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please don't remove this and above from the top
HTML special characters:
`a < b`
`a < b`
`a < < b`
Dollar vs apostrophe vs double dollar vs dollar inside apostrophe:
$a < < b$
`a < < b`
$$a < < b$$
`$a < < b$`
Test:
`frac{df}{dx}=lim_(h->0) frac{f(x+h)-f(x)}{h}=lim_(\Deltax->0) frac{f(x+\Deltax)-f(x)}{\Deltax}`
How do we wrap the math to the next line:
`
frac{d}{dx}(m(x)*n(x))=lim_(h->0) frac{m(x+h)n(x+h)-m(x)n(x)}{h}
=lim_(h->0) frac{m(x+h)n(x+h)-m(x)n(x+h)+m(x)n(x+h)-m(x)n(x)}{h}
=lim_(h->0) frac{m(x+h)-m(x)}{h}* n(x+h)+ m(x)*lim_(h->0) frac{n(x+h)-n(x)}{h}
`
How do we wrap the math to the next line:
$frac{d}{dx}(m(x)*n(x))=lim_(h->0) frac{m(x+h)n(x+h)-m(x)n(x)}{h}$
$=lim_(h->0) frac{m(x+h)n(x+h)-m(x)n(x+h)+m(x)n(x+h)-m(x)n(x)}{h}$
$=lim_(h->0) frac{m(x+h)-m(x)}{h}* n(x+h)+ m(x)*lim_(h->0) frac{n(x+h)-n(x)}{h}$
Invisible grouping:
`{:[2/3, y],[w, z]:}`
Quantification:
`forall x in E, x=x`
`exists x:E, x=x`
Pythagorean theorem:
`0+a^2+b^2=c^2`
`AA x in CC (sin^2x+cos^2x=1)`
Dijkstra style:
`(AA x: x in CC: sin^2x+cos^2x=1)`
Test:
$A$, $B$ and $C$ are on the same line $iff$ there exists a real number $lambda$ such that $vec (AC) = lambda . vec(AB)$
`sum_(i=1)^ni^3=(sum_(i=1)^ni^2)^2`
Definition of the Riemann integral: If `f` is continuous on the interval `(a,b)`, except perhaps at finitely many points, then
`int_a^b f(x)dx=lim_(n->oo)sum_[i=1]^n f(x_i^(**))Delta x` where `Delta x=(b-a)/n`, `x_i=a+iDeltax` and `x_i^(**)in[x_[i-1],x_i]`.
And then after extending the Riemann integral to improper integrals (unbounded domains) you get the magic formula:
$\int_0^oo e^{-x^2}dx = 1/2\sqrt{pi}.$
`x/x=(1 if x!=0)`
`int_0^pi sinxdx=-cosx]_0^pi=-cospi-(-cos0)=-(-1)-(-1)=2`
Decimal numbers (right-click on the expression to see the MathML code): `epsilon=.001 quad h=-.01 quad pi~~3.14159 quad` weird number `-0.123.456` and dot product `u.v`
Test:
`RR = uuu_{n=0}^oo[-n,n]` and `{0} = nnn_{n=1}^oo(- 1/n,1/n)`
`^^^_{i=1}^nphi_i = phi_1 ^^ phi_2 ^^ cdots ^^ phi_n` and `vvv_{i=1}^nphi_i = phi_1 vv phi_2 vv cdots vv phi_n`
`pi~~3.141592653589793`
`int_-1^1 sqrt(1-x^2)dx = pi/2`
`lim_(x->a) f(x)=l <=> AA epsi > 0 EE delta > 0 : 0 < {:|x-a|:} < delta => {:|f(x) - l|:} < epsi`
Chain fractions:
`1/(1+1/(1+...))`
$x := y$
$int vec{A} cdot vec{dl} = int int vec{B} cdot vec{dS}$
Equation numbers:
`1/(1+1/(1+1/(1+1/(1+...))))=(sqrt5-1)/2`
(1.1)
RyanVolpe was here to ramble about finding offset `o` from indices `[a_0, a_1...a_(n-1)]` in an `n`-dimensional field bounded by a list of dimension deltas `[d_0, d_1...d_(n-1)]`:
`o = sum_(i=0)^n(a_i * prod_(j=0)^(i-1)d_j)`
`a = [1,1,1];
n = 3;
d = [2,3,2];
o = sum_(i=0)^3(a_i * prod_(j=0)^(i-1)d_j) = (1) + (1 * (2)) + (1 * (2*3)) = 1 + 2 + 6 = 9`
`o = i_0 + d_0[i_1 + d_1[...[i_(n-1)]]]` does
`g o f = Id_(e)`
`max`
`e^( i pi ) = -1`
`cos(pi/4)=sqrt(2)/2`
`x^2 + y^2 = z^2`
`x_1 , x_2, ... x_n`
test:
`o = sum_(i=0)^3(a_i * prod_(j=0)^(i-1)d_j) = (1) + (1 * (2)) + (1 * (2*3)) = 1 + 2 + 6 = 9`
test:
$int_0^oo \hbar e^{-x^2}dx = 1/2\sqrt{pi}.$
`grad * vec A = beta phi`
$\hbar$
$a$ $a!=b$ ${[a,b],[\,d]:}
$(\preceq_i)_{i \in N}$
$\pi pi \frac{1}{2} {1}/{2} + {1} + 2 |==_frI a => b$
\`sinx\` fa `sinx`
`-0.4093 * cos(2 * pi * (gDoy + 10) /365)`