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1.6: The Theory of Vector Fields

1.6.1: The Helmholtz Theorem

Ever since Faraday, the laws of electricity and magnetism have been expressed in terms of electric and magnetic fields, E and B. Like many physical laws, these are most compactly expressed as differential equations. Since E and B are vectors, the differential equations naturally involve vector derivatives: divergence and curl. Indeed, Maxwell reduced the entire theory to four equations, specifying respectively the divergence and the curl of E and B.

Maxwell's formulation raises an important mathematical question: To what extent is a vector function determined by its divergence and curl? In other words, if I tell you that the divergence of F (which stands for E or B, as the case may be) is a specified (scalar) function D, $$\div \vec{F} = D$$ and the curl of F is a specified (vector) function C, $$\curl \vec{F} = \vec{C}$$ can you then determine the function F?

Well... not quite. For example, as you may have discovered in Prob. 1.20, there are many functions whose divergence and curl are both zero everywhere - the trivial case $\vec{F} = 0$, of course, but also $\vec{F} = yz \vu{x} + zx \vu{y} + xy \vu{z}, \vec{F} = \sin x \cosh y \vu{x} - \cos x \sinh y \vu{y}$, etc. To solve a differential equation you must also be supplied with appropriate boundary conditions. In electrodynamics we typically require that the fields go to zero "at infinity" (far away from all charges). With that extra information, the Helmholtz theorem guarantees that the field is uniquely determined by its divergence and curl. (The Helmholtz theorem is discussed in Appendix B.)

1.6.2: Potentials

If the curl of a vector field (F) vanishes (everywhere), then F can be written as the gradient of a scalar potential (V): $$\curl \vec{F} = 0 \Longleftrightarrow \vec{F} = - \grad V \tagl{1.103}$$ (The minus sign is purely conventional.) That's the essential burden of the following theorem:

Theorem 1. Curl-less (or 'irrotational') fields. The following conditions are equivalent (that is, F satisfies one if and only if it satisfies all the others):

$$\tag{a} \curl \vec{F} = 0 \text{ everywhere }$$ $$\tag{b} \int_a ^b \vec{F} \cdot \dd \vec{l} \text{ is independent of path, for any given end points}$$ $$\tag{c} \oint \vec{F} \cdot \dd \vec{l} = 0 \text{ for any closed loop}$$ $$\tag{d} \vec{F} \text{ is the gradient of some scalar function: } \vec{F} = - \grad V$$

The potential is not unique, and any constant can be added to V with impunity, since this will not affect its gradient.

If the divergence of a vector field (F) vanishes (everywhere), then F can be expressed as the curl of a vector potential (A): $$\div \vec{F} = 0 \Longleftrightarrow \vec{F} = \curl \vec{A} \tagl{1.104}$$ That's the main conclusion of the following theorem:

Theorem 2. Divergence-less (or 'solenoidal') fields. The following conditions are equivalent:

$$\tag{a} \div \vec{F} = 0 \text{ everywhere}$$ $$\tag{b} \int \vec{F} \cdot \dd \vec{a} \text{ is independent of surface, for any given boundary line}$$ $$\tag{c} \oint \vec{F} \cdot \dd \vec{a} = 0 \text{ for any closed surface.}$$ $$\tag{d} \vec{F} \text{ is the curl of some vector function: } \vec{F} = \curl \vec{A}$$

The vector potential is not unique - the gradient of any scalar function can be added to A without affecting the curl, since the curl of a gradient is zero.

Incidentally, in all cases (whatever its curl and divergence may be), a vector field F can be written as the gradient of a scalar plus the curl of a vector: $$\vec{F} = - \grad V + \curl \vec{A} \quad \text{(always)} \tagl{1.50}$$

In physics, the word field denotes generically any function of position (x, y, z) and time (t). But in electrodynamics two particular fields (E and B) are of such paramount importance as to preempt the term. Thus technically the potentials are also "fields," but we never call them that.