TeX Notation
When a≠0a≠0, there are two solutions to ax2+bx+c=0ax2+bx+c=0 and they are
x=−b±b2−4ac‾‾‾‾‾‾‾‾√2a.x=−b±b2−4ac2a.
The Lorenz Equations
x˙=σ(y−x)y˙=ρx−y−xzz˙=−βz+xyx˙=σ(y−x)y˙=ρx−y−xzz˙=−βz+xy
The Cauchy-Schwarz Inequality
(∑k=1nakbk)2≤(∑k=1na2k)(∑k=1nb2k)(∑k=1nakbk)2≤(∑k=1nak2)(∑k=1nbk2)
A Cross Product Formula
V1×V2=∣∣ijk ∂X∂u∂Y∂u0 ∂X∂v∂Y∂v0 ∣∣V1×V2=|ijk ∂X∂u∂Y∂u0 ∂X∂v∂Y∂v0 |
The probability of getting kk heads when flipping nn coins is:
P(E)=(nk)pk(1−p)n−kP(E)=(nk)pk(1−p)n−k
An Identity of Ramanujan
1(ϕ5‾√‾‾‾‾‾√−ϕ)e25π=1+e−2π1+e−4π1+e−6π1+e−8π1+…1(ϕ5−ϕ)e25π=1+e−2π1+e−4π1+e−6π1+e−8π1+…
A Rogers-Ramanujan Identity
1+q2(1−q)+q6(1−q)(1−q2)+⋯=∏j=0∞1(1−q5j+2)(1−q5j+3),for |q|<1.1+q2(1−q)+q6(1−q)(1−q2)+⋯=∏j=0∞1(1−q5j+2)(1−q5j+3),for |q|<1.
Maxwell's Equations
∇×B⃗ −1c∂E⃗ ∂t=4πcj⃗ ∇⋅E⃗ =4πρ∇×E⃗ +1c∂B⃗ ∂t=0⃗ ∇⋅B⃗ =0∇×B→−1c∂E→∂t=4πcj→∇⋅E→=4πρ∇×E→+1c∂B→∂t=0→∇⋅B→=0
In-line Mathematics
Finally, while display equations look good for a page of samples, the ability to mix math and text in a paragraph is also important. This expression 3x−1‾‾‾‾‾‾√+(1+x)23x−1+(1+x)2 is an example of an inline equation. As you see, MathJax equations can be used this way as well, without unduly disturbing the spacing between lines.