<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