}=a_{11}b_{11}+\cdots+a_{1L}b_{L1} \\ \\
c_{12}=a_{11}b_{12}+\cdots+a_{1L}b_{L2} \\ \\
\quad \cdots \\ \\
c_{1M}=a_{11}b_{1M}+\cdots+a_{1L}b_{LM}
\end{array}\right. \\ \\
\left\{\begin{array}{l}
c_{21}=a_{21}b_{11}+\cdots+a_{2L}b_{L1} \\ \\
c_{22}=a_{21}b_{12}+\cdots+a_{2L}b_{L2} \\ \\
\quad \cdots \\ \\
c_{2M}=a_{21}b_{1M}+\cdots+a_{2L}b_{LM}
\end{array}\right. \\ \\
\left\{\begin{array}{l}
c_{N1}=a_{N1}b_{11}+\cdots+a_{NL}b_{L1} \\ \\
c_{N2}=a_{N1}b_{12}+\cdots+a_{NL}b_{L2} \\ \\
\quad \cdots \\ \\
c_{NM}=a_{N1}b_{1M}+\cdots+a_{NL}b_{LM}
\end{array}\right.
\end{array}
<6-8> (6.1.5)
\mbox{When }
A=\left(\begin{array}{ccc}
1 & 2 & 3 \\ \\ 4 & 5 & 6
\end{array}\right),\mbox{ then }
A^{T}=\left(\begin{array}{cc}
1 & 4 \\ \\ 2 & 5 \\ \\ 3 & 6
\end{array}\right)
<6-9> (6.1.6)
(AB)^T=B^TA^T
<6-10> (6.1.7)
E=\left(\begin{array}{cccc}
1 & 0 & 0 & \cdots \\ \\
0 & 1 & 0 & \cdots \\ \\
0 & 0 & 1 & \cdots \\ \\
\cdots & \cdots & \cdots & \cdots
\end{array}\right)
<6-11> (6.1.8)
AE=EA=A
<6-12> (6.1.9)
O=\left(\begin{array}{ccc}
0 & 0 & \cdots \\ \\
0 & 0 & \cdots \\ \\
\cdots & \cdots & \cdots
\end{array}\right)
<6-13>
\frac{1}{a}=a^{-1}a=1
<6-14>
AB=BA=E
<6-15> (6.1.10)
AA^{-1}=A^{-1}A=E
<6-16>
A=\left(\begin{array}{cc}
a_{11} & a_{12} \\ \\
a_{21} & a_{22}
\end{array}\right)
<6-17>
A^{-1}=\frac{1}{\mbox{det}(A)}\left(\begin{array}{cc}
a_{22} & -a_{21} \\ \\
-a_{12} & a_{11}
\end{array}\right)
<6-18>
\mbox{det}(A)=a_{11}a_{22}-a_{12}a_{21}
<6-19> (6.1.11)
M=\left(\begin{array}{cccc}
p_{11} & p_{12} & \cdots & p_{1m} \\ \\
p_{21} & p_{22} & \cdots & p_{2m} \\ \\
\cdots & \cdots & \cdots & \cdots \\ \\
p_{m1} & p_{m2} & \cdots & p_{mm}
\end{array}\right)
<6-20> (6.1.12)
\mbox{det}(M)=\left|\begin{array}{cccc}
p_{11} & p_{12} & \cdots & p_{1m} \\ \\
p_{21} & p_{22} & \cdots & p_{2m} \\ \\
\cdots & \cdots & \cdots & \cdots \\ \\
p_{m1} & p_{m2} & \cdots & p_{mm}
\end{array}\right|
<6-21> (6.1.13)
\mbox{det}(M_2)=\left|\begin{array}{cc}
p_{11} & p_{12} \\ \\
p_{21} & p_{22}
\end{array}\right|=p_{11}p_{22}-p_{12}p_{21}
<6-22> (6.1.14)
\begin{array}{rl}
\mbox{det}(M_3)&=\left|\begin{array}{ccc}
p_{11} & p_{12} & p_{13} \\ \\
p_{21} & p_{22} & p_{23} \\ \\
p_{31} & p_{32} & p_{33}
\end{array}\right| \\ \\
&=p_{11}p_{22}p_{33}+p_{12}p_{23}p_{31}+p_{13}p_{21}p_{32}
\\ \\
&-p_{13}p_{22}p_{31}-p_{12}p_{21}p_{33}-p_{11}p_{23}p_{32}
\end{array}
<6-23> (6.1.15)
A^*=(\overline{A})^T=\overline{(A^T)}
<6-24>
A=\left(\begin{array}{cc}
\cos\theta & \sin\theta \\ \\
-\sin\theta & \cos\theta
\end{array}\right)
<6-25>
A^*=\left(\begin{array}{cc}
\cos\theta & -\sin\theta \\ \\
\sin\theta & \cos\theta
\end{array}\right)
<6-26>
H=\left(\begin{array}{cc}
\cos\theta & i\sin\theta \\ \\
-i\sin\theta & \cos\theta
\end{array}\right)
<6-27>
H^*=\left(\begin{array}{cc}
\cos\theta & i\sin\theta \\ \\
-i\sin\theta & \cos\theta
\end{array}\right)
<6-28>
B=\left(\begin{array}{cc}
0 & 2 \\ \\
1 & 0
\end{array}\right)
<6-29>
\left\{\begin{array}{l}
AB=\left(\begin{array}{cc}
\sin\theta & 2\cos\theta \\ \\
\cos\theta & -2\sin\theta
\end{array}\right) \\ \\
BA=\left(\begin{array}{cc}
-2\sin\theta & \cos\theta \\ \\
\cos\theta & \sin\theta
\end{array}\right)
\end{array}\right.
<6-30>
\begin{array}{l}
A^T=\left(\begin{array}{cc}
\cos\theta & -\sin\theta \\ \\
\sin\theta & \cos\theta
\end{array}\right) \\ \\
B^T=\left(\begin{array}{cc}
0 & 1 \\ \\
2 & 0
\end{array}\right)
\end{array}\right.
<6-31>
\left\{\begin{array}{l}
(AB)^T=\left(\begin{array}{cc}
\sin\theta & \cos\theta \\ \\
2\cos\theta & -2\sin\theta
\end{array}\right) \\ \\
B^TA^T=\left(\begin{array}{cc}
\sin\theta & \cos\theta \\ \\
2\cos\theta & -2\sin\theta
\end{array}\right)
\end{array}\right.
<6-32>
(AB)^T=B^TA^T
<6-33> (6.1.16)
\begin{array}{l}
\left\{\begin{array}{l}
a_{11}x+a_{12}y=b_1 \\ \\
a_{21}x+a_{22}y=b_2
\end{array}\right., \\ \\
\mbox{with }\quad
a_{11}a_{22}-a_{12}a_{21}\ne0
\quad\mbox{assumed.}
\end{array}
<6-34> (6.1.17)
\begin{array}{l}
\displaystyle{x=
\frac{a_{22}b_{1}-a_{12}b_2}
{a_{11}a_{22}-a_{12}a_{21}}}
\\ \\
\displaystyle{y=
\frac{a_{11}b_{2}-a_{21}b_1}
{a_{11}a_{22}-a_{12}a_{21}}}
\end{array}
<6-35> (6.1.18)
\begin{array}{l}
A=\left(\begin{array}{cc}
a_{11} & a_{12} \\ \\ a_{21} & a_{22}
\end{array}\right) \\ \\
X=\left(\begin{array}{c}
x \\ \\ y
\end{array}\right) \\ \\
B=\left(\begin{array}{c}
b_1 \\ \\ b_2
\end{array}\right)
\end{array}
<6-36> (6.1.19)
AX=B
<6-37> (6.1.20)
X=A^{-1}B
<6-38> (6.1.21)
A^{-1}=\left[\mbox{det}(A)\right]^{-1}
\left(\begin{array}{cc}
a_{22} & -a_{12} \\ \\ -a_{21} & a_{11}
\end{array}\right)
<6-39>
\mbox{det}(A)=a_{11}a_{22}-a_{12}a_{21}
<6-40> (6.1.22)
X=\frac{1}{a_{11}a_{22}-a_{12}a_{21}}
\left(\begin{array}{c}
a_{22}b_1-a_{12}b_2 \\ \\ -a_{21}b_1+a_{11}b_2
\end{array}\right)
<6-41> (6.1.23)
\begin{array}{l}
\displaystyle{x=
\frac{a_{22}b_{1}-a_{12}b_2}
{a_{11}a_{22}-a_{12}a_{21}}}
\\ \\
\displaystyle{y=
\frac{a_{11}b_{2}-a_{21}b_1}
{a_{11}a_{22}-a_{12}a_{21}}}
\end{array}
<6-42> (6.1.24)
\left\{\begin{array}{l}
a_{11}x_1+a_{12}x_2+\cdots+a_{1N}x_N=b_1 \\ \\
a_{21}x_1+a_{22}x_2+\cdots+a_{2N}x_N=b_2 \\ \\
\cdots \\ \\
a_{N1}x_1+a_{N2}x_2+\cdots+a_{NN}x_N=b_N
\end{array}\right.
