About the lunar eclipse of April 15 , 2014

Here are images of the lunar eclipse  which took place on April 15 , 2014 . I chose as an example location the city of Miami in Florida. The  astronomy software I used is Starry Night. Click on the images for more info and to view enlarged images in a photo gallery.

In a lunar eclipse , the position of the Earth is between the Sun and the Moon.The Moon revolving in its orbit around Earth passes through Earth’s shadow.

Here is a heliocentric view of the inner planets of the solar system for April 15 , 2014 (done with Starry Night).


The same view is zoomed in and centered on the Earth and the Moon (the Earth and Moon dimensions are not to scale).The yellow lines represent the approximate direction of the Sun’s rays (done with Starry Night and Photoshop):


Click here to see an enlarged and detailed view of the solar system for April 15 , 2014. 

And click here to see an article explaining the reddish color of the moon during the eclipse.

Equations in 3 D and a surface

I have made two 3D drawings ,   one representing the Einstein field equations of General Relativity and the other for the Dirac equation . The drawings were made using MathType , Photoshop , Illustrator and Mathematica . The Dirac and Einstein equations are two of the most important equations of advanced physics of the twentieth century.

I have already inserted a less complex drawing of the Einstein equations in the slideshow for the books page .

Here is the drawing for the Einstein field equations:


I tried to represent at the bottom  the curvature of space-time by a massive object (the sphere or spherical body in the middle of the picture).

The equation at the top is the geodesic equation of motion in General Relativity.

I have studied General Relativity and Tensor Calculus in the past and I’m going to describe some elements of the equations in the two drawings . For more details one can refer to books , courses and works about advanced physics and about  these subjects .

In the geodesic equation of motion , the Christoffel symbols of the second kind are related to the Christoffel symbols of the first kind [ρσ,ν] and are given by


I think the field equations of General Relativity are the most important physics equations elaborated by Einstein . These equations were part of a successful attempt to unify Gravity and electromagnetism . They contain within them as special cases and at certain conditions other known equations such as the mass-energy equivalence equation ( E = m c² ) , Newton’s law of universal gravitation F = -\frac {\left (G m_ 1 m_ 2 \right)} {r^2}\hat {r}  , and Maxwell’s equations

The Einstein equations with the cosmological constant added are: eins-eq-cosmo-cstThe left-hand side of the field equations without the cosmological constant is the Einstein tensor , and the right-hand side is the constant χ multiplied by the stress-energy tensor.

Here are some more details about the elements of the equation (which represents a set of partial differential  equations):


 And here is the drawing for the Dirac equation:


I’ve placed in the middle of the drawing the condensed simple form of the Dirac equation with the partial differential symbol ∂ in Feynman slash notation . The background of the picture pertains to particle physics , particle accelerators and quantum mechanics , which are domains where the Dirac equation is used and applied.

Here is a brief description of this equation :


At the top one can see the expanded  Dirac equation ( in terms of the gamma matrices ) written around the 3 D ring . Any similarity with the ring from ‘The lord of the rings’ novel and movies is unintentional.

Finally , here is a  nice looking 3D surface I obtained with Mathematica and customized with Photoshop. It took a little more time to render but I think it was worth it. I was inspired by the book  ‘Graphics with Mathematica  Fractals, Julia Sets, Patterns and Natural Forms ‘ , and I made my own modifications and changes to the Mathematica input and output . Click on the picture to see an enlarged version.


 The Mathematica code for the surface is:


Enhanced by Zemanta

About the gamma function , math , and language dictionaries

I have a good knowledge of  English (see my literary books page) , and I have a number of  reliable language dictionaries I consult from time to time. I also know about science and mathematics (see my science books page and the problem solvers page) , and I sometimes find mathematical expressions or physics formulas interspersed between scientific definitions in dictionaries. But occasionally I have noticed that math expressions or formulas published in these reference works contain errors and inaccuracies or seem to have been published hastily and without enough care. Whether they have been published using a math writing software or html related coding , the lack of accuracy or completeness is visible.

One example from physics is that of the Schrodinger equation (HΨ = EΨ)  inserted in a scientific definition in a renowned English language dictionary .The summation sigma symbol ( ∑) in the expanded Schrodinger equation is shown followed by the index of summation  and by the lower and upper bounds of summation as if they were multiplied by it . At other times parentheses are lacking  or a plus (+) sign , a letter or variable is omitted.

In one known language dictionary there was an error in the integral definition of the gamma function. I will not name any dictionaries  here , but in any case the error remained for a few years and  was later corrected in newer editions. The gamma function is related to the factorial by :gamma-def-1

and is defined by the improper integral

gamma-true-defIn the dictionary the definition was :

gamma-changedSo the variable of integration changed from dt to dx . In the online version of the dictionary ,the same integral expression is shown with the correct variable of integration , but it is missing a minus (-) sign and a letter variable.

Out of curiosity , I had the idea to find out what the value of the integral would be with a change of variable or if more than one letter or parameter were changed.

For the integral above with dx as integration variable , exp(-t) is constant , and the solution of the integral is :

integ-sol-01A plot of the solution gives (done with Mathematica):


Here is a 3D plot of the absolute value of the function  fct-solution in the complex plane (with the help of Mathematica) :

A generalized solution of the integral with dx and with arbitrary bounds of integration is the following:

integ-sol-02Another definition of the gamma function is :

gamma-true-def-logIf we change the integration variable we get:


One could go on making variations to the gamma function definitions and finding the corresponding solutions.

That was an excursion caused by a mistake in a math definition in a dictionary . Errors can be sometimes found even in science dictionaries and they are often corrected , but I think  if language dictionaries are able to write mathematical expressions and formulas more carefully and clearly , they would become more accurate and subsequently more appreciated.

Enhanced by Zemanta