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.
Johann Carl Friedrich Gauss (German, 1777 - 1855) was one of the formulators of the theory of magnetism. He is considered one of the three greatest mathematicians of all time along with Newton and Archimedes. To reflect on a famous theorem known in the West as the Gauss Divergence Theorem, we will intuitively examine the equations below:
òòs D . ds = what
òòs G . ds = - M
In Russia this important theorem is known as the Ostrogradski Theorem (Mikhail Ostrogradski, 1801-1861). Indeed, the fuller history of this subject must be related to the history of Green's Theorem (George Green, 1793-1841) and Stokes's Theorem (George Gabriel Stokes, 1819-1903). The theorem discovered by Green in England was also discovered by Ostrogradski in Russia and today we conceive of these theorems all together in a generalized form called Green - Gauss - Stokes Theorem and do not mention Ostrogradski. These theorems are of extraordinary importance in mathematics and physics, but here we are just wanting to give some thought to the version known as the Gauss Divergence Theorem.
The first equation above, known as Gauss's Law, refers to a positive electric charge. what placed inside a closed surface s and the electric flow emanating from it through s.
The charge produces an electric field represented by a vector AND at each point in space and in particular at each point of s. Let's not go into mathematical details about this field, just imagine that it gives us a vector AND at every point in space. This idea is analogous to the idea of gravitational field. F. The only difference is that the vector AND get out of s at one point P while F comes into s at one point P.
The second equation refers to a mass M placed inside a closed surface s and the gravitational flow towards M crossing this surface. We can think of the sun at the center of a sphere with the earth at a point on that sphere. On earth there is always a vector F imaginary pointed to the sun which is the gravitational force that the sun exerts on the earth and M it is the mass of the sun. Of course we are imagining all this from a Newtonian rather than an Einsteinian perspective. Our goal here is just to reflect on symmetries and anti-symmetries that can be understood vectorically.
In the first equation, D is the electrical flux density at a point on the surface s, that is, the amount of electrical flow that passes in a small piece dS from the surface s around the point P. How are we cutting a surface s in a lot of little pieces dS, and we intend to add numbers associated with these dS´s, we use the symbols double-sum over s. In practice, this sum can be separated into two partial sums, one for each surface dimension. Similarly, below we will use the symbol òòòs to indicate that, in practice, the sum of numbers associated with dV´s can be separated into three partial sums, one for each space dimension. In fact, we could just use the symbols of sum over s in both cases because we will not go into the technical details of each sum.
In the second equation we can imagine G as the gravitational flow density at a point in the orbit of the earth. This orbit is an ellipse we can imagine as a line on an imaginary ellipsoid. There will be no loss of understanding if we assume that this ellipsoid is a sphere. In a little piece of ellipsoid around the earth there is an amount of gravitational flow associated with the force F, due to the mass M, which attracts the earth towards the sun. The gravitational force also depends on the mass of the earth, but we are only thinking about the gravitational flow produced by the mass. M and therefore the Earth's mass does not enter the amount of this flux.
Let us try to analyze the above equations as simply as possible. Take the electrical case. There are three elements to look at. The surface s, the right side of the equation and the left side.
The right side is very easy: it is the amount of electrical charge contained within a closed surface. s.
Speaking of surface sLet us first clarify its role in the equation. Imagine a sphere of radius R, and take the opportunity to put in your center Ç an electric charge what. Suppose that what be positive to define the idea of electric flow emanating from it. Thus, this emanation comes out of the charge, because we define it this way when the charge is positive, and passes through the sphere. The geometry of this situation can be visualized as straight lines extending from the center of the sphere outwards. We say that this flow emanates radially from the center Ç out of the sphere. This surface encompasses a space that we will call solid B wrapped by the surface s - in this case B It is a ball.
Using our current conception of vector field we can try to understand one of Gauss's brilliant discoveries. The key idea to keep track of your intelligence is the idea that in every little bit dS from the surface s there's a little bit of flow dy that emanates from the electrical charge as a vector field, and therefore there is a flux density D = dy/ dS at each point of s.
However, to master this density mathematically, one thing is missing. At each point of s the flow can be thought of as an arrow pointing out of s in a certain direction. Since these directions are generally different, we need to give a mathematical form to this difference. We then assume that at one point P in s, the size of the arrow is D = dy/ dS and its direction will be implied when we write D.