<6-43> (6.1.25)
X=A^{-1}B
<6-44> (6.1.26)
\begin{array}{l}
X=\left(\begin{array}{c}
x_1 \\ \\ x_2 \\ \\ \cdots \\ \\ x_N
\end{array}\right) \\ \\
A=\left(\begin{array}{cccc}
a_{11} & a_{12} & \cdots & a_{1N} \\ \\
a_{21} & a_{22} & \cdots & a_{2N} \\ \\
\cdots & \cdots & \cdots & \cdots \\ \\
a_{N1} & a_{N2} & \cdots & a_{NN}
\end{array}\right),\quad B=\left(\begin{array}{c}
b_1 \\ \\ b_2 \\ \\ \cdots \\ \\ b_N
\end{array}\right)
\end{array}
<6-45> (6.2.1)
\left(\begin{array}{c}
2 \\ \\ 3
\end{array}\right)=\vec{a}
<6-46> (6.2.2)
\mbox{unit vector }(\vec{i},\vec{j}):\quad
\vec{i}=\left(\begin{array}{c}
1 \\ \\ 0
\end{array}\right),\quad\vec{j}=
\left(\begin{array}{c}
0 \\ \\ 1
\end{array}\right)
<6-47> (6.2.3)
\mbox{unit vector }(\vec{e}_r,\vec{e}_\theta):\quad
\vec{e}_r=\displaystyle{\frac{1}{\sqrt{2}}}
\left(\begin{array}{c}
1 \\ \\ i
\end{array}\right),\quad
\vec{e}_\theta=
\displaystyle{\frac{1}{\sqrt{2}}}
\left(\begin{array}{c}
1 \\ \\ -i
\end{array}\right)
<6-48> (6.2.4)
\begin{array}{rl}
\vec{a} &=\vec{i}2+\vec{j}3 \\ \\
&=\displaystyle{
\vec{e}_r\left(\frac{2-3i}{\sqrt{2}}\right)
+\vec{e}_\theta\left(\frac{2+3i}{\sqrt{2}}}\right)
\end{array}
<6-49> (6.2.5)
\vec{a}=\vec{i}a_1+\vec{j}a_2
<6-50> (6.2.6)
\left\{\begin{array}{l}
a_1=a\cos\theta \\ \\
a_2=a\sin\theta
\end{array}\right.
<6-51> (6.2.7)
\left\{\begin{array}{l}
\displaystyle{a=\sqrt{a_1^2+a_2^2}} \\ \\
\displaystyle{\tan\theta=\frac{a_2}{a_1}}\quad
\mbox{or}\quad
\displaystyle{\theta
=\tan^{-1}\left(\frac{a_2}{a_1}\right)}
\end{array}\right.
<6-52> (6.2.8)
\vec{a}=\left(\begin{array}{c}
a_1 \\ \\ a_2
\end{array}\right),\quad \vec{b}
=\left(\begin{array}{c}
b_1 \\ \\ b_2
\end{array}\right)
<6-53> (6.2.9)
\left\{\begin{array}{l}
\vec{a}=\vec{i}a_1+\vec{j}a_2 \\ \\
\vec{b}=\vec{i}b_1+\vec{j}b_2
\end{array}\right.
<6-54> (6.2.10)
\vec{a}\pm\vec{b}=\vec{i}(a_1\pm b_1)+
\vec{j}(a_2\pm b_2)
<6-55> (6.2.11)
\vec{a}=\left(\begin{array}{c}
1 \\ \\ 2
\end{array}\right),\quad \vec{b}
=\left(\begin{array}{c}
2 \\ \\ 1
\end{array}\right)
<6-56> (6.2.12)
\vec{c}=\vec{a}+\vec{b}=\left(\begin{array}{c}
3 \\ \\ 3
\end{array}\right)
<6-57> (6.2.13)
\vec{a}-\vec{b}=\vec{a}+(-\vec{b})
<6-58> (6.2.14)
\vec{a}\cdot\vec{b}=a_1b_1+a_2b_2
<6-59> (6.2.15)
\left\{\begin{array}{l}
(\vec{i}\cdot\vec{i})=(\vec{j}\cdot\vec{j})=1
\\ \\
(\vec{i}\cdot\vec{j})=(\vec{j}\cdot\vec{i})=0
\end{array}\right.
<6-60>
\begin{array}{rl}
(\vec{a}\cdot\vec{b})&=(\vec{i}a_1
+\vec{j}a_2)\cdot(\vec{i}b_1+\vec{j}b_2) \\ \\
&=(\vec{i}\cdot\vec{i})a_1b_1
+(\vec{i}\cdot\vec{j})a_1b_2
+(\vec{j}\cdot\vec{i})a_2b_1
+(\vec{j}\cdot\vec{j})a_2b_2 \\ \\
&=a_1b_1+a_2b_2
\end{array}
<6-61> (6.2.16)
\left\{\begin{array}{l}
(a_1=a\cos\theta_a,a_2=a\sin\theta_a) \\ \\
(b_1=b\cos\theta_b,b_2=b\sin\theta_b)
\end{array}\right.
<6-62> (6.2.17)
\begin{array}{rl}
(\vec{a}\cdot\vec{b})
&=ab(\cos\theta_a\cos\theta_b
+\sin\theta_a\sin\theta_b \\ \\
&=ab\cos(\theta_a-\theta_b) \\ \\
&=ab\cos\theta
\end{array}
<6-63> (6.2.18)
\cos(\theta_1\pm\theta_2)=
\cos\theta_1\cos\theta_2
\mp\sin\theta_1\sin\theta_2
<6-64> (6.2.19)
\vec{a}\cdot\vec{b}=a(b\cos\theta)
=b(a\cos\theta)
<6-65> (6.2.20)
a=\sqrt{\vec{a}\cdot\vec{a}}
<6-66> (6.2.21)
\vec{a}=\vec{i}a_1+\vec{j}a_2
<6-67> (6.2.22)
\left\{\begin{array}{l}
(\vec{i}\cdot\vec{i})=(\vec{j}\cdot\vec{j})
=1 \\ \\
(\vec{i}\cdot\vec{j})=(\vec{j}\cdot\vec{i})
=0
\end{array}\right.
<6-68> (6.2.23)
\left\{\begin{array}{l}
\vec{a}=\vec{i}a_1+\vec{j}a_2 \\ \\
\vec{b}=\vec{i}b_1+\vec{j}b_2
\end{array}\right.
<6-69> (6.2.24)
\begin{array}{rl}
\vec{a}\cdot\vec{b}&=
(\vec{i}a_1+\vec{j}a_2)\cdot
(\vec{i}b_1+\vec{j}b_2) \\ \\
&=a_1b_1+a_2b_2
\end{array}\right.
<6-70> (6.2.25)
\vec{r}=\vec{i}x+\vec{j}y
<6-71> (6.2.26)
\left\{\begin{array}{l}
x'=x\cos\theta-y\sin\theta \\ \\
y'=x\sin\theta+y\cos\theta
\end{array}\right.
<6-72>
\mbox{(a)}\quad\left\{\begin{array}{l}
x=r\cos\theta_0 \\ \\
y=r\sin\theta_0
\end{array}\right.
<6-73>
\mbox{(b)}\quad\left\{\begin{array}{rl}
x'&=r\cos(\theta_0+\theta) \\ \\
&=r(\cos\theta_0\cos\theta-
\sin\theta_0\sin\theta) \\ \\
&=x\cos\theta-y\sin\theta \\ \\
y'&=r\sin(\theta_0+\theta) \\ \\
&=r(\sin\theta_0\cos\theta-
\cos\theta_0\sin\theta) \\ \\
&=y\cos\theta+x\sin\theta
\end{array}\right.
<6-74>
\mbox{(c)}\quad\left\{\begin{array}{l}
\sin(a\pm b)=\sin a\cos b\pm\cos a\sin b \\ \\
\cos(a\pm b)=\cos a\cos b\mp\sin a\sin b
\end{array}\right.
<6-75> (6.2.27)
\left(\begin{array}{c}
x' \\ \\ y'
\end{array}\right)=\left(\begin{array}{cc}
\cos\theta & -\sin\theta \\ \\
\sin\theta & \cos\theta
\end{array}\right)\left(\begin{array}{c}
x \\ \\ y
\end{array}\right)
<6-76> (6.2.28)
R(\theta)=\left(\begin{array}{cc}
\cos\theta & -\sin\theta \\ \\
\sin\theta & \cos\theta
\end{array}\right)
<6-77> (6.2.29)
\left(\begin{array}{c}
x \\ \\ y
\end{array}\right)\rightarrow
\left(\begin{array}{c}
x' \\ \\ y'
\end{array}\right)=R(\theta_1)
\left(\begin{array}{c}
x \\ \\ y
\end{array}\right)
<6-78> (6.2.30)
\left(\begin{array}{c}
x' \\ \\ y'
\end{array}\right)\rightarrow
\left(\begin{array}{c}
x'' \\ \\ y''
\end{array}\right)=R(\theta_2)
\left(\begin{array}{c}
x' \\ \\ y'
\end{array}\right)=R(\theta_2)R(\theta_1)
\left(\begin{array}{c}
x \\ \\ y
\end{array}\right)
<6-79> (6.2.31)
R(\theta_2)R(\theta_1)=
\left(\begin{array}{cc}
\cos\theta_2 & -\sin\theta_2 \\ \\
\sin\theta_2 & \cos\theta_2
\end{array}\right)\left(\begin{array}{cc}
\cos\theta_1 & -\sin\theta_1 \\ \\
\sin\theta_1 & \cos\theta_1
\end{array}\right)
<6-80> (6.2.32)
\begin{array}{l}
R(\theta_2)R(\theta_1) \\ \\
=\left(\begin{array}{cc}
(\cos\theta_1\cos\theta_2-
\sin\theta_1\sin\theta_2)
& (-\sin\theta_1\cos\theta_2-
\cos\theta_1\sin\theta_2) \\ \\
(\cos\theta_1\sin\theta_2+
\sin\theta_1\cos\theta_2)
& (-\sin\theta_1\sin\theta_2+
\cos\theta_1\cos\theta_2)
\end{array}\right)
<6-81> (6.2.33)
\begin{array}{l}
R(\theta_2)R(\theta_1) \\ \\
=\left(\begin{array}{cc}
\cos(\theta_1+\theta_2) &
-\sin(\theta_1+\theta_2) \\ \\
\sin(\theta_1+\theta_2) &
\cos(\theta_1+\theta_2)
\end{array}\right)
<6-82> (6.2.34)
\vec{r}(t)=\vec{i}x(t)+\vec{j}y(t)
<6-83> (6.2.35)
\vec{v}(t)=\vec{i}\frac{dx(t)}{dt}
+\vec{j}\frac{dy(t)}{dt}
<6-84> (6.2.36)
\vec{v}(t)=\frac{d\vec{r}(t)}{dt}
<6-85> (6.2.37)
\vec{v}(t)=\vec{i}v_x(t)+\vec{j}v_y(t)
<6-86> (6.2.38)
v(t)=\sqrt{v_x(t)^2+v_y(t)^2}
<6-87> (6.2.39)
\vec{a}(t)=\frac{d\vec{v}(t)}{dt}=
\frac{d^2\vec{r}(t)}{dt^2}
<6-88> (6.2.40)
\vec{a}(t)=\vec{i}a_x(t)+\vec{j}a_y(t)
<6-89>
a(t)=\sqrt{a_x(t)^2+a_y(t)^2}
<6-90>
v(t)=\sqrt{v_x(t)^2+v_y(t)^2}
<6-91>
\vec{a}(t)=\frac{d\vec{v}(t)}{dt}
<6-92>
a(t)\ne\frac{dv(t)}{dt}
<6-93> (6.2.41)
\vec{r}=\vec{i}x+\vec{j}y
<6-94> (6.2.42)
\begin{array}{rl}
\displaystyle{\vec{i}\frac{\partial A}{\partial x}+
\vec{j}}\frac{\partial A}{\partial y}
&=\displaystyle{\left(
\vec{i}\frac{\partial}{\partial x}
+\vec{j}\frac{\partial}{\partial y}\right)
A(\vec{r},t)}\\ \\
&\displaystyle{\equiv\nabla A\quad\mbox{or}
\quad \mbox{grad}A}
\end{array}
<6-95> (6.2.43)
\nabla=\vec{i}\frac{\partial}{\partial x}+
\vec{j}\frac{\partial}{\partial y}
<6-96> (6.2.44)
\left\{\begin{array}{l}
T_x(\vec{r})=T(\vec{r})\cos\theta \\ \\
T_y(\vec{r})=T(\vec{r})\sin\theta
\end{array}\right.
