This is the third part of a multipart series on Maxwell's equations of electromagnetism. The discussion leading up to the first equation got so long that we had to separate it into two parts. However, the ultimate goal of the series is a definitive explanation of these four equations; readers will be left to judge how definitive it is. As a reminder, figures are being numbered sequentially throughout this series, which is why the first figure in this column is Figure 16. I hope this does not cause confusion. Another note: this is going to get a bit mathematical. It can't be helped: models of the physical universe, like Newton's second law F = ma, are based in math. So are Maxwell's equations.
In the previous installment, we started introducing the concepts of slope and discussed how calculus deals with slopes of curved lines. Here, we'll start with calculus again, with our ultimate goal being the understanding of Maxwell's first equation.
More Advanced Calculus
We have already discussed the derivative, which is a determination of the slope of a function (straight or curved). The other fundamental operation in calculus is integration, whose representation is called an integral:
where the symbol ∫ is called the integral sign and represents the integration operation; f
(x) is called the integrand and is the function to be integrated; dx is the infinitesimal of the dimension of the function; and a and b are the limits between which the integral is numerically evaluated, if it is to be numerically evaluated. (If the integral sign looks like an elongated "s", it should — Leibniz, one of the cofounders of calculus [with Newton], adopted it in 1675 to represent "sum", since an integral is a limit of a sum.) A statement called the fundamental theorem of calculus establishes that integration and differentiation are the opposites of each other, a concept that allows us to calculate the numerical value of an integral. For details of the fundamental theorem of calculus, consult a calculus text. For our purposes, all we need to know is that the two are related and calculable.
Figure 16: The geometric interpretation of a simple integral is the area under a function and bounded on the bottom by the x-axis (that is, y = 0). (a) For the function f(x) = x, the areas as calculated by geometry and integration are equal. (b) For the function f(x) = x2, the approximation from geometry is not a good value for the area under the function. A series of rectangles can be used to approximate the area under the curve, but in the limit of an infinite number of infinitesimally-narrow rectangles, the area is equal to the integral.