Of all time???

Space bending has been proved not possible but why do people still believe whatever the scientist tell them ?

BigBang is one such example that was shattered with observations, hope they can observe space bendings.

1+1=2
Don't laugh I have something with simplicity,

My favorite equation would be...

1^(0.5) = {-1,1}

This equation means a lot to me, because I once thought that equations having multiple roots were really unusable abberations in mathematics. Then when I graphed a unit circle for the first time, I saw that the double solution was integral to the curve.

Mathematical beauty.

I think that there's a fascinating symmetry in the manifestly covariant form of Maxwell's equations.

\partial_{\mu}F^{\mu\nu}=J^{\nu}
\epsilon_{\mu\nu\tau\sigma}\partial^{\nu}F^{\tau \sigma}=0

The two Maxwell equations with source terms are neatly wrapped up in the first equation, and the two sourceless equations are contained in the second. Another neat way to look at these equations is in the language of differential forms, in which the second equation is seen to contain an exterior derivative, and the first is seen to contain its dual. It's kind of like looking at two sides of the same coin.

Speaking of differential forms, I might as well post my favorite mathematical theorem, the Generalized Stokes' Theorem.

\int_Bd\omega=\int_{\partial B}\omega

The integral of the derivative of a differential form over a chain is equal to the integral of the form over the boundary of the chain. From this elegant theorem, the following theorems of classical calculus are easily proved corollaries.

* The Fundamental Theorem of Calculus
* The Divergence Theorem
* Stokes' Theorem
* Green's Theorem

Talk about getting more bang for your buck!

\vec{F} = \frac{d \vec{p}}{dt}

The most beautiful equation in the world! It is simplicity juxtaposed with amazing power!

hmmmm....Just because Tom2 took mine...

F \equiv dA + A \wedge A .

Then the action for \mathcal{N} = 4 Super Yang-Mills theory is...

S_{YM} = \int d^Dx \sqrt{-g} \frac{-1}{4 g_{YM}^2} \rm{Tr} F \wedge F.

I can't believe nobody's mentioned this one:

e^{i\pi} = -1

x^n + y^n = z^n

General form (Fermat)
Pythagoras (n=2)

This is my favorite for several reasons. First, it is the most revolutionary step forward in the advancement of geometrical understanding. Second, it is well-known by the populace. Third, it is the foundation of all trigonometry. Fourth, it has strange limitations by Fermat's last theorem, whose proof involves non-planar non-Euclidean geometry way beyond my scope. Fifth, but not least, I know one of its proofs.

I remember a South American poster mentioning the Biot-Savart equation as one of the top ten most important in all of science, or something along those lines (could've also been a poll by scientists). What is important about Biot-Savart?

Fifth, but not least, I know one of its proofs.

We should start a thread to see how many proofs of the Pythagorean Theorem we can come up with.

\vec{F} = \frac{d \vec{p}}{dt}

The most beautiful equation in the world! It is simplicity juxtaposed with amazing power!

Damn! I should have quoted this post before you edited out the part about how looking at Newton's second law makes you want to cry. I bet your real life buddies would never have let you live that one down!

Damn! I should have quoted this post before you edited out the part about how looking at Newton's second law makes you want to cry. I bet your real life buddies would never have let you live that one down!

Haha, it does make me want to cry!

Quick, quote before he edits :D

E=hc/delta

Anyone like this one?

\mathcal{L} = \bar\psi(i\gamma^\mu D_\mu-m_e)\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}

Whats that?

It's the QED Lagrangian for the interaction of electrons and positrons through photons. I don't understand QED that well but I think it's one of the most fundamental equations in science at present (along with E = mc2 etc.)

Yes it is the QED lagrangian, except you have a typo.

\mathcal{L} = \bar{\psi}\left(i \gamma^{\mu} D_{\mu} -m_e\right)\psi - \frac{1}{4} F^{\mu \nu}F_{\mu \nu}

This is the most accurately tested theory in all of physics, tested to something like one part in ten trillion (10^-13).

I almost chose this one as my favorite equation:) The SYM lagrangian does reduce to this equation in the specific case of QED.

except you have a typo.

Thanks Ben. This LaTeX writing style is very demanding on the concentration!

You get used to it after a while. TONS better than microsof equation editor.

How could you integrate the LaTeX capabilities in a program like Word though? Is it possible?

The only way I can think of is to do it is to post code on a LaTeX-enabled forum (such as this one), then copy and paste the generated LaTeX images into your MS Word doc.

How could you integrate the LaTeX capabilities in a program like Word though? Is it possible?

With TexPoint:

http://texpoint.necula.org/

Thanks quad. That's exactly what I was looking for

The science equation I like the most is F = \frac{dp}{dt}

However, from the math side... I have a few.

i^i = e^{-\pi/2}\

Gamma constant: \gamma = \lim_{n \rightarrow \infty } \left( \left( \sum_{k=1}^n \frac{1}{k} \right) - \ln (n) \right)=\int_1^\infty\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx

And of course, the Collatz Conjecture:
f(n) = \begin{cases} n/2 &\mbox{if } n \equiv 0 \\ 3n+1 & \mbox{if } n\equiv 1 \end{cases} \pmod{2}.

a_i = \begin{cases}n & \mbox{for } i = 0 \\ f(a_{i-1}) & \mbox{for } i > 0\end{cases}

\forall n \in \mathbb{N} > 0 \ \exists i \in \mathbb{N}: (a_0 = n \Rightarrow a_i = 1)

(note... TeX stolen from Wiki since I don't really know TeX)

ewww. Number theory. God I hate mathematics for the sake of mathematics :D