University Mathematics Equation Database

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$(x+y{)}^n={\sum }_{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})x^{n-k}{y}^k$

${\log }_{a}b=\frac{{\log }_{c}b}{{\log }_{c}a}$

$S_n=\frac{n}{2}(a_1+a_n)$

$S_n=a\frac{1-r^n}{1-r}$

$e^{i\pi }+1=0$

$(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta )$

${\sin }^2\theta +{\cos }^2\theta =1$

$\sin (2\theta )=2\sin \theta \cos \theta$

$\cos (2\theta )={\cos }^2\theta -{\sin }^2\theta$

$a^2-b^2=(a-b)(a+b)$

$a^3-b^3=(a-b)(a^2+ab+b^2)$

$z^{1/n}=r^{1/n}{e}^{i\frac{\theta +2k\pi }{n}}$

$\frac{x_1+x_2}{2}\ge \sqrt{x_1{x}_2}$

$A=P{(1+\frac{r}{n})}^{nt}$

$A=P{e}^{rt}$

$\sinh x=\frac{e^x-e^{-x}}{2}$

$\cosh x=\frac{e^x+e^{-x}}{2}$

$\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$

$\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta$

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

$c^2=a^2+b^2-2ab\cos C$

${\log }_b(xy)={\log }_{b}x+{\log }_{b}y$

${\log }_b\left(\frac{x}{y}\right)={\log }_{b}x-{\log }_{b}y$

$\varphi =\frac{1+\sqrt{5}}{2}$

$f^{\prime }(x)={lim}_{h\to 0}\frac{f(x+h)-f(x)}{h}$

$\frac{dy}{dx}=\frac{dy}{du}·\frac{du}{dx}$

$\int udv=uv-\int vdu$

$\frac{d}{dx}{\int }_a^{x}f(t)dt=f(x)$

${\int }_a^{b}f(x)dx=F(b)-F(a)$

$f(x)={\sum }_{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$

$e^x={\sum }_{n=0}^{\infty }\frac{x^n}{n!}$

$\sin x={\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}$

$\cos x={\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n}}{(2n)!}$

${lim}_{x\to 0}\frac{\sin x}{x}=1$

$e={lim}_{n\to \infty }{(1+\frac{1}{n})}^n$

$\frac{d}{dx}(\ln x)=\frac{1}{x}$

$\int \frac{1}{x}dx=\ln |x|+C$

${\int }_{-\infty }^{\infty }{e}^{-x^2}dx=\sqrt{\pi }$

$x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}$

$L={\int }_a^b\sqrt{1+[f^{\prime }(x){]}^2}dx$

$V=\pi {\int }_a^b[f(x){]}^{2}dx$

${lim}_{x\to c}\frac{f(x)}{g(x)}={lim}_{x\to c}\frac{f^{\prime }(x)}{g^{\prime }(x)}$

$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$

${\left(\frac{f}{g}\right)}^{\prime }=\frac{f^{\prime }g-f{g}^{\prime }}{g^2}$

${\sum }_{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}$

${\int }_a^{\infty }f(x)dx={lim}_{t\to \infty }{\int }_a^{t}f(x)dx$

$\frac{d}{dx}(\sec x)=\sec x\tan x$

$\frac{d}{dx}(\arctan x)=\frac{1}{1+x^2}$

$A=\frac{1}{2}{\int }_{\alpha }^{\beta }[r(\theta ){]}^{2}d\theta$

$\nabla f=\frac{\partial f}{\partial x}𝐢+\frac{\partial f}{\partial y}𝐣+\frac{\partial f}{\partial z}𝐤$

$\nabla ·𝐅=\frac{\partial F_x}{\partial x}+\frac{\partial F_y}{\partial y}+\frac{\partial F_z}{\partial z}$

$\nabla \times 𝐅=det\left(\begin{array}{ccc}𝐢& 𝐣& 𝐤\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ F_x& F_y& F_z\end{array}\right)$

$D_{𝐮}f=\nabla f·𝐮$

${\oint }_C(Pdx+Qdy)={\iint }_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dA$

${\oint }_C𝐅·d𝐫={\iint }_S(\nabla \times 𝐅)·d𝐒$

${\iint }_S𝐅·d𝐒={\iiint }_E(\nabla ·𝐅)dV$

$J(u,v)=det\left(\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial x}{\partial v}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}\right)$

