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\left\{{\begin{aligned}4x+y&=6x\\6x+3y&=6y\end{aligned}}\right.}      that is      { − 2 x + y = 0 6 x − 3 y = 0 {\displaystyle \left\{{\begin{aligned...

cos ⁡ θ . {\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \,\\y'&=x\sin \theta +y\cos \theta \,\end{aligned}}.} For example, when the vector...

y d x d y d z = ∂ A x ∂ x + ∂ A y ∂ y + ∂ A z ∂ z {\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} =\lim _{V\to 0}{\frac {\iint _{\partial...

{\begin{aligned}f(x)&=C_{1}\,e^{-\pi {\frac {x^{2}}{\sigma ^{2}}}}\\\therefore {\hat {f}}(\xi )&=\sigma C_{1}\,e^{-\pi \sigma ^{2}\xi ^{2}}\end{aligned}}}...

{\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}} where the sign...

{\displaystyle {\begin{aligned}\nu _{0}'&=\nu _{0}+n\\\nu _{0}'{\sigma _{0}^{2}}'&=\nu _{0}\sigma _{0}^{2}+\sum _{i=1}^{n}(x_{i}-\mu )^{2}\end{aligned}}} or equivalently...

( x ) , s ′ ( x ) = c ( x ) , {\displaystyle {\begin{aligned}c'(x)&=s(x),\\s'(x)&=c(x),\\\end{aligned}}} with the initial conditions s ( 0 ) = 0 , c (...

{\displaystyle {\begin{aligned}A_{0}&=C_{0}&\\A_{n}&=C_{n}+C_{-n}\qquad &{\textrm {for}}~n>0\\B_{n}&=i(C_{n}-C_{-n})\qquad &{\textrm {for}}~n>0\end{aligned}}} Substituting...

{\displaystyle {\begin{aligned}\varphi ^{n}&=\varphi ^{n-1}+\varphi ^{n-2},\\[3mu]\psi ^{n}&=\psi ^{n-1}+\psi ^{n-2}.\end{aligned}}} It follows that...

{\displaystyle {\begin{aligned}\vartheta _{00}(x)&=1+2\sum _{n=1}^{\infty }x^{n^{2}}\\\vartheta _{01}(x)&=1+2\sum _{n=1}^{\infty }(-1)^{n}x^{n^{2}}\end{aligned}}} The...

∇ × E = − ∂ B ∂ t ∇ × B = μ 0 ( J + ε 0 ∂ E ∂ t ) {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} \,\,\,&={\frac {\rho }{\varepsilon _{0}}}\\\nabla...

tr ⁡ σ j = 0 , {\displaystyle {\begin{aligned}\det \sigma _{j}&=-1,\\\operatorname {tr} \sigma _{j}&=0,\end{aligned}}} from which we can deduce that...

22 = x y . {\displaystyle {\begin{aligned}w_{11}&=(1-x)(1-y),\\w_{12}&=(1-x)y,\\w_{21}&=x(1-y),\\w_{22}&=xy.\end{aligned}}} Alternatively, the interpolant...

{\displaystyle {\begin{aligned}{\hat {k}}{\hat {\theta }}&={\bar {x}}&&{\text{and}}&\psi ({\hat {k}})+\ln {\hat {\theta }}&={\overline {\ln x}}.\end{aligned}}} For...

{\begin{aligned}\sin(\theta )&=\cos \left({\frac {\pi }{2}}-\theta \right),\\\cos(\theta )&=\sin \left({\frac {\pi }{2}}-\theta \right).\end{aligned}}}...

σ ( X ) . {\displaystyle {\begin{aligned}\sigma (c)&=0\\\sigma (X+c)&=\sigma (X),\\\sigma (cX)&=|c|\sigma (X).\end{aligned}}} The standard deviation of...

{\displaystyle {\begin{aligned}F(x)=f(x)-\sum _{k=0}^{n-1}{\frac {f^{(k)}(a)}{k!}}(x-a)^{k}\end{aligned}}} G ( x ) = ( x − a ) n {\displaystyle {\begin{aligned...

{\displaystyle {\begin{aligned}\varphi ^{2}&=\varphi +1=2.618033\dots ,\\[5mu]{\frac {1}{\varphi }}&=\varphi -1=0.618033\dots .\end{aligned}}} The sequence...

{\displaystyle {\begin{aligned}&&J_{a}(\lambda )&=\lambda I+N\\&\Rightarrow &e^{J_{a}(\lambda )}&=e^{\lambda I+N}=e^{\lambda }e^{N}.\end{aligned}}} where N...

( f ) = ∫ − ∞ ∞ v ( x ) e − i 2 π f x d x , f ∈ R {\displaystyle {\begin{aligned}U(f)&\triangleq {\mathcal {F}}\{u\}(f)=\int _{-\infty }^{\infty }u(x)e^{-i2\pi...