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  • Beautiful Equations

    Euler's Identity

    e to the power of (i * pi) + 1 = 0

    e is Euler's number, the base of the natural logarithm,
    i is the imaginary unit, one of the two complex numbers whose square is negative one (the other is ), and
    pi, the ratio of the circumference of a circle to its diameter.


    http://en.wikipedia.org/wiki/Euler%27s_identity

    Proof of a fundamental interconnectivity of all things, and the unreasonable effectiveness of mathematics .
    How to Talk to a Climate Skeptic: http://grist.org/series/skeptics/
    Global Warming & Climate Change Myths: https://www.skepticalscience.com/argument.php

  • #2
    The most beautiful AND understandable to normal mortals like me, IMO is:

    Pythagoras' theorem:
    BoRG

    You may not be interested in War, but War is interested in You - Leon Trotski, June 1919.

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    • #3
      F=MA

      The net force on a body = Mass of the body X acceleration of the body

      This is consistent with/summarizes Newton's three laws

      In the absence of a net force, a body maintains a constant velocity
      A body experiencing a net force F experiences an acceleration A
      When Body one exerts a force F on Body tow, Body two exerts a force -F on Body one

      http://en.wikipedia.org/wiki/Newton's_laws_of_motion

      Comment


      • #4
        Originally posted by Nick the Noodle View Post
        Euler's Identity

        e to the power of (i * pi) + 1 = 0

        e is Euler's number, the base of the natural logarithm,
        i is the imaginary unit, one of the two complex numbers whose square is negative one (the other is ), and
        pi, the ratio of the circumference of a circle to its diameter.


        http://en.wikipedia.org/wiki/Euler%27s_identity

        Proof of a fundamental interconnectivity of all things, and the unreasonable effectiveness of mathematics .
        A couple of great essays on the "unreasonable effectiveness of mathematics"...

        Eugenge Wigner's The Unreasonable Effectiveness of Mathematics in the Natural Sciences

        Richard W. Hamming's The Unreasonable Effectiveness of Mathematics

        Wigner was a Nobel Prize winning physicist (The Nobel Prize for Physics actually meas something as opposed to the Nobel Peace prize).

        Hamming was a pioneer in the field of signal processing.

        Some of my “favorite” equations...

        The Dix Equation:
        An equation used to calculate the interval velocity within a series of flat, parallel layers, named for American geophysicist C. Hewitt Dix (1905 to 1984). Sheriff (1991) cautions that the equation is misused in situations that do not match Dix's assumptions. The equation is as follows:



        LINK
        Snell's Law:

        In optics and physics, Snell's law (also known as Descartes' law, the Snell–Descartes law, and the law of refraction), is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water and glass. The law says that the ratio of the sines of the angles of incidence and of refraction is a constant that depends on the media...





        LINK
        The Wave Equation:
        The wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange...





        LINK

        Hooke's Law:
        In mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load added to it as long as this load does not exceed the elastic limit. Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials.

        Mathematically, Hooke's law states that



        where

        x is the displacement of the end of the spring from its equilibrium position;
        F is the restoring force exerted by the material; and
        k is the force constant (or spring constant).

        LINK
        Watts Up With That? | The world's most viewed site on global warming and climate change.

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        • #5
          The number e

          The number e is found over and over again, especially in calculus and differential equations. it can be used to characterize something increasing or decreasing geometrically, like compound interest or bacteria growing at a constant temperature with no with no constraints like nutrients, air, no poisoning, etc. to throttle it back. The two simplest expressions are:

          e= the limit as n goes to infinity of (1 + 1/n)**n

          ie when n=1, e'=1
          when n=2, e''=9/4, =2.25
          when n=3, e'''=64/27=2.37
          when n=4, e''''=625/256=2.44

          when n=32, e evaluates as 2.692
          when n=100, e evaluates as 2.7048

          There are other methods which converge much quicker, but this when really represents reality better.

          and e is approx. 2.718... when n goes to infinity

          an equivalent expression is as x goes to 0 (1+x)**1/x

          http://en.wikipedia.org/wiki/E_(mathematical_constant))
          Last edited by lakechampainer; 08 Mar 10, 14:08.

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          • #6
            There must be plenty of people out there to give a brief description of, for example, the AC circuit equation, the tendency of distributions to be well approximated by the Gaussian distribution as the number of samples increases, the entropy/information equation, etc.

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            • #7
              Ah, reading this thread is like going to a bar full of old friends!


