In any case, it is intriguing and at the same time fascinating this obstruction to the existence of divisive real algebras. Our question in the future will be: What is the ability of octonions to describe the fundamental patterns of the universe? There is a long way to go, but we hope that tiredness will be well compensated for by the landscape.
Symmetry, Anti-Symmetry, and Symmetry Breaking VI
It is impossible, on our way to describing Nature's patterns through octonions, not to stop at Maxwell Station.
A vector field F It is a set of points in space where at each point there is a vector acting. For example, at each point near the Earth there is the vector gravitational force acting. This means that if one is at this point, the gravitational force will act on him and he will fall toward Earth if nothing can hold him back. The moon, for example, is falling, but due to its speed it is propelled into a resulting circular motion around the earth. Another example is the magnetic field near a magnet. Or let us remember that an electric current produces a magnetic field around its path. An electric charge produces a radial electric field analogous to the gravitational field of the earth.
Michael Faraday, (English, 1791 - 1867) discovered electromagnetic induction. The constitution of matter was a mystery and the relationship between light and electromagnetism was unsuspected.
James Clerk Maxwell (Scottish, 1831 - 1879) published his famous Treatise on electricity and electromagnetismin 1873. Maxwell delved into Faraday's electrical research and set out to formulate a mathematical theory of electricity and magnetism. Maxwell's mathematical approach was so brilliant that it allowed him to theoretically discover the electromagnetic waves. He introduced the notion of rotational rot F of a vector field F.
Gottfried Wilhelm Leibniz (German, 1646 - 1716) one of the discoverers of Differential and Integral Calculus saw derivation as an “operator D that acts on functions ”.
Only two hundred years later, the brilliant English engineer Heaviside discovered that this idea had profound practical consequences in human life by solving important electro-technical problems that did not allow treatment by conventional methods of calculus known at the time.
Oliver Heaviside (English, 1850 - 1925) was the first to make extensive and effective use of the derivation operator. D in your famous Operational Calculation. Heaviside alone studied Maxwell's treatise on electromagnetism and simplified Maxwell's twenty equations into four.
Maxwell, who knows mathematics and physics, unveiled a remarkable mathematical antisymmetry present in nature. Probably his mathematical ability was the main factor that allowed him this genius discovery. In particular, Maxwell used his own ideas of vector analysis to develop his imagination about the electric field. AND and the magnetic field H.
Maxwell accepted that one does not live without the other but, surprisingly, obeys a simple mathematical relationship and perfect antisymmetry. Either of them may vary over time, but it may not vary at all. They can only vary conditioned on each other. In other words, the speed of each is closely linked to the other.
Surprise is the mathematical form that relates them as time goes by. So when the electric field AND varies, your speed ¶AND/¶t - or its rate of instantaneous temporal variation - is not double, not half, not even a fraction, or even a multiple of the other.
Maxwell deciphered the puzzle with the notion of rotational of a vector field. The speed of one is a multiple of the rotational of the other. Like this, ¶AND/¶t is a multiple of rotational rot H of the magnetic field H. And, astonishingly all, the factor required by Nature is nothing more, nothing less than speed. ç from light! That is,
¶AND/¶t = ç rot H.
We will not worry at the moment about spelling out the rotational formula, because we just need to know its perpendicularity to the vector H.
Imagine the vector H as a force circling the circumference Ç of a small disk s. That is, at each point of Ç have a vector H representing the magnetic field. A very useful visualization is that of an arrow coming out of the point of application of the vector. As the point slides over the circumference Ç this arrow changes, but without sudden changes. In mathematical language it changes continuously. If we project the arrow H over the tangent line to Ç at each point of application of Hthen we have the tangential component of H who does work on the way Ç.
The work t accomplished by this tangential force the Ç it's called circulation of H in Ç. If we divide this workt through the area of the small disk sthen we will have the circulation density per m2.
In fact, we assume the radius of the disk tending to zero, and therefore, at the limit, we get an amount called the perpendicular projection of rot. H. The perpendicular projection of the rot vector H is applied to the center of the discs s and therefore is perpendicular to the vectors H that circulate around its circumferences, and its magnitude, or intensity, is the circulation density of H. To find out if the perpendicular component of the rot vector H points to “up” or “down” the disc, think of the right hand rule, that is, think that H have the direction of the four fingers of your right hand and that rot H it is the thumb perpendicular to them.
If H circular at the border line Ç of a surface s in space, such as a half sphere, that is, like a half of an orange's peel, then the work that force performs Ç, is exactly the rot flow H across s. The perpendicular component of the rot arrow H pierce the small discs s perpendicularly forming a flow.
Once understood the notion of rot Hwe can imagine a surface s having vectors H circling at your border Ç and the perpendicular projections to s of the rot vectors H crossing S. The amount of this flow is exactly the amount of work given by the circulation of H in Ç. This is what guarantees the famous Stokes theorem. However, the perpendicular projection to s rot vector H at a particular point is just the density of circulation per m2 in H around that point.
For example, imagine s as the cross section of a cylindrical wire and Ç as the circumference of s. In this case, H circulates in the cylinder wall while an electric current passes through the wire with exactly the same direction of rotation. H. That is, the tangential component of the arrow H slides over Ç always tangent to it and the rot arrow H is parallel to the electric current vector i.
Maxwell envisioned that when the magnetic field varies over time, its rate of instantaneous temporal variation is related to the electric field, and surprisingly behaves in an antisymmetrical manner, say, with respect to what happened to the rate. of instantaneous temporal variation of the electric field AND.
¶H/¶t = - ç rot AND,
formula known as Faraday's Law, while:
¶AND/¶t = ç rot H,
formula known as Ampere's Law. These names are due to the experimental discoveries of the great scientists Faraday and Ampère in their laboratories. However, the above mathematical formulations are from Maxwell and simplified by the English engineer Heaviside.
Could one imagine greater simplicity and beauty than this? Who wants you to submit your proposal.
Even more striking is the fact that nature behaves according to the algebraic operations possible with these vectors.
The instantaneous rate of change of the electric flow density is given by the divergent of AND. Without having to spell out the div formula AND, we remember that such an operation has the following property:
div rot F = 0
for any vector field F which admits at least the second derivative. We could intuitively say here that F has "acceleration" as expected from "things present in nature". So we have:
div (¶AND/¶t) = ç div (rot H) = 0.
Note the vector algebra working here, that is, the div changing position with the constant ç. Moreover, div also shifts positions with the derivation, that is, we can first calculate the div of AND and then the speed ¶_ / ¶t. So the div of speed ¶AND/¶t is the same as speed ¶_ / ¶t from div AND:
¶ (div AND)/¶t = div (¶AND/¶t) = ç div (rot H) = 0.
Therefore, we conclude from Maxwell's equations that ¶ (div AND)/¶t = 0. This means that the
velocity ¶_/¶t from div AND is zero, that is, the div AND it does not vary over time and is therefore stalled. Who is stopped can always have only the same value, or a constant value. That means div AND = K = constant.
If an electric field has zero divergent at any time, that is, if div AND = 0 for a certain instant t, so it will always be zero after that because it has to be constant according to the last equation above.
To continue our appreciation of Maxwell Station, we will need to draw on a small portion of the work of one of the three greatest geniuses in mathematics: Carl F. Gauss. This is what we will do in the next column.
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