<6-97> (6.2.45)
\vec{r}=\vec{i}T_x(\vec{r})+\vec{j}T_y(\vec{r})
<6-98>
\left(\begin{array}{cc}
\displaystyle{\frac{\partial T_x}{\partial x}}
& \displaystyle{\frac{\partial T_x}{\partial y}} \\ \\
\displaystyle{\frac{\partial T_y}{\partial x}}
& \displaystyle{\frac{\partial T_y}{\partial y}}
\end{array}\right)
<6-99> (6.2.46)
\begin{array}{rl}
\mbox{div}\vec{T}(\vec{r})
&=\displaystyle{
\frac{\partial T_x(\vec{r})}{\partial x}+
\frac{\partial T_y(\vec{r})}{\partial y}}\\ \\
&=\displaystyle{
\frac{\partial}{\partial x}T_x(\vec{r})+
\frac{\partial}{\partial y}T_y(\vec{r})} \\ \\
&=\displaystyle{
\left(\vec{i}\frac{\partial}{\partial x}+
\vec{j}\frac{\partial}{\partial y}\right)}
\cdot\left(\vec{i}T_x+\vec{j}T_y\right)
\end{array}
<6-100> (6.2.47)
\mbox{div}\vec{T}(\vec{r})
=\nabla\cdot\vec{T}
<6-101> (6.3.1)
\left\{\begin{array}{l}
\vec{i}\cdot\vec{i}=\vec{j}\cdot\vec{j}
=\vec{k}\cdot\vec{k}=
1 \\ \\
\vec{i}\cdot\vec{j}=\vec{j}\cdot\vec{k}
=\vec{k}\cdot\vec{i}=0
\end{array}\right.
<6-102> (6.3.2)
\left\{\begin{array}{l}
\vec{i}\times\vec{i}=\vec{j}\times\vec{j}
=\vec{k}\times\vec{k}=
0 \\ \\
\vec{i}\times\vec{j}=\vec{k},
\;\vec{j}\times\vec{k}=\vec{i},\;
\vec{k}\times\vec{i}=\vec{j}
\end{array}\right.
<6-103>
\vec{b}\times\vec{a}=-\vec{a}\times\vec{b}
<6-104> (6.3.3)
\vec{T}=\vec{i}T_1+\vec{j}T_2+\vec{k}T_3
<6-105> (6.3.4)
\left\{\begin{array}{rl}
\vec{a}\cdot\vec{b}&=(\vec{i}a_1
+\vec{j}a_2+\vec{k}a_3)\cdot
(\vec{i}b_1+\vec{j}b_2+\vec{k}b_3) \\ \\
&=a_1b_1+a_2b_2+a_3b_3
\end{array}\right.
<6-106> (6.3.5)
\vec{a}\cdot\vec{b}=ab\cos\theta
<6-107> (6.3.6)
\left\{\begin{array}{rl}
\vec{a}\times\vec{b}&=(\vec{i}a_1
+\vec{j}a_2+\vec{k}a_3)\times
(\vec{i}b_1+\vec{j}b_2+\vec{k}b_3) \\ \\
&=\vec{i}(a_2b_3-a_3b_2)
+\vec{j}(a_3b_1-a_1b_3)+
\vec{k}(a_1b_2-a_2b_1)
\end{array}\right.
<6-108> (6.3.7)
\vec{r}=\vec{i}x+\vec{j}y+\vec{k}z
<6-109> (6.3.8)
\nabla A(\vec{r})
=\vec{i}\frac{\partial A}{\partial x}
+\vec{j}\frac{\partial A}{\partial y}}
+\vec{k}\frac{\partial A}{\partial z}
<6-110> (6.3.9)
\vec{T}(\vec{r})=\vec{i}T_1(\vec{r})
+\vec{j}T_2(\vec{r})+\vec{k}T_3(\vec{r})
<6-111> (6.3.10)
\mbox{div}\vec{T}(\vec{r})=
\frac{\partial T_1}{\partial x}+
\frac{\partial T_2}{\partial y}}+
\frac{\partial T_3}{\partial z}
<6-112> (6.3.11)
\nabla=\vec{i}\frac{\partial}{\partial x}+
\vec{j}\frac{\partial}{\partial y}+
\vec{k}\frac{\partial}{\partial z}
<6-113> (6.3.12)
\mbox{div}\vec{T}(\vec{r})
=\nabla\cdot\vec{T}(\vec{r})
<6-114>
\begin{array}{rl}
\mbox{rot}\vec{T}(\vec{r})&=
\nabla\times\vec{T}(\vec{r}) \\ \\
&=\left(\vec{i}\frac{\partial}{\partial x}+
\vec{j}\frac{\partial}{\partial y}
+\vec{k}\frac{\partial}{\partial z}\right) \\ \\
&\quad\times\left(\vec{i}T_x}+\vec{j}T_y
+\vec{k}T_z\right) \\ \\
&=(\vec{i}\times
\vec{i})\frac{\partial T_x}{\partial x}
+(\vec{i}\times
\vec{j})\frac{\partial T_y}{\partial x}
+(\vec{i}\times
\vec{k})\frac{\partial T_z}{\partial x} \\ \\
&=(\vec{j}\times\vec{i})
\frac{\partial T_x}{\partial y}+
(\vec{j}\times\vec{j})
\frac{\partial T_y}{\partial y}
+(\vec{j}\times\vec{k})
\frac{\partial T_z}{\partial y} \\ \\
&=(\vec{k}\times\vec{i})
\frac{\partial T_x}{\partial z}+
(\vec{k}\times\vec{j})
\frac{\partial T_y}{\partial z}
+(\vec{k}\times\vec{k})
\frac{\partial T_z}{\partial z}
\end{array}
<6-115> (6.3.13)
\mbox{rot}\vec{T}(\vec{r})=\vec{i}\left[
\frac{\partial T_z}{\partial y}-
\frac{\partial T_y}{\partial z}\right]
+\vec{j}\left[\frac{\partial T_x}{\partial z}
-\frac{\partial T_z}{\partial x}\right]
+\vec{k}\left[\frac{\partial T_y}{\partial x}
-\frac{\partial T_x}{\partial y}\right]
<6-116>
\mbox{grad}f(\vec{r})=\nabla f(\vec{r})
=\vec{i}\frac{\partial f}{\partial x}
+\vec{j}\frac{\partial f}{\partial y}
+\vec{k}\frac{\partial f}{\partial z}
<6-117>
\mbox{div}\vec{a}(\vec{r})
=\nabla\cdot\vec{a}(\vec{r})
=\frac{\partial a_x}{\partial x}
+\frac{\partial a_y}{\partial y}
+\frac{\partial a_z}{\partial z}
<6-118>
\begin{array}{rl}
\mbox{rot}\vec{a}(\vec{r})&=
\nabla\times\vec{a}(\vec{r}) \\ \\
&=\displaystyle{
\vec{i}\left(\frac{\partial a_z}{\partial y}-
\frac{\partial a_y}{\partial z}\right)
+\vec{j}\left(\frac{\partial a_x}{\partial z}-
\frac{\partial a_z}{\partial x}\right)
+\vec{k}\left(\frac{\partial a_y}{\partial x}-
\frac{\partial a_x}{\partial y}\right)}
\end{array}
<6-119> (6.3.14)
\frac{d}{dt}(f\vec{a})
=\frac{df}{dt}\vec{a}+f\frac{d\vec{a}}{dt}
<6-120> (6.3.15)
\frac{d(\vec{a}\cdot\vec{b})}{dt}
=\frac{d\vec{a}}{dt}\cdot\vec{b}+
\vec{a}\cdot\frac{d\vec{b}}{dt}
<6-121> (6.3.16)
\frac{d[\vec{a}\times\vec{b}]}{dt}=
\frac{d\vec{a}}{dt}\times\vec{b}
+\vec{a}\times\frac{d\vec{b}}{dt}
<6-122>
\vec{a}\cdot\vec{a}
<6-123> (6.3.17)
\vec{a}\cdot\vec{a}=\mbox{constant}
\quad\Rightarrow\quad
\vec{a}\cdot\frac{\partial\vec{a}}{\partial t}
=0
<6-124> (6.3.18)
\mbox{rot}\vec{r}=\nabla\times\vec{r}=0
<6-125>
\vec{a}(\vec{r},t)=\nabla f(\vec{r},t)
<6-126> (6.3.19)
\mbox{rot}\vec{a}=\nabla\times\vec{a}=0
