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Additional info for Mathematical Methods for Physicists: A Concise Introduction

Example text

In ¯uid dynamics, the curl of the velocity ®eld v…r† is called its vorticity (for example, the whirls that one creates in a cup of co€ee on stirring it). If the velocity ®eld is derivable from a potential v…r† ˆ Àr…r† it must be irrotational (see Eq. 51)). For this reason, an irrotational ¯ow is also called a potential ¯ow, which describes a steady ¯ow of the ¯uid, free of vortices and eddies. One of Maxwell's equations of electromagnetism (AmpeÁre's law) states that r  B ˆ 0 j; where B is the magnetic induction, j is the current density (per unit area), and 0 is the permeability of free space.

We now consider the inverse-square central force of gravitational and electrostatics. Newton's second law then gives n; mdv=dt ˆ À…k=r2 †” …1:36† where n” ˆ r=r is a unit vector in the r-direction, and k ˆ Gm1 m2 for the gravitational force, and k ˆ q1 q2 for the electrostatic force in cgs units. First we note that v ˆ dr=dt ˆ dr=dt” n ‡ rd n”=dt: Then L becomes n  …d n”=dt†Š: L ˆ r  …mv† ˆ mr2 ‰” …1:37† Now consider d dv k k …v  L† ˆ  L ˆ À 2 …” n  L† ˆ À 2 ‰” n  mr2 …” n  d n”=dt†Š dt dt mr mr ˆ Àk‰” n…d n”=dt Á n”† À …d n”=dt†…” n Á n”†Š: Since n” Á n” ˆ 1, it follows by di€erentiation that n” Á d n”=dt ˆ 0.

In particular, if both divergence and curl are speci®ed everywhere and if they both disappear at in®nity suciently rapidly, then the vector ®eld can be written as a unique sum of an irrotational part and a solenoidal part. In other words, we may write V…r† ˆ Àr…r† ‡ r  A…r†; …1:89† where Àr is the irrotational part and r  A is the solenoidal part, and (r) and A…r† are called the scalar and the vector potential, respectively, of V…r). If both A and  can be determined, the theorem is veri®ed.

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