${\iint }_{R}f(x,y)dA={\iint }_{S}f(r\cos \theta ,r\sin \theta )rdrd\theta$

$dV={\rho }^2\sin \varphi d\rho d\varphi d\theta$

$dV=rdzdrd\theta$

$a(x-x_0)+b(y-y_0)+c(z-z_0)=0$

$\frac{\partial z}{\partial u}=\frac{\partial z}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial z}{\partial y}\frac{\partial y}{\partial u}$

$\frac{{\partial }^{2}f}{\partial x\partial y}=\frac{{\partial }^{2}f}{\partial y\partial x}$

$z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)$

$\kappa =\frac{\Vert {𝐓}^{\prime }(t)\Vert }{\Vert {𝐫}^{\prime }(t)\Vert }$

$𝐍(t)=\frac{{𝐓}^{\prime }(t)}{\Vert {𝐓}^{\prime }(t)\Vert }$

$s(t)={\int }_a^t\Vert {𝐫}^{\prime }(u)\Vert du$

$\nabla f=\lambda \nabla g$

$D=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b){]}^2$

$(x-a{)}^2+(y-b{)}^2+(z-c{)}^2=R^2$

$W={\int }_C𝐅·d𝐫$

$A(S)={\iint }_D\sqrt{1+{\left(\frac{\partial z}{\partial x}\right)}^2+{\left(\frac{\partial z}{\partial y}\right)}^2}dA$

${\nabla }^{2}f=\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2}+\frac{{\partial }^{2}f}{\partial z^2}=0$

$\Phi ={\iint }_S𝐅·𝐧dS$

$c_{ij}={\sum }_{k=1}^n{a}_{ik}{b}_{kj}$

$det(A-\lambda I)=0$

$A𝐯=\lambda 𝐯$

$A^{-1}=\frac{1}{detA}\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)$

$𝐯=c_1{𝐯}_1+c_2{𝐯}_2+\dots +c_k{𝐯}_k$

$𝐮·𝐯={\sum }_{i=1}^n{u}_i{v}_i$

$\Vert 𝐯\Vert =\sqrt{v_1^2+v_2^2+\dots +v_n^2}$

$\cos \theta =\frac{𝐮·𝐯}{\Vert 𝐮\Vert \Vert 𝐯\mathrm{\backslash shallow}}$

${\text{proj}}_{𝐯}𝐮=\frac{𝐮·𝐯}{\Vert 𝐯{\Vert }^2}𝐯$

${𝐮}_k={𝐯}_k-{\sum }_{j=1}^{k-1}\frac{{𝐯}_k·{𝐮}_j}{\Vert {𝐮}_j{\Vert }^2}{𝐮}_j$

$A^{T}A𝐱=A^T𝐛$

$Q^{T}Q=I$

$A=PD{P}^{-1}$

$A=U\Sigma V^T$

$\text{tr}(A)={\sum }_{i=1}^n{a}_{ii}$

$\text{rank}(A)+\text{nullity}(A)=n$

$detA={\sum }_{j=1}^n(-1{)}^{i+j}{a}_{ij}det{M}_{ij}$

$x_i=\frac{det{A}_i}{detA}$

$𝐮\times 𝐯=det\left(\begin{array}{ccc}𝐢& 𝐣& 𝐤\\ u_1& u_2& u_3\\ v_1& v_2& v_3\end{array}\right)$

$[T{]}_B=(\begin{array}{ccc}T(b_1)& \dots & T(b_n)\end{array})$

$q(𝐱)={𝐱}^{T}A𝐱$

$|𝐮·𝐯|\le \Vert 𝐮\Vert \Vert 𝐯\Vert \$$

$\Vert 𝐮+𝐯\Vert \le \Vert 𝐮\Vert +\Vert 𝐯\Vert \$$

$A=LU$

$A=QR$

$\frac{dy}{dx}=g(x)h(y)$

$\frac{dy}{dx}+P(x)y=Q(x)$

$I(x)=e^{\int P(x)dx}$

$a{y}^{″}+b{y}^{\prime }+cy=0$

$a{r}^2+br+c=0$

$x^2{y}^{″}+\alpha x{y}^{\prime }+\beta y=0$

$ℒ\{f(t)\}={\int }_0^{\infty }{e}^{-st}f(t)dt$

$ℒ\{f^{\prime }(t)\}=sℒ\{f(t)\}-f(0)$

$ℒ\{f^{″}(t)\}=s^2ℒ\{f(t)\}-sf(0)-f^{\prime }(0)$

$f(t)=\frac{a_0}{2}+{\sum }_{n=1}^{\infty }[a_n\cos (n{\omega }_{0}t)+b_n\sin (n{\omega }_{0}t)]$