<6-127> (6.3.20)
\begin{array}{rl}
\nabla\cdot(\nabla f)
&=\displaystyle{\frac{\partial^2f}{\partial x^2}
+\frac{\partial^2f}{\partial y^2}+
\frac{\partial^2f}{\partial z^2}} \\ \\
&=\displaystyle{
\left(\frac{\partial^2}{\partial x^2}
+\frac{\partial^2}{\partial y^2}
+\frac{\partial^2}{\partial z^2}\right)f} \\ \\
&\displaystyle{\equiv\nabla f}
\end{array}
<6-128> (6.3.21)
\Delta\left(\frac{1}{r}\right)=0
<6-129> (6.3.22)
\nabla(fg)=f(\nabla g)+(\nabla f)g
<6-130> (6.3.23)
\nabla\cdot(f\vec{a})=f(\nabla g)+(\nabla f)g
<6-131> (6.3.24)
\nabla\times(f\vec{a})=(\nabla f)\times\vec{a}
+f(\nabla\times\vec{a})
<6-132> (6.3.25)
\nabla\cdot(\vec{a}\times\vec{b})=
(\nabla\tiems\vec{a})\cdot\vec{b}-
\vec{a}\cdot(\nabla\times\vec{b})
<6-133> (6.3.26)
\nabla\times[\vec{a}\times\vec{b}]
=(\vec{b}\cdot\nabla)\vec{a}
-\vec{b}(\nabla\cdot\vec{a})
-(\vec{a}\cdot\nabla)\vec{b}
+\vec{a}(\nabla\cdot\vec{b})
<6-134> (6.3.27)
\nabla(\vec{a}\cdot\vec{b})
=(\vec{b}\cdot\nabla)\vec{a}
+(\vec{a}\cdot\nabla)\vec{b}+
\vec{b}\times(\nabla\times\vec{a})
+\vec{a}\times(\nabla\times\vec{b})
<6-135> (6.3.28)
\nabla\times(\nabla f)
=\mbox{rot}(\mbox{grad}f)=0
<6-136> (6.3.29)
\nabla\cdot(\nabla\times\vec{a})
=\mbox{div}(\mbox{rot}\vec{a})=0
<6-137> (6.3.30)
\begin{array}{l}
\displaystyle{\nabla\times[\nabla\times\vec{a}]
=\nabla(\nabla\cdot\vec{a})-\nabla^2\vec{a}} \\ \\
\mbox{where}\quad\displaystyle{\nabla^2\vec{a}
=\frac{\partial^2\vec{a}}{\partial x^2}
+\frac{\partial^2\vec{a}}{\partial y^2}
+\frac{\partial^2\vec{a}}{\partial z^2}}
\end{array}
<7-1> (7.1.1)
F(x)=\int f(x)dx
<7-2> (7.1.2)
f(x)=f(g(y))
<7-3> (7.1.3)
dx=\frac{dx}{dy}dy=\frac{dg(y)}{dy}dy
<7-4> (7.1.4)
F(x)=\int\left[f(g(y))\frac{dg(y)}{dy}\right]dy
<7-5> (7.1.5)
dx=\frac{dx}{dy}dy
<7-6>
\begin{array}{rl}
I&=\displaystyle{\int x^3dx} \\ \\
&=\displaystyle{\int(\pm\sqrt{y})^3
\left(\pm\frac{1}{2\sqrt{y}}\right)dy} \\ \\
&=\displaystyle{\frac{1}{2}\int ydy} \\ \\
&=\displaystyle{\frac{1}{2}\frac{y^2}{2}} \\ \\
&=\displaystyle{\frac{y^2}{4}} \\ \\
&=\displaystyle{\frac{x^4}{4}}
\end{array}
<7-7> (7.1.6)
\left\{\begin{array}{rl}
\sin 2x&=2\sin x\cos x \\ \\
\cos 2x&=\cos^2x-\sin^2x \\ \\
&=2\cos^2x-1 \\ \\
&=1-2\sin^2x
\end{array}\right.
<7-8> (7.1.7)
\sin^2x=\displaystyle{\frac{1}{2}(1-\cos2x)}
<7-9>
\begin{array}{rl}
I&=\displaystyle{\int\sin^2xdx} \\ \\
&=\displaystyle{
\frac{1}{2}\int\left[1-\cos(2x)\right]dx} \\ \\
&=\displaystyle{
\frac{1}{2}\left[\int1dx-\int\cos(2x)dx\right]}
\end{array}
<7-10>
\begin{array}{rl}
\displaystyle{\int\cos(2x)dx}&=
\displaystyle{\int\cos y\frac{dx}{dy}dy} \\ \\
&=\displaystyle{\frac{1}{2}\int\cos ydy} \\ \\
&=\displaystyle{\frac{1}{2}\sin y}
\end{array}
<7-11>
I=\frac{x}{2}-\frac{1}{4}\sin 2x
<7-12> (7.1.8)
I=\int\sqrt{a^2+x^2}dx
<7-13>
1+\tan^2y=\frac{1}{\cos^2y}
<7-14>
\begin{array}{rl}
\displaystyle{\sqrt{a^2+x^2}} &=
a\displaystyle{\sqrt{1+\tan^2y}} \\ \\
&=\displaystyle{\frac{a}{\cos y}}
\end{array}
<7-15>
\begin{array}{rl}
\displaystyle{\frac{dx}{dy}}&=
a\displaystyle{\frac{d\tan y}{dy}} \\ \\
&=\displaystyle{\frac{a}{\cos^2y}}
\end{array}
<7-16>
I=a^2\int\frac{1}{\cos^3y}dy
<7-17>
\frac{1}{\cos^3y}=
\displaystyle{\frac{1}{\left(\sqrt{1-z^2}\right)^3}}
<7-18>
\begin{array}{rl}
\displaystyle{\frac{dy}{dz}}&=
\displaystyle{
\frac{1}{\displaystyle{\frac{dz}{dy}}}} \\ \\
&=\displaystyle{\frac{1}{\cos y}} \\ \\
&=\displaystyle{\frac{1}{\sqrt{1-z^2}}}
\end{array}
<7-19>
\begin{array}{rl}
I&=\displaystyle{
a^2\int\frac{1}{(\sqrt{1-z^2})^3}
\frac{1}{\sqrt{1-z^2}}dz} \\ \\
&=\displaystyle{
a^2\int\frac{1}{(1-z^2)^2}dz}
\end{array}
<7-20>
\begin{array}{rl}
\displaystyle{\frac{1}{(1-z^2)^2}}
&=\displaystyle{\left[\frac{1}{2}
\left(\frac{1}{1+z}
+\frac{1}{1-z}\right)\right]^2} \\ \\
&=\displaystyle{
\frac{1}{4}\left[\frac{1}{(1+z)^2}+
\frac{1}{(1-z)^2}+\frac{2}{1-z^2}\right]}
\end{array}
<7-21>
\begin{array}{rl}
\displaystyle{\int\frac{1}{(1+z^2)}dz
+\int\frac{1}{(1-z^2)}dz}
&=\displaystyle{-\frac{1}{1+z}
+\frac{1}{1-z}\right]} \\ \\
&=\displaystyle{\frac{2z}{1-z^2}}
\end{array}
<7-22>
\begin{array}{rl}
\displaystyle{\frac{2}{1-z^2}}
&=\displaystyle{\frac{1}{1-z}
+\frac{1}{1+z}} \\ \\
&=\displaystyle{\frac{1}{z+1}
-\frac{1}{z-1}}
\end{array}
<7-23>
\begin{array}{rl}
\displaystyle{\int\frac{2}{1-z^2}dz}
&=\displaystyle{\int\frac{1}{z+1}dz
-\int\frac{1}{z-1dz} \\ \\
&=\displaystyle{\ln|z+1|-\ln|z-1|} \\ \\
&=\displaystyle{\ln\left|
\frac{z+1}{z-1}\right|}
\end{array}
<7-24>
I=\frac{a^2}{4}\frac{2z}{1-z^2}+
\frac{a^2}{4}
\ln\left|\frac{z+1}{z-1}\right|
<7-25>
\begin{array}{rl}
x^2&=a^2\displaystyle{
\frac{\sin^2y}{\cos^2y}} \\ \\
&=a^2\displaystyle{\frac{\frac{z^2}{1-z^2}}
\end{array}
<7-26>
z=\frac{x}{\sqrt{a^2+x^2}}
<7-27>
\left\{\begin{array}{l}
\displaystyle{\frac{2z}{1-z^2}=
\frac{2x\sqrt{a^2+x^2}}{a^2}} \\ \\
\displaystyle{\frac{z+1}{z-1}=
\frac{\left(\sqrt{a^2+x^2}+x\right)^2}{a^2}}
\end{array}\right.
