Differentiation of Functions With More Than One Independent Variable

Consider the characteristics of the electromagnetic field as itʼs being energized by the moving charge in the antenna. The field is propagating outwards from the longitudinal axis of the antenna and itʼs rotating (as per the left-hand rule) around the antenna. The vector potential has x, y, and z components that are all changing. Itʼs not, therefore, possible to simply differentiate a function containing Ax with respect to y (d/dy) without taking the fact that Az is changing and affecting the situation too. Here, then, are situations where differentiation must be performed on three variables (the three components of A, for example) all at the same time.

When multiple variables are present (as is the case with the x-, y-, and z-axis values of a vector quantity) the equations must be differentiated on the basis of each variable separately while the other variables are temporarily held constant. This operation is called partial differentiation and itʼs represented by the symbol ∂, which is a stylized, curved letter “d” (and not a strange letter of the Greek alphabet). A partial differential equation is an equation for an unknown function of more than one independent variable expressed in terms of a relation between the partial derivatives of the function.

One of the most basic of Maxwellʼs wave equations sets forth the relationship between vector potential (A) and the magnetic flux density (B) and serves as an example of how calculus, the language of electromagnetic field theory, represents the physical characteristics of the field.

Magnetic Flux Density (B) and the Vector Potential (A)

Maxwell described how each of the three vector components of B could be determined based on the components of A. He set forth the following description of the Magnetic Flux Density (B):

•Bx can be determined from AZ and AY

•By can be determined from AX and AZ

•Bz can be determined from AY and AX

We are now ready to express the most basic of the relationships put forth in the Maxwell Wave Equations. The three equations below show how the various partial derivatives of the components of A interrelate to produce the components of B. The cyclic operation of computing one, then another, and finally the other component of a vector in this manner is called the cross product. In this case weʼre calculating the cross product of the partial derivative of each of the vector components.

Figure 3.11 Partial Differentiation to Compute the Components of B

The mathematical relationship between the three vector components in Figure 3.11 is called the “curl of A”. Computing the curl of a vector means processing each of the components of each of the vectors in the manner shown. The group of mathematical operations expressed by the three equations shown above would be written using either of the two following notations:

Copyright 2003 - Joseph Bardwell

The upside-down triangle ( ) is called the “del” operator and refers to the partial differentiation that takes place between the components of the vectors. The notation on the left is read “B equals the derivative cross product of A” and on the right, simply “B equals curl A”. Both notations mean the same thing.

This equation (using the curl operator) means that the magnetic flux density (B) has a rotational relationship to the vector potential (A). This is the basis for the famous Maxwell Equations in Electrodynamics. Maxwellʼs equations express the relationships between the fundamental quantities related to electrical charge and magnetic fields. The full set of equations includes two additional special vector operators grad (gradient) and div (divergence). As with curl, these two operators take a series of vectors, component-by-component, and perform operations on them. The div operator takes a vector field and represents how much the field is expanding or being created at each point. Divergence allows calculation of how much a field is moving in or out and curl calculates the rotational characteristics of the field. The grad operator takes individual scalar components

and equates them to a vector field. The gradient of change over the field shows where the field is changing most rapidly and would give the direction of field movement.

There are various forms in which Maxwellʼs equations are presented, some using curl, div, and grad and others expanding the equations so that they can either be solved or so that other characteristics can be emphasized. The basic equations, in vector form, are shown below.

Figure 3.12 Basic Maxwell Wave Equations in Vector Form

In the wave equations (Figure 3.12) you see the del operator ( ) and, on the right, the cross product symbol (X). On the left, the dot between the del and B or E does not mean multiply. It refers to an operation called the dot product. The dot product is an operation whereby a product is calculated between the length of two vectors and the cosine of the angle between them. Therefore, given two vectors X and Y: X · Y = xy cos θ.

The electromagnetic field not only obeys the mathematical laws expressed in the field equations, it also manifests the strange mathematical consequences of the equations. One of these consequences relates to the variable t (time) in Maxwellʼs equations. By the laws of simple arithmetic it can be deduced that t could be either positive or negative and the equations would not break down. There are no equations that yield infinite or undefined results (like y=1/x when x=0) and there are no equations that are forced into the exclusive realm of imaginary numbers (the square root of negative one). Nothing strange happens arithmetically when t is a negative number. This aspect of the field equations gives rise to a time symmetric characteristic of electromagnetic fields called the Reciprocity Theorem, discussed next.

Copyright 2003 - Joseph Bardwell