              Here's one of my buds, the KASNER SPACE-TIME METRIC

              "The Kasner metric is an exact solution to Einstein's theory of general relativity. It describes an anisotropic universe without matter (i.e., it is a vacuum solution).
              ...
              The metric describes a spacetime whose equal-time slices are spatially flat, however space is expanding or contracting at different rates in different directions, depending on the values of the pj."

              http://en.wikipedia.org/wiki/Kasner_metric

              Back in my grad school days I did a cosmological simulation on a Cray Y-MP computer using the Kasner Metric. The challenge was that you start with a uniform 3-D grid where each grid cell expands at a different rate but remain directly connected to the adjacent cells as time evolves. Since the Space-Time is "etheric" it has no "shear" which would destroy a normal physical system of the same sort.

              Bear in mind that Math is simply a language which can describe things with great precision. Likewise, just because you can demonstrate something with math doesn't mean that math has any physical reality related to the mechanisms of physics. e.g. we know F=ma, but the equation doesn't tell you "why" the universe behaves in this manner. It just IS.

              Good stuff!
              Battles are dangerous affairs... Wang Hsi

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              • #8
                Get too much past the Quadratic Equation, and I am kind of lost.

                Some of you guys are insane.

                Comment


                • #9
                  "Ask not what your country can do for you"

                  Left wing, Right Wing same bird that they are killing.

                  you’re entitled to your own opinion but not your own facts.

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                  • #10
                    Ideal Gas Law

                    The first equation taught in the first chemical engineering class is the ideal glass law.

                    PV=NRT

                    both sides of the equation have units of energy, such as Joules

                    Where P is Pressure, V is volume, N is the factor accounting for the quantity of gas, T is temperature (in absolute terms), and R is the gas constant which is basically a constant of proportionality that makes the units and numbers match. The equation describes average, macroscopic properties, whereas an actual gas has a wide range of distributions.

                    This equation lends itself to use during the use of assumptions such as constant pressure, constant volume, constant temperature, no heat flow etc.,

                    The equation assumes the particles take up no volume, and that there are no interactions among the particles. As the gas is under higher pressure (packed closer), and as the particles become larger/not monotonic these factors become important, and correction factors are taken into account, such as the Van Der Waals equation.

                    http://en.wikipedia.org/wiki/Ideal_gas_law
                    Last edited by lakechampainer; 09 Mar 10, 07:09.

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                    • #11
                      ... I hated PV=NRT almost as much as I hated Q=mc(deltaT)
                      Watts Up With That? | The world's most viewed site on global warming and climate change.

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                      • #12
                        not an equation, but a good parlor trick. The story is that when Gauss was very young in school, the teacher said, we'll figure out what all the numbers from 1 to 100 add up to. A second later Gauss said 5,050. He simply paired the numbers:

                        1 + 100 =101, 2 + 99=101 etc. 100/2=50 times

                        101 X 50 = 5,050

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                        • #13
                          Originally posted by lakechampainer View Post
                          not an equation, but a good parlor trick. The story is that when Gauss was very young in school, the teacher said, we'll figure out what all the numbers from 1 to 100 add up to. A second later Gauss said 5,050. He simply paired the numbers:

                          1 + 100 =101, 2 + 99=101 etc. 100/2=50 times

                          101 X 50 = 5,050
                          3+98=101
                          4+97=101
                          5+96=101

                          [...]

                          13+88=101

                          [...]

                          37+64=101

                          [...]

                          49+52=101
                          50+51=101

                          That is so cool!

                          50 pairs, each with a sum of 101.

                          Gauss was a real [email protected]$$!
                          Watts Up With That? | The world's most viewed site on global warming and climate change.

                          Comment


                          • #14
                            Originally posted by The Doctor View Post
                            3+98=101
                            4+97=101
                            5+96=101

                            [...]

                            13+88=101

                            [...]

                            37+64=101

                            [...]

                            49+52=101
                            50+51=101

                            That is so cool!

                            50 pairs, each with a sum of 101.

                            Gauss was a real [email protected]$$!
                            Nice one Lake and Doc! Never heard of that one, but makes perfect sense...
                            Last edited by Pirate-Drakk; 09 Mar 10, 23:15.
                            Battles are dangerous affairs... Wang Hsi

                            Comment


                            • #15
                              A better description of Euler's Identity



                              And an obligitory cartoon

                              How to Talk to a Climate Skeptic: http://grist.org/series/skeptics/
                              Global Warming & Climate Change Myths: https://www.skepticalscience.com/argument.php

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