<7-28>
\begin{array}{rl}
\displaystyle{\int\sqrt{a^2+x^2}dx}&=
\displaystyle{\frac{x}{2}\sqrt{a^2+x^2}} \\ \\
&\quad+\displaystyle{
\frac{a^2}{2}\ln\left|\sqrt{a^2+x^2}
+x\right|-\frac{a^2}{2}\ln a}
\end{array}
<7-29> (7.1.9)
\frac{d\{f(x)g(x)\}}{dx}=\frac{df(x)}{dx}g(x)
+f(x)\frac{dg(x)}{dx}
<7-30>
f(x)g(x)=\int\frac{df(x)}{dx}g(x)dx
+\int f(x)\frac{dg(x)}{dx}dx
<7-31>
\int f(x)\frac{dg(x)}{dx}dx=f(x)g(x)
-\int\frac{df(x)}{dx}g(x)dx
<7-32>
I=\int x\cos xdx
<7-33>
\begin{array}{rl}
\displaystyle{\int x\cos xdx}&=
\displaystyle{x\sin x-\int\sin xdx} \\ \\
&=\displaystyle{x\sin x+\cos x}
\end{array}
<7-34>
I=\int x\ln xdx
<7-35>
\begin{array}{rl}
\displaystyle{\int x\ln xdx}&=
\displaystyle{\frac{x^2}{2}\ln x-
\int\frac{1}{x}\frac{x^2}{2}dx} \\ \\
&=\displaystyle{\frac{x^2}{2}\ln x
-\frac{1}{2}\int xdx} \\ \\
&=\displaystyle{\frac{x^2}{2}\ln x
-\frac{x^2}{4}}
\end{array}
<7-36> (7.2.1)
\int_b^a f(x)dx=F(a)-F(b)
<7-37> (7.2.2)
I=\int_C\vec{A}(\vec{r})\cdot d\vec{s}
<7-38> (7.2.3)
\vec{A}(\vec{r})=
\vec{i}A_x(x,y,z)+\vec{j}A_y(x,y,z)
+\vec{k}A_z(x,y,z)
<7-39> (7.2.4)
d\vec{s}=\vec{i}dx+\vec{j}dy+\vec{k}dz
<7-40>
\left\{\begin{array}{l}
dx=(\vec{i}\cdot d\vec{s}) \\ \\
dy=(\vec{j}\cdot d\vec{s}) \\ \\
dz=(\vec{k}\cdot d\vec{s})
\end{array}\right.
<7-41> (7.2.5)
I=\int_CA_x(x,y,z)dx+\int_CA_y(x,y,z)dy
+\int_CA_z(x,y,z)dz
<7-42> (7.2.6)
\vec{e}_r=\frac{\vec{r}}{r}
=\vec{i}\frac{x}{r}+\vec{j}\frac{y}{r}
<7-43>
\vec{A}(\vec{r})=\vec{e}_r\sqrt{x}
<7-44>
\left\{\begin{array}{l}
A_x=\displaystyle{\frac{x\sqrt{x}}{r}
=\frac{x}{\sqrt{x+1}}} \\ \\
A_y=\displaystyle{\frac{y\sqrt{x}}{r}
=\frac{y}{\sqrt{x+1}}
=\frac{y}{\sqrt{y^2+1}}}
\end{array}\right.
<7-45>
I=\int_0^3\frac{x}{\sqrt{x+1}}dx
+\int_0^{\sqrt{3}}\frac{y}{\sqrt{y^2+1}}dy
<7-46>
\left\{\begin{array}{l}
I_1=\displaystyle{
\int_0^3\frac{x}{\sqrt{x+1}}dx} \\ \\
I_2=\displaystyle{
\int_0^{\sqrt{3}}\frac{y}{\sqrt{y^2+1}}dy}
\end{array}\right.
<7-47>
dy=\frac{dy}{dx}dx=\frac{1}{2\sqrt{x}}dx
<7-48>
I_2=\int_0^3\frac{\sqrt{x}}{\sqrt{x+1}}
\frac{1}{2\sqrt{x}}dx=
\frac{1}{2}\int_0^3\frac{1}{\sqrt{x+1}}dx
<7-49>
\left\{\begin{array}{l}
\displaystyle{
\int\frac{x}{\sqrt{x+1}}dx}=
\displaystyle{\frac{2}{3}}(x+1)^{3/2}
-2(x+1)^{1/2}\equiv J_1(x) \\ \\
\displaystyle{
\int\frac{1}{\sqrt{x+1}}dx}=
2(x+1)^{1/2}\equiv J_2(x)
\end{array}\right.
<7-50>
\left\{\begin{array}{l}
I_1=J_1(3)-J_1(0)
=\displaystyle{\frac{8}{3}} \\ \\
I_2=\displaystyle{
\frac{1}{2}\left[J_2(3)-J_2(0)\right]
=1
\end{array}\right.
<7-51>
I=I_1+I_2=\frac{11}{3}
<7-52>
I_1=\int_{C_1}\vec{A}\cdot d\vec{s}
=\int_0^1xdx+\int_0^1xdy
<7-53>
\begin{array}{rl}
I_1&=\displaystyle{
\int_0^1xdx+\int_0^1ydy} \\ \\
&=\displaystyle{
\left[\frac{x^2}{2}\right]_0^1
+\left[\frac{y^2}{2}\right]_0^1=1}
\end{array}
<7-54>
\begin{array}{rl}
I_2&=\displaystyle{\int_{C_2}
\vec{A}(\vec{r})\cdot d\vec{s}} \\ \\
&=\displaystyle{\int_0^1 xdx+\int_0^1 xdy}
\end{array}
<7-55>
I_2=\int_0^1 xdx+\int_0^1 \sqrt{y}dy
<7-56>
\begin{array}{rl}
I_2&=\displaystyle{
\left[\frac{x^2}{2}\right]_0^1+
\frac{2}{3}\left[y^{3/2}\right]_0^1} \\ \\
=\displaystyle{\frac{7}{6}}
\end{array}
<7-57> (7.2.7)
\begin{array}{rl}
I&=\displaystyle{
\int_C\vec{A}\cdot d\vec{s}} \\ \\
&=\displaystyle{
\int_C P(x,y)dx+\int_C Q(x,y)dy}
\end{array}
<7-58> (7.2.8)
\frac{\partial P(x,y)}{\partial y}=
\frac{\partial Q(x,y)}{\partial x}
<7-59>
\left\{\begin{array}{l}
\displaystyle{\frac{\partial P(x,y)}{\partial y}=2xy} \\ \\
\displaystyle{\frac{\partial Q(x,y)}{\partial x}=2xy}
\end{array}\right.
<7-60>
\begin{array}{rl}
I_1&=\displaystyle{
\int_{C_1}\vec{A}\cdot d\vec{s}} \\ \\
&=\displaystyle{
\int_{C_1} A_xdx+\int_{C_1} A_ydy} \\ \\
&=\displaystyle{
\int_0^1xy^2dx+\int_0^1x^2ydy}
\end{array}
<7-61>
\begin{array}{rl}
I_1&=\displaystyle{
\int_0^1x^3dx+\int_0^1y^3dy} \\ \\
&=\displaystyle{\left[\frac{x^4}{4}\right]_0^1+
\left[\frac{y^4}{4}\right]_0^1
=\frac{1}{2}}
\end{array}
<7-62>
\begin{array}{rl}
I_2&=\displaystyle{
\int_{C_2}\vec{A}\cdot d\vec{s}} \\ \\
&=\displaystyle{
\int_0^1xy^2dx+\int_0^1x^2ydy}
\end{array}
<7-63>
\begin{array}{rl}
I_1&=\displaystyle{
\int_0^1x^5dx+\int_0^1y^2dy} \\ \\
&=\displaystyle{\left[\frac{x^6}{6}\right]_0^1
+\left[\frac{y^3}{3}\right]_0^1=\frac{1}{2}}
\end{array}
<7-64> (7.2.9)
\frac{\partial P(x,y)}{\partial y}=
\frac{\partial Q(x,y)}{\partial x}
<7-65> (7.2.10)
df(x,y)=P(x,y)dx+Q(x,y)dy
<7-66> (7.2.11)
\int_a^b df(x,y)=
\int_a^b P(x,y)dx+\int_a^b Q(x,y)dy
<7-67> (6.2.12)
\begin{array}{l}
f(x_2,y_2)-f(x_1,y_1) \\ \\
\quad=\displaystyle{
\int_a^b P(x.y)dx+\int_a^b Q(x.y)dy}
\end{array}
<7-68>
\vec{A}=\vec{i}x+\vec{j}x
<7-69>
P(x,y)=x
<7-70>
Q(x,y)=x
<7-71>
\frac{\partial P}{\partial y}=0
<7-72>
\frac{\partial Q}{\partial x}=1
<7-73>
\frac{\partial P}{\partial y}\ne
\frac{\partial Q}{\partial x}
<7-74>
\vec{A}=\vec{i}xy^2+\vec{j}x^2y
<7-75>
P(x,y)=xy^2
<7-76>
Q(x,y)=x^2y
<7-77>
\frac{\partial P}{\partial y}=2xy
<7-78>
\frac{\partial Q}{\partial x}=2xy
<7-79>
\frac{\partial P}{\partial y}=
\frac{\partial Q}{\partial x}
<7-80> (7.3.1)
A=\iint_S\phi(x,y,z)dS
<7-81>
2x+2y+z=2
<7-82>
I=\iint_S(\vec{r}\cdot\vec{n})dS
<7-83>
\vec{r}=\vec{i}x+\vec{j}y+\vec{k}z
<7-84>
\left\{\begin{array}{l}
\vec{a}=-\vec{i}+\vec{j} \\ \\
\vec{b}=-\vec{j}+2\vec{k}
\end{array}\right.