$a_n=\frac{2}{T}{\int }_0^{T}f(t)\cos (n{\omega }_{0}t)dt$

$x^2{y}^{″}+x{y}^{\prime }+(x^2-p^2)y=0$

$(1-x^2)y^{″}-2x{y}^{\prime }+n(n+1)y=0$

$y_p(x)=-y_1(x)\int \frac{y_2(x)g(x)}{W(x)}dx+y_2(x)\int \frac{y_1(x)g(x)}{W(x)}dx$

$W(y_1,y_2)=det\left(\begin{array}{cc}{y}_1& y_2\\ y_1^{\prime }& y_2^{\prime }\end{array}\right)$

$\frac{\partial u}{\partial t}=\alpha \frac{{\partial }^{2}u}{\partial x^2}$

$\frac{{\partial }^{2}u}{\partial t^2}=c^2\frac{{\partial }^{2}u}{\partial x^2}$

$ℱ\{f(t)\}={\int }_{-\infty }^{\infty }f(t)e^{-i\omega t}dt$

$f(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }F(\omega )e^{i\omega t}d\omega$

$u(x,t)=\frac{1}{2}[f(x-ct)+f(x+ct)]+\frac{1}{2c}{\int }_{x-ct}^{x+ct}g(s)ds$

$ℒ\{\delta (t-t_0)\}=e^{-s{t}_0}$

$ℒ\{u(t-t_0)\}=\frac{e^{-s{t}_0}}{s}$

$k_1=hf(x_n,y_n)$

$y_{n+1}=y_n+hf(x_n,y_n)$

$\frac{T^{\prime }(t)}{\alpha T(t)}=\frac{X^{″}(x)}{X(x)}=-\lambda$

$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$

$e^{iz}=\cos z+i\sin z$

$f(z_0)=\frac{1}{2\pi i}{\oint }_C\frac{f(z)}{z-z_0}dz$

${\oint }_{C}f(z)dz=2\pi i{\sum }_{k=1}^n\text{Res}(f,z_k)$

$\text{Res}(f,z_0)={lim}_{z\to z_0}\frac{1}{(m-1)!}\frac{d^{m-1}}{d{z}^{m-1}}[(z-z_0{)}^{m}f(z)]$

$f(z)={\sum }_{n=-\infty }^{\infty }{a}_n(z-z_0{)}^n$

$f^{(n)}(z_0)=\frac{n!}{2\pi i}{\oint }_C\frac{f(z)}{(z-z_0{)}^{n+1}}dz$

$\exp (z)=e^x(\cos y+i\sin y)$

$\sin z=\frac{e^{iz}-e^{-iz}}{2i}$

$\cos z=\frac{e^{iz}+e^{-iz}}{2}$

$f(z)=\frac{az+b}{cz+d}$

$ℒ\{(f*g)(t)\}=ℒ\{f(t)\}·ℒ\{g(t)\}$

$ℱ\{f^{\prime }(t)\}=i\omega F(\omega )$

$ℱ\{e^{-a{t}^2}\}=\sqrt{\frac{\pi }{a}}{e}^{-\frac{{\omega }^2}{4a}}$

$ℱ\{u(t)e^{-at}\}=\frac{1}{a+i\omega }$

$X(z)={\sum }_{n=-\infty }^{\infty }x[n]z^{-n}$

$𝒵\{x[n-n_0]\}=z^{-n_0}X(z)$

$𝒵\{a^{n}u[n]\}=\frac{z}{z-a}$

$s=\frac{2}{T_s}\frac{z-1}{z+1}$

$\log z=\ln |z|+i\mathrm{\backslash arg}z$

$|a_n|\le \frac{M}{R^n}$

$u(r,\theta )=\frac{1}{2\pi }{\int }_0^{2\pi }\frac{R^2-r^2}{R^2-2Rr\cos (\theta -\varphi )+r^2}u(R,\varphi )d\varphi$

