What is your favourite scientific equation?

Discussion in 'Physics & Math' started by John Connellan, Feb 19, 2007.

  1. John Connellan Valued Senior Member

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    Of all time???
     
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  3. Singularity Banned Banned

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    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.
     
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  5. orcot Valued Senior Member

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    1+1=2
    Don't laugh I have something with simplicity,
     
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  7. Kron Maxwell's demon Registered Senior Member

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    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.
     
  8. Tom2 Registered Senior Member

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    726
    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!
     
    Last edited: Feb 20, 2007
  9. Physics Monkey Snow Monkey and Physicist Registered Senior Member

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    869
    \( \vec{F} = \frac{d \vec{p}}{dt} \)

    The most beautiful equation in the world! It is simplicity juxtaposed with amazing power!
     
  10. BenTheMan Dr. of Physics, Prof. of Love Valued Senior Member

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    8,967
    hmmmm....Just because Tom2 took mine...

    \( F \equiv dA + A \wedge A \).

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

    \( S_{YM} = \int d^Dx \sqrt{-g} \frac{-1}{4 g_{YM}^2} \rm{Tr} F \wedge F\).
     
  11. quadraphonics Bloodthirsty Barbarian Valued Senior Member

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

    \( e^{i\pi} = -1 \)
     
  12. Facial Valued Senior Member

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    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.
     
  13. Facial Valued Senior Member

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    2,223
    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?
     
  14. BenTheMan Dr. of Physics, Prof. of Love Valued Senior Member

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    We should start a thread to see how many proofs of the Pythagorean Theorem we can come up with.
     
  15. Tom2 Registered Senior Member

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    726
    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!
     
  16. Physics Monkey Snow Monkey and Physicist Registered Senior Member

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    Haha, it does make me want to cry!
     
  17. John Connellan Valued Senior Member

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    Quick, quote before he edits

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  18. draqon Banned Banned

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    E=hc/delta
     
  19. John Connellan Valued Senior Member

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    Anyone like this one?

    \(\mathcal{L} = \bar\psi(i\gamma^\mu D_\mu-m_e)\psi-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\)
     
    Last edited: Feb 21, 2007
  20. Nickelodeon Banned Banned

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    Whats that?
     
  21. John Connellan Valued Senior Member

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    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 = mc[sup]2[/sup] etc.)
     
  22. BenTheMan Dr. of Physics, Prof. of Love Valued Senior Member

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    8,967
    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).
     
    Last edited: Feb 21, 2007
  23. BenTheMan Dr. of Physics, Prof. of Love Valued Senior Member

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    I almost chose this one as my favorite equation

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    The SYM lagrangian does reduce to this equation in the specific case of QED.
     

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