<7-85>
\begin{array}{rl}
\vec{a}\times\vec{b}
&=(-\vec{i})\times(-\vec{j})+
(-\vec{i})\times(2\vec{k}) \\ \\
&\quad+(\vec{j})\times(-\vec{j})+
(\vec{j})\times(2\vec{k}) \\ \\
&=\vec{k}+2\vec{j}+2\vec{i}
\end{array}
<7-86>
\left|2\vec{i}+2\vec{j}+\vec{k}\right|=
\sqrt{2^2+2^2+1^1}=3
<7-87> (7.3.2)
\vec{n}=\frac{1}{3}(2\vec{i}+2\vec{j}+\vec{k})
<7-88> (7.3.3)
\vec{n}(x,y,z)=
\frac{\nabla f(x,y,z)}{|\nabla f(x,y,z)|}
<7-89>
f(x,y,z)=2x+2y+z-2=0
<7-90>
\begin{array}{l}
\nabla f(x,y,z) \\ \\
\quad=\displaystyle{
\left(\vec{i}\frac{\partial}{\partial x}+
\vec{j}\frac{\partial}{\partial y}+
\vec{k}\frac{\partial}{\partial z}\right)}
(2x+2y+z-2) \\ \\
\quad=2\vec{i}+2\vec{j}+\vec{k}
\end{array}
<7-91>
\left|\nabla f(x,y,z)\right|=
\sqrt{2^2+2^2+1^1}=3
<7-92>
\vec{r}=\vec{i}x+\vec{j}y+\vec{k}z
<7-93>
\begin{array}{rl}
\vec{r}\cdot\vec{n}
&=\displaystyle{(\vec{i}x+
\vec{j}y+
\vec{k}z)\cdot\frac{1}{3}(2\vec{i}+
2\vec{j}+\vec{k})} \\ \\
&=\displaystyle{\frac{2x+2y+z}{3}}
\end{array}
<7-94>
2x+2y+z=2
<7-95>
Z=2-2x-2y
<7-96>
\vec{r}\cdot\vec{n}=\frac{2}{3}
<7-97>
\sqrt{(x_1-x_2)^2+(y_1-y_2)^2+
(z_1-z_2)^2}
<7-98>
I=\frac{2}{3}\times\frac{3}{2}=1
<7-99> (7.3.4)
\int_C\vec{A}(\vec{r})\cdot d\vec{s}=
\iint_S
\left[\mbox{rot}\vec{A}(\vec{r})\right]_ndS
<7-100>
\displaystyle{
\int_A^B\vec{A}(\vec{r})\cdot d\vec{s}=
I_C(A\rightarrow B)}
<7-101>
I_{C_1}(A\rightarrow B)=I_{C_2}(A\rightarrow B)
<7-102>
I_{C_1}(A\rightarrow B)=-I_{C_2}(B\rightarrow A)
<7-103>
I_{C_1}(A\rightarrow B)+
I_{C_2}(B\rightarrow A)=0
<7-104>
\displaystyle{
\oint_C\vec{A}(\vec{r})\cdot d\vec{s}=0}
<7-105>
\displaystyle{
\iint_S\left[\vec{A}(\vec{r})\right]_ndS=0}
<7-106>
\mbox{rot}\vec{A}(\vec{r})=0
<7-107>
\mbox{rot}\cdot\mbox{grad}f(\vec{r})=0
<7-108>
\vec{A}(\vec{r})=\mbox{grad}f(\vec{r})
<7-109> (7.3.5)
I=\iint_{S}f(x,y)dxdy
<7-110>
x_1\le x\le x_2
<7-111>
y_1\le y\le y_2
<7-112> (7.3.6)
I=\int_{x_1}^{x_2}dx\int_{y_1}^{y_2}dyf(x,y)
<7-113>
\frac{d}{dx}f(x)
<7-114>
\frac{df(x)}{dx}
<7-115>
\int_{x_1}^{x_2}dxf(x)
<7-116>
\int_{x_1}^{x_2}f(x)dx
<7-117> (7.3.7)
\left\{\begin{array}{l}
x=r\cos\theta \\ \\
y=r\sin\theta
\end{array}\right.
<7-118> (7.3.8)
I=\int_{-\infty}^{+\infty}dx
\int_{-\infty}^{+\infty}dyf(x,y)
<7-119> (7.3.9)
I=\int_{r_1}^{r_2}dr
\int_{\theta_1}^{\theta_2}d\theta
|J|f(r\cos\theta,r\sin\theta )
<7-120> (7.3.10)
\left(\begin{array}{cc}
\displaystyle{
\frac{\partial x}{\partial r}} &
\displaystyle{
\frac{\partial x}{\partial \theta}} \\ \\
\displaystyle{
\frac{\partial y}{\partial r}} &
\displaystyle{\frac{
\partial y}{\partial \theta}}
\end{array}\right)
<7-121>
\left(\begin{array}{cc}
\displaystyle{
\frac{\partial x}{\partial r}} &
\displaystyle{
\frac{\partial x}{\partial \theta}} \\ \\
\displaystyle{
\frac{\partial y}{\partial r}} &
\displaystyle{
\frac{\partial y}{\partial \theta}}
\end{array}\right)=
\left(\begin{array}{cc}
\cos\theta & -r\sin\theta \\ \\
\sin\theta &
r\cos\theta
\end{array}\right)
<7-122>
\begin{array}{rl}
J&=\left|\begin{array}{cc}
\cos\theta & -r\sin\theta \\ \\
\sin\theta &
r\cos\theta
\end{array}\right| \\ \\
&=r\cos^2\theta+r\sin^2\theta \\ \\
&=r
\end{array}
<7-123>
\left(\begin{array}{l}
x=-\infty\sim+\infty \\ \\
y=-\infty\sim+\infty
\end{array}\right)\quad\rightarrow\quad
\left(\begin{array}{l}
r=0\sim+\infty \\ \\
\theta=0\sim2\pi
\end{array}\right)
<7-124> (7.3.11)
I=\int_{0}^{\infty}dr\int_{0}^{2\pi}d\theta
rf(r\cos\theta,r\sin\theta)
<7-125> (7.3.12)
\int_{-\infty}^{+\infty}e^{-x^2}dx
=\sqrt{\pi}
<7-126>
\begin{array}{rl}
I^2&=\left(\int_{-\infty}^{+\infty}
e^{-x^2}dx\right)\times
\left(\int_{-\infty}^{+\infty}
e^{-y^2}dy\right) \\ \\
&=\int_{-\infty}^{+\infty}dx
\int_{-\infty}^{+\infty}dy
e^{-(x^2+y^2)}
\end{array}
<7-127>
x^2+y^2=r^2
<7-128>
\begin{array}{rl}
I^2&=\displaystyle{
\left(\int_{0}^{\infty}e^{-r^2}rdr\right)
\left(\int_{0}^{2\pi}d\theta\right)} \\ \\
&=\displaystyle{
2\pi\int_{0}^{\infty}e^{-r^2}rdr}
\end{array}
<7-129>
J=\frac{dr}{ds}=\frac{1}{2\sqrt{s}}
<7-130>
\begin{array}{rl}
I^2&=\displaystyle{
2\pi\int_{0}^{\infty}
e^{-s}|J|\sqrt{s}ds} \\ \\
&=\displaystyle{
\pi\int_{0}^{\infty}e^{-s}ds}
\end{array}
<7-131>
\int e^{-s}ds=-e^{-s}
<7-132>
\begin{array}{rl}
I^2&=\displaystyle{
-\pi\left[e^{-s}\right]_0^\infty} \\ \\
&=\pi
\end{array}
<7-133>
I=\sqrt{\pi}
<7-134>
\lim_{R\rightarrow\infty}\int_{0}^{R}rdr
\int_{0}^{2\pi}d\theta f\equiv
\lim_{R\rightarrow\infty}I(R)
<7-135>
\int_{0}^{R}rdr=\frac{R^2}{2}
<7-136>
\int_{0}^{2\pi}d\theta=2\pi
<7-137>
I(R)=\pi R^2
<7-138> (7.4.1)
I=\iiint_V\phi(x,y,z)dV
<7-139> (7.4.2)
\iint_S\vec{E}(\vec{r})\cdot\vec{n}(\vec{r})dS=
\iiint_V\mbox{div}\vec{E}(\vec{r})dV
<7-140>
\left\{\begin{array}{l}
x=r\sin\theta\cos\phi \\ \\
y=r\sin\theta\sin\phi \\ \\
z=r\cos\theta
\end{array}\right.
<7-141> (7.4.4)
\begin{array}{rl}
I&=\displaystyle{\int_{-\infty}^{+\infty}dx
\int_{-\infty}^{+\infty}dy
\int_{-\infty}^{+\infty}dzf(x,y,z)} \\ \\
&=\displaystyle{\int_{r_1}^{r_2}dr
\int_{\theta_1}^{\theta_2}d\theta
\int_{\phi_1}^{\phi_2}d\phi
|J|\tilde{f}(r,\theta,\phi)}
\end{array}
<7-142> (7.4.5)
J=\left|
\begin{array}{ccc}
\displaystyle{
\frac{\partial x}{\partial r}} &
\displaystyle{
\frac{\partial x}{\partial\theta}} &
\displaystyle{
\frac{\partial x}{\partial\phi}} \\ \\
\displaystyle{
\frac{\partial y}{\partial r} &
\displaystyle{
\frac{\partial y}{\partial\theta}} &
\displaystyle{
\frac{\partial y}{\partial\phi}} \\ \\
\displaystyle{
\frac{\partial z}{\partial r}} &
\displaystyle{
\frac{\partial z}{\partial\theta}} &
\displaystyle{
\frac{\partial z}{\partial\phi}}
\end{array}\right|
<7-143>
\begin{array}{l}
\left\{\begin{array}{l}
\displaystyle{
\frac{\partial x}{\partial r}=
\sin\theta\cos\phi} \\ \\
\displaystyle{
\frac{\partial x}{\partial\theta}=
r\cos\theta\cos\phi} \\ \\
\displaystyle{
\frac{\partial x}{\partial\theta}=
-r\sin\theta\sin\phi}
\end{array}\right. \\ \\
\left\{\begin{array}{l}
\displaystyle{
\frac{\partial y}{\partial r}=
\sin\theta\sin\phi} \\ \\
\displaystyle{
\frac{\partial x}{\partial\theta}=
r\cos\theta\sin\phi} \\ \\
\displaystyle{
\frac{\partial y}{\partial\theta}=
r\sin\theta\cos\phi}
\end{array}\right. \\ \\
\left\{\begin{array}{l}
\displaystyle{
\frac{\partial z}{\partial r}=
\cos\theta} \\ \\
\displaystyle{
\frac{\partial z}{\partial\theta}=
-r\sin\theta} \\ \\
\displaystyle{
\frac{\partial z}{\partial\theta}=0}
\end{array}\right.