${\int }_{-\infty }^{\infty }|f(t){|}^{2}dt=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }|F(\omega ){|}^{2}d\omega$

${\int }_{C}f(z)dz={\int }_a^{b}f(z(t))z^{\prime }(t)dt$

$\{ℳf\}(s)={\int }_0^{\infty }{x}^{s-1}f(x)dx$

$P(A|B)=\frac{P(B|A)P(A)}{P(B)}$

$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$

$P(n,k)=\frac{n!}{(n-k)!}$

$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$

$P(X=k)=\frac{{\lambda }^k{e}^{-\lambda }}{k!}$

$f(x)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}$

$𝔼[X]={\sum }_i{x}_{i}P(X=x_i)$

$𝔼[X]={\int }_{-\infty }^{\infty }xf(x)dx$

$\text{Var}(X)=𝔼[X^2]-(𝔼[X]{)}^2$

$\text{Var}(X)={\int }_{-\infty }^{\infty }(x-\mu {)}^{2}f(x)dx$

$\overline{X}=\frac{1}{n}{\sum }_{i=1}^n{X}_i$

$S^2=\frac{1}{n-1}{\sum }_{i=1}^n(X_i-\overline{X}{)}^2$

$Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}$

$\text{Cov}(X,Y)=𝔼[(X-{\mu }_X)(Y-{\mu }_Y)]$

${\rho }_{X,Y}=\frac{\text{Cov}(X,Y)}{{\sigma }_X{\sigma }_Y}$

$f(x)=\lambda e^{-\lambda x}(x\ge 0)$

$P(A|B)=\frac{P(A\cap B)}{P(B)}$

$P(A)={\sum }_{i}P(A|B_i)P(B_i)$

$𝔼[X]=\frac{a+b}{2}$

$P(X=k)=\frac{\left(\genfrac{}{}{0ex}{}{K}{k}\right)\left(\genfrac{}{}{0ex}{}{N-K}{n-k}\right)}{\left(\genfrac{}{}{0ex}{}{N}{n}\right)}$

${\beta }_1=\frac{\sum (X_i-\overline{X})(Y_i-\overline{Y})}{\sum (X_i-\overline{X}{)}^2}$

${\beta }_0=\overline{Y}-{\beta }_1\overline{X}$

${\chi }^2=\sum \frac{(O_i-E_i{)}^2}{E_i}$

$M_X(t)=𝔼[e^{tX}]$

$P(|X-\mu |\ge k\sigma )\le \frac{1}{k^2}$

$\overline{A\cap B}=\bar{A}\cup \bar{B}$

$\overline{A\cup B}=\bar{A}\cap \bar{B}$

$\neg (p\wedge q)⟺\neg p\vee \neg q$

$\neg (p\vee q)⟺\neg p\wedge \neg q$

$A⧵B=A\cap \bar{B}$

$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$

$A\times B=\{(a,b)\mid a\in A\wedge b\in B\}$

$|𝒫(S)|=2^{|S|}$

$\forall x_1,x_2\in A,f(x_1)=f(x_2)⟹x_1=x_2$

$\forall y\in B,\exists x\in A\text{s.t.}f(x)=y$

$F_n=F_{n-1}+F_{n-2}(F_0=0,F_1=1)$

$a_n=r{a}_{n-1}$

$P(k)⟹P(k+1)$

${\sum }_{i=1}^{n}i=\frac{n(n+1)}{2}$

${\sum }_{i=1}^n{i}^2=\frac{n(n+1)(2n+1)}{6}$

$k>n⟹\lceil k/n\rceil \ge 2$

$V-E+F=2$

${\sum }_{v\in V}\mathrm{deg}(v)=2|E|$

$\frac{n!}{n_1!n_2!\dots n_r!}$

$C_n=\frac{1}{n+1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$

$!n=n!{\sum }_{i=0}^n\frac{(-1{)}^i}{i?}$

$A_{ij}=\{\begin{array}{cc}1\hfill & \text{if }(v_i,v_j)\in E\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}$

$S(n,k)=\frac{1}{k!}{\sum }_{j=0}^k(-1{)}^{k-j}(\genfrac{}{}{0ex}{}{k}{j})j^n$

$p⟹q⟺\neg p\vee q$

$\neg (\forall x\in S,P(x))⟺\exists x\in S\text{s.t.}\neg P(x)$