It is easier to repeat this reasoning for the surface. s. The unitary vector u perpendicular to s in P It is extremely useful. For example, we can describe the little piece dS in s as a vector whose direction is perpendicular to s in P and whose magnitude is the area dS. Here's a trick: We use the length of a vector to represent an area. We will say that the superficial element of s in P is the infinitesimal vector ds = dS u.
Similarly we can conceive of the flux density vector D in P. We just need to remember that the direction of D is not necessarily perpendicular to s. Let's explain this vector a little further.
Let's use a fundamental idea associated with the idea of flow. It is the idea that the flow part dy that goes through s is the perpendicular part of it the s in P.
How are we going to deal mathematically with this fundamental idea about the flow that goes through s? We already have at our disposal the mathematical elements that easily solve this question. Just consider the product to scale D . ds. This number gives us D . ds = |D| . |ds| . waistband what = |dy/ dS| . dS . waistband what = |dy| . waistband what which is the absolute amount of flow at the point P cos factor what due to the slope at which the flow hits the surface at this point. So the amount of flow that goes through s é D . ds = |dy| . waistband what.
Note that the more inclined the flow at one point Pthat is, the closer to a direction parallel to the surface at P, nearest to zero is its part that crosses it in Pbecause cos 90O = 0! On the other hand, when the flow is perpendicular to s in P he is max because cos 0O = 1! Therefore we have:
D . ds = |D| . |ds| . cos 90O = |dy/ dS| . dS . 0 = 0
in the event that the flow is parallel to s in P and therefore does not cross it, and
D . ds = |D| . |ds| . cos 0O = |dy / dS| . dS . 1 = |dy|.
in the case that the flow is perpendicular to s in P and therefore everything that comes in P through s fully.
Between these two extremes we have an intermediate amount of flow |dy| . waistband what crossing s in P.
Thus mathematics easily solves this situation of great physical importance and needs only the notion of vector and scalar product of vectors.
We immediately interpreted the first member of the famous Gauss equation. Just imagine the sum of all the bits of flow |dy| . waistband what in a lot of points P scattered all over the surface s one point always has many others extremely close to it. At each point P the angle what has a value that depends only on P. The total flow that goes through s is the sum of these infinitesimal flows:
flow emanating from what and crosses s = ys = òòs |dy| . waistband what = òòs D . ds.
We would then like to know how much this sum is. In the case of the sphere s radius R with a load what centrally located Ç in s, Gauss calculated this sum and obtained the value what! This is:
flow emanating from what and crosses s = ys = òòs |dy| . waistband what = òs D . ds = what.
In the gravitational case, we have:
gravitational flow that addresses M and crosses s = ys = òòs |dy| . waistband what = òòs G . ds = - M.
Maxwell showed that the electrical charge density r = dq / dV at one point P, that is, the ratio between the infinitesimal amount of electric charge dqcontained in an infinitesimal volume dV, and the volume dV, is the divergence of density D.
div D = r.
The divergent is a kind of derivative. He operates in D taking its partial derivatives and their sum. This equality is one of Maxwell's celebrated equations.
Imagine, then, the load what no longer concentrated in the center of the sphere sbut distributed over the ball that has the sphere s like shell. In a little piece dV of volume of the ball there is a little load dq. As the charge density r is given by
r = dq/dV
we have, of course:
what = òòòV dq = òòòV r dV.
So supposing that Gauss's calculation also applies in this situation of the load what distributed by the solid B in little pieces dq, by Gauss's Law we can conclude that
òòs D . ds = what = òòòV dq = òòòV r dV = òòòV div D dV.
Applied to gravitational flow, noting that in this case the density r is the mass given by r = dM / dV, and div G = -r (think of this negative sign as indicating that the flow is going to the mass M that produces it), this reasoning gives us:
òòs G . ds = - M = -òòòV dM = òòòV (-r) dV = òòòV div G dV.
We could thus reflect on two examples of the famous result known in the West as the Gauss Divergence Theorem for a flow density. D
òòs D . ds = òòòV div D dV,
but seen with the eyes of those who live after Maxwell. Both Gauss's Law and Maxwell's approach are direct consequences of a point of view that seeks to explore symmetries and anti-symmetries in nature and mathematics, and the analogies between them.
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