\end{array}
<7-144>
\begin{array}{rl}
J&=\left|\begin{array}{ccc}
\sin\theta\cos\phi
& r\cos\theta\cos\phi
& -r\sin\theta\sin\phi \\ \\
\sin\theta\sin\phi
& r\cos\theta\sin\phi
& r\sin\theta\cos\phi \\ \\
\cos\theta
& -r\sin\theta & 0
\end{array}\right| \\ \\
&=r^2\sin\theta
\end{array}
<7-145>
\left\{\begin{array}{l}
x=-\infty\sim+\infty \\ \\
y=-\infty\sim+\infty \\ \\
z=-\infty\sim+\infty
\end{array}\right\}\Rightarrow
\left\{\begin{array}{l}
r=0\sim+\infty \\ \\
\theta=-\pi\sim\pi \\ \\
\phi=0\sim2\pi
\end{array}\right\}
<7-146> (7.4.6)
I=\int_{0}^{\infty}r^2dr
\int_{-\pi}^{\pi}\sin\theta d\theta
\int_{0}^{2\pi}d\phi\tilde{f}(r,\theta,\phi)
<7-147>
\int_{0}^{R}r^2dr
\int_{-\pi}^{\pi}\sin\theta d\theta
\int_{0}^{2\pi}d\phi
=\frac{4\pi}{3}R^3
<8-1> (8.1.1)
\left\{\begin{array}{l}
x=r\cos\theta \\ \\ y=r\sin\theta
\end{array}\right.
<8-2> (8.1.2)
z=re^{i\theta}
<8-3>
\frac{f(z)}{z-z_0}
<8-4>
I=\oint_C\frac{f(z)}{z-z_0}dz
<8-5> (8.1.3)
\frac{1}{2\pi i}\oint_C\frac{f(z)}{z-z_0}dz
=f(z_0)
<8-6> (8.1.4)
\int_{-\infty}^{\infty}\frac{1}{x^2+1}dx
=\pi
<8-7> (8.1.5)
\begin{array}{rl}
I(X)&=\displaystyle{
\int_{-X}^{+X}\frac{1}{x^2+1}dx}+
\displaystyle{
\int_{C'}\frac{1}{z^2+1}dz} \\ \\
&=\displaystyle{
\int_{C(X)}\frac{1}{z^2+1}dz}
\end{array}
<8-8>
x^2+a^2=(x+ia)(x-ia)
<8-9>
z^2+1=(z+i)(z-i)
<8-10> (8.1.6)
\begin{array}{rl}
I(X)&=\displaystyle{
\int_{C(X)}\frac{1}{(z+i)(z-i)}dz} \\ \\
&=\displaystyle{
\int_{C(X)}\frac{f(z)}{z-i}dz}
\end{array}
<8-11>
f(z)=\frac{1}{z+i}
<8-12>
\sqrt{(-i)\times(-i)^*}=\sqrt{-i\times i}
=1
<8-13> (8.1.7)
\begin{array}{rl}
I(X)&=2\pi i\times f(z=i) \\ \\
&=2\pi i\times\displaystyle{
\frac{1}{2i}}=\pi
\end{array}
<8-14> (8.1.8)
\begin{array}{rl}
I(X)&=\displaystyle{
\int_{-X}^{X}\frac{1}{x^2+1}dx+
\int_{C'}\frac{1}{z^2+1}dz} \\ \\
&\rightarrow\displaystyle{
\int_{-\infty}^{\infty}\frac{1}{x^2+1}dx}
\end{array}
<8-15>
\int_{-\infty}^{\infty}\frac{1}{x^2+1}dx
=\pi
<8-16>
\int_{-\infty}^{\infty}\frac{1}{x^2+2}dx
=\frac{\pi}{\sqrt{2}}
<8-17>
\int_{-\infty}^{\infty}\frac{1}{x^2+2x+3}dx
=\frac{\pi}{\sqrt{2}}
<8-18>
\int_{0}^{\infty}\frac{\cos x}{x^2+1}dx
=\frac{\pi}{\sqrt{2e}}
<8-19>
\int_{0}^{\infty}\frac{1}{x^3+1}dx
=\frac{2\pi}{3\sqrt{3}}
<8-20>
y=A_1\sin\left(\frac{\pi}{\ell}x\right)
\equiv y_1
<8-21>
y=A_2\sin\left(\frac{2\pi}{\ell}x\right)
\equiv y_2
<8-22>
y=A_3\sin\left(\frac{3\pi}{\ell}x\right)
\equiv y_3
<8-23> (8.2.1)
y_n(x)=A_n\sin\left(\frac{n\pi}{\ell}x\right),
\quad(n=1,2,3,\cdots)
<8-24>
\lambda_n=\frac{2\ell}{n},
\quad(n=1,2,3,\cdots)
<8-25> (8.2.2)
y_n(x)=
A_n\sin\left(\frac{2\pi x}{\lambda_n}\right),
\quad(n=1,2,3,\cdots)
<8-26> (8.2.3)
\begin{array}{rl}
f(x)&=\displaystyle{
\sum_{n=1}^{\infty}F_ny_n(x)} \\ \\
&=\displaystyle{
\sum_{n=1}^{\infty}F_n
\sin\left(\frac{2\pi x}{\lambda_n}\right)}
\end{array}
<8-27> (8.2.4)
\sum_{n=1}^{\infty}S_n=S_1+S_2+\cdots
<8-28>
\begin{array}{ll}
\mbox{(*)}&\quad\displaystyle{\int_{0}^{\ell}
\sin\left(\frac{m\pi}{\ell}x\right)f(x)dx}
\\ \\
&\quad=\displaystyle{\sum_{n=1}^{\infty}F_n
\int_{0}^{\ell}\sin\left(\frac{m\pi}{\ell}x\right)
\sin\left(\frac{n\pi}{\ell}x\right)dx}
\end{array}
<8-29> (8.2.5)
\left\{\begin{array}{l}
\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B \\ \\
\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B
\end{array}\right.
<8-30>
\sin A\sin B=\frac{1}{2}[\cos(A-B)-\cos(A+B)]
<8-31>
\begin{array}{l}
\displaystyle{\int_{0}^{\ell}
\sin\left(\frac{m\pi}{\ell}x\right)
\sin\left(\frac{n\pi}{\ell}x\right)dx} \\ \\
&\quad=\displaystyle{\frac{1}{2}\int_{0}^{\ell}
\left[\cos\left(\frac{(m-n)\pi}{\ell}x\right)
-\cos\left(\frac{(m+n)\pi}{\ell}x\right)\right]dx}
\end{array}
<8-32>
\begin{array}{rl}
\mbox{(**)}&\quad\displaystyle{\int_{0}^{\ell}
\sin\left(\frac{m\pi}{\ell}x\right)f(x)dx} \\ \\
&\quad=\displaystyle{
\frac{1}{2}\sum_{n=1}^{\infty}F_n
\left[\int_{0}^{\ell}
\cos\left(\frac{(m-n)\pi}{\ell}x\right)dx
\right.}\\ \\
&\quad\quad\left.-\displaystyle{
\int_{0}^{\ell}
\cos\left(\frac{(m+n)\pi}{\ell}x\right)dx
\right]} \\ \\
&\quad=\displaystyle{
\frac{1}{2}\sum_{n=1}^{\infty}F_n
\left[\frac{\ell}{(m-n)\pi}
\sin\left(\frac{(m-n)\pi}{\ell}x\right)
\right.}\\ \\
&\quad\quad\left.-\displaystyle{
\frac{\ell}{(m+n)\pi}
\sin\left(\frac{(m+n)\pi}{\ell}x\right)
\right]_{x=0}^{x=\ell}}}
\end{array}
<8-33>
\int_{0}^{\ell}dx=\ell
<8-34>
\lim_{n\rightarrow m}\left[\frac{\ell}{(m-n)\pi}
\sin\left(\frac{(m-n)\pi}{\ell}\right)\right]
\begin{array}{l}
\displaystyle{
\int_{0}^{\ell}\sin\left(\frac{m\pi}{\ell}x\right)
\sin\left(\frac{n\pi}{\ell}x\right)dx} \\ \\
\quad=\left\{\begin{array}{ll}
\displaystyle{
\frac{\ell}{2}}, & \mbox{(if $n=m$).} \\ \\
0, & \mbox{(if $n\ne m$).}
\end{array}\right.
\end{array}
<8-36> (8.2.7)
\begin{array}{l}
\displaystyle{F_m=\frac{2}{\ell}\int_{0}^{\ell}
\sin\left(\frac{m\pi}{\ell}x\right)f(x)dx}, \\ \\
\quad (\mbox{where $m$ is a natural number.})
\end{array}
<8-37> (8.2.8)
f(x)=\sum_{n=1}^{\infty}
F_n\sin\left(\frac{n\pi}{\ell}x\right)
<8-38> (8.2.9)
F_n=\frac{2}{\ell}\int_{0}^{\ell}
\sin\left(\frac{n\pi}{\ell}x\right)f(x)dx
<8-39> (8.2.10)
u_n(x)=\sqrt{\frac{2}{\ell}}
\sin\left(\frac{n\pi}{\ell}x\right),
\quad(n=1,2,\cdots)
<8-40> (8.2.11)
f(x)=\sqrt{\frac{\ell}{2}}\sum_{n=1}^{\infty}
F_nu_n(x)
<8-41>
F_n=\sqrt{\frac{2}{\ell}}\int_{0}^{\ell}
u_n(x)f(x)dx
<8-42> (8.2.12)
\int_{0}^{\ell}u_n(x)u_m(x)dx=\left\{
\begin{array}{ll}
1 & \mbox{when $n=m$.} \\ \\
0 & \mbox{when $n\ne m$.}
\end{array}\right.
<8-43> (8.2.13)
f(x)=F_0+\sum_{n=1}^{\infty}
\left[F_n\sin\left(\frac{n\pi}{\ell}x\right)+
G_n\cos\left(\frac{n\pi}{\ell}x\right)\right]
<8-44> (8.2.14)
\left\{\begin{array}{l}
\displaystyle{
F_0=\frac{1}{\ell}\int_{0}^{\ell}f(x)dx} \\ \\
\displaystyle{
F_n=\frac{2}{\ell}\int_{0}^{\ell}
\sin\left(n\pi}{\ell}x\right)f(x)dx} \\ \\
\displaystyle{
G_n=\frac{2}{\ell}\int_{0}^{\ell}
\cos\left(n\pi}{\ell}x\right)f(x)dx}
\end{array}\right.
<8-45> (8.2.15)
\left\{\begin{array}{l}
\displaystyle{
F_0=\frac{1}{L}\int_{-L/2}^{L/2}f(x)dx} \\ \\
\displaystyle{
F_n=\frac{2}{L}\int_{-L/2}^{L/2}
\sin\left(\frac{n\pi}{L}x\right)f(x)dx} \\ \\
\displaystyle{
G_n=\frac{2}{L}\int_{-L/2}^{L/2}
\cos\left(\frac{n\pi}{L}x\right)f(x)dx}
\end{array}\right.
<8-46>
f(-x)=+f(x)
<8-47>
f(-x)=-f(x)
<8-48>
\frac{dx}{dy}=-1
<8-49>
\int_{b}^{a}f(x)dx=-\int_{a}^{b}f(x)
<8-50>
\sin(-\theta)=-\sin\theta
<8-51>
\left\{\begin{array}{rl}
F_n&=\displaystyle{
\frac{2}{L}\int_{L/2}^{-L/2}
\sin\left(-\frac{n\pi}{L}y\right)
f(-y)(-1)dy} \\ \\
&=\displaystyle{
-\frac{2}{L}\int_{-L/2}^{L/2}
\sin\left(\frac{n\pi}{L}y\right)
f(y)dy} \\ \\
&=-F_n
\end{array}\right.
<8-52> (8.2.16)
f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}
g(k)e^{ikx}dk
<8-53> (8.2.17)
g(k)=\int_{-\infty}^{\infty}e^{-ikx}f(x)dx
<8-54> (8.2.18)
e^{\pm ikx}=\cos(kx)\pm i\sin(kx)
<8-55>
|x|=\left\{\begin{array}{ll}
+x, & \mbox{when $x\ge0$.} \\ \\
-x, & \mbox{when $x\le0$.}
\end{array}\right.
<8-56>
\begin{array}{rl}
g(k)&=\displaystyle{
\int_{-\infty}^{\infty}
e^{-ikx}e^{-a|x|}dx}} \\ \\
&=\displaystyle{
\int_{-\infty}^{0}e^{-ikx}e^{-a|x|}dx}}
+\displaystyle{
\int_{0}^{\infty}e^{-ikx}e^{-a|x|}dx}}
\end{array}\right.
<8-57>
\begin{array}{rl}
g(k)&=\displaystyle{\int_{-\infty}^{\infty}
e^{-ikx}e^{-a|x|}dx}} \\ \\
&=\displaystyle{
\int_{-\infty}^{0}e^{-ikx}e^{ax}dx}}
+\displaystyle{
\int_{0}^{\infty}e^{-ikx}e^{-ax}dx}} \\ \\
&=\displaystyle{
\int_{-\infty}^{0}e^{(a-ik)x}dx}}
+\displaystyle{
\int_{0}^{\infty}e^{(-a-ik)x}dx}}
\end{array}\right.
<8-58>
\int e^{px}dx=\frac{e^{px}}{p}
<8-59>
g(k)=\left[
\frac{e^{(a-ik)x}}{a-ik}
\right]_{-\infty}^{0}
+\left[
\frac{e^{(-a-ik)x}}{-a-ik}
\right]_{0}^{\infty}
<8-60>
g(k)=\frac{1}{a-ik}-\frac{1}{-a-ik}
=\frac{2a}{a^2+k^2}
<8-61>
g(k)=\frac{\sqrt{\pi}}{a}e^{-k^2/(4a^2)}
<8-62>
F(s)=\int_{0}^{\infty}f(x)e^{-sx}dx
<8-63> (8.3.2)
Z(\beta)=\int_{0}^{\infty}
g(E)e^{-\beta E}dE
<8-64>
\mbox{any constant $a$}
<8-65>
\frac{a}{s}
<8-66>
x^n\quad(n>0)
<8-67>
\frac{n!}{s^{n+1}}
<8-68>
e^{-\lambda x}\quad
\mbox{with a constant $\lambda$.}
<8-69>
\frac{1}{s+\lambda}
<8-70>
\sin(\lambda x)\quad
\mbox{with a constant $\lambda$.}
<8-71>
\frac{\lambda}{s^2+\lambda^2}
<8-72>
\cos(\lambda x)\quad
\mbox{with a constant $\lambda$.}
<8-73>
\frac{s}{s^2+\lambda^2}
<8-74> (8.3.3)
L\left\{af(x)+bg(x)\right\}=
aL\left\{f(x)\right\}
+bL\left\{g(x)\right\}
<9-1>
J[f]=\int_{0}^{1}f(x)dx
<9-2>
J=\int_{0}^{1}xdx
=\left[\frac{1}{2}x^2\right]_{0}^{1}
=\frac{1}{2}
<9-3>
J=\int_{0}^{1}x^2dx
=\left[\frac{1}{3}x^3\right]_{0}^{1}
=\frac{1}{3}
<9-4>
J=\int_{0}^{1}x^3dx
=\left[\frac{1}{4}x^4\right]_{0}^{1}
=\frac{1}{4}
<9-5>
J[f]=\int_{0}^{1}F[f(x)]dx
<9-6>
F[f(x)]=f(x)+1
<9-7>
J=\int_{0}^{1}(x+1)dx
=\left[\frac{1}{2}x^2+x\right]_{0}^{1}
=\frac{3}{2}
<9-8>
J=\int_{0}^{1}(x^2+1)dx
=\left[\frac{1}{3}x^3+x\right]_{0}^{1}
=\frac{4}{3}
<9-9>
J=\int_{0}^{1}(x^3+1)dx
=\left[\frac{1}{4}x^4+x\right]_{0}^{1}
=\frac{5}{4}
<9-10> (9.1.1)
J[f]=\int_{A}^{B}F[f(x)]dx
<9-11> (9.1.2)
J[f+\Delta]-J[f]
=\int_{A}^{B}F[f(x)+\Delta(x)]dx
-\int_{A}^{B}F[f(x)]dx
<9-12> (9.1.3)
\delta J[f]=
\int_{A}^{B}\left(
\frac{\partial F}{\partial f}\delta f
\right)dx
<9-13> (9.1.4)
\frac{d}{dx}\frac{\partial F}{\partial f'}
-\frac{\partial F}{\partial f}=0
<9-14>
s=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}
<9-15>
\Delta s=\sqrt{(\Delta x)^2+(\Delta y)^2}
=\sqrt{1+\left(\frac{\Delta y}{\Delta x}\right)^2}
\Delta x
<9-16>
L=\int_{0}^{a}\sqrt{1+y'^2}dx
<9-17> (9.1.6)
\frac{dF[y'(x)]}{dy'}
=\frac{d\sqrt{1+{y'}^2}}{d{y'}}
=\frac{y'}{\sqrt{1+{y'}^2}}
<9-18> (9.1.7)
\frac{d}{dx}\frac{y'}{\sqrt{1+{y'}^2}}=0
<9-19> (9.1.8)
y'=A
<9-20> (9.1.9)
y=Ax+B
<9-21> (9.1.10)
\left\{\begin{array}{l}
0=B \\ \\ b=Aa
\end{array}\right.
<9-22> (9.1.11)
B=0 \\ \\ \displaystyle{A=\frac{b}{a}}
\end{array}\right.
<9-23> (9.1.12)
y=\frac{b}{a}x
<10-1> (10.1.1)
\int_{-\infty}^{+\infty}\delta(x)f(x)dx=f(0)
<10-2> (10.1.2)
\int_{-\infty}^{+\infty}\delta(x-a)f(x)dx=f(a)
<10-3> (10.1.3)
\delta(x)
=\lim_{h\rightarrow0}\frac{1}{\sqrt{2\pi}h}
e^{-x^2/(2h^2)}
<10-4> (10.1.4)
\delta(x)=\lim_{h\rightarrow0}\frac{1}{\pi}
\frac{h}{x^2+h^2}
<10-5> (10.1.5)
\delta(x)=\lim_{n\rightarrow\infty}
\frac{\sin(nx)}{\pi x}
<10-6> (10.1.6)
\int_{-\infty}^{+\infty}\delta(x)dx=1