V15. Relation to Physics The three theorems we have studied: the divergence theorem and Stokes' theorem in space, and Green's theorem in the plane (which is really just a special case of Stokes' theo- rem) are widely used in physics and continuum mechanics, in the study of fields, potentials, heat flow, wave motion in liquids, gases, and solids, and thermodynamics, to name some of the uses. Often partial differential equations which model some physical situation are de- rived using the vector integral calculus theorems. This section is devoted to a brief account of where you will first meet the theorems: in electromagnetic theory. 1. Symbolic notation: the del operator To have a compact notation, wide use is made of the symbolic operator "del" (some call it "nabla") : a aMRecall that the "product" of -and the function M(x, y, z) is understood to be -. Thenax ax we have af afgrad f = Vf = -i+-jaf +-kax ay az The divergence is sort of a symbolic scalar product: if F = M i + Nj + Pk , while the curl, as we have noted, as a symbolic cross-product: i j k acurlF = VxF= a . MNP Notice how this notation reminds you that V .F is a scalar function, while V x F is a vector function. We may also speak of the Laplace operator (also called the "Laplacian"), defined by Thus, Laplace's equation may be written: v2f = 0. (This is for example the equation satisfied by the potential function for an electrostatic field, in any region of space where there are no charges; or for a gravitational field, in a region of space where there are no masses.) In this notation, the divergence theorem and Stokes' theorem are respectively2 V. VECTOR INTEGRAL CALCLUS Two important relations involving the symbolic operator are: (7) curl (grad f) = 0 div curl F = 0 (7') VxVf = 0 V-VxF = 0 The first we have proved (it was part of the criterion for gradient fields); the second is an easy exercise. Note however how the symbolic notation suggests the answer, since we know that for any vector A, we have AxA = 0, A.AxF = 0, and (7') says this is true for the symbolic vector V as well. 2. Application to Maxwell's equations. Each of Maxwell's equations in electromagnetic theory can be written in two equivalent forms: a differential form which involves only partial derivatives, and an integrated form involving line, surface, and other multiple integrals. In a sense we have already seen this with our criterion for conservative fields; we assume F is continuously differentiable in all of 3-space. Then the integrated form of the criterion is on the left, and the differential form is on the right: And we know that it is Stokes' theorem which provides the bridge between these two equiv- alent forms of the criterion. The situation with respect to Maxwell's equations is similar. We consider here two of them, as typical. Gauss-Coulomb Law. Let E be an electrostatic field, arising from a distribution in space of positive and negative electric charge. Then the Gauss-Coulomb Law may be written in either of the two forms (8) V . E = 4np, p = charge density; (differential form)/LE . dS(8') = 4x4, Q = total net charge inside S. (integrated form) These are two equivalent statements of the same physical law. The integrated form is perhaps a little easier to understand, since the left hand side is the flux of E through S, which is a more intuitive idea than div E. On the other hand, quite a lot of technique is required actually to calculate the flux, whereas very little is needed to calculate the divergence. Neither (8) nor (8') is mathematics -both are empirically established laws of physics. But their equivalence is a purely mathematical statement that can be proved by using the divergence theorem. Proof that (8) +-(8'). Let D be the interior of the closed surface S. Then/LE . dS //LV . E dV= by the divergence theorem; = 4nQ, by definition of p and Q.V15. RELATION TO PHYSICS 3 Proof that (8') + (8). We reason by contraposition: that is, we show that if (8) is false, then (8') must also be false. If (8) is false, this means that we can find some point Po: (xo,yo, zo) where E is defined and such that V . E # 47rp at Po; we write this inequality as Say the quantity on the left is positive at Po. Then by continuity, it is also positive in the interior of a small sphere Socentered at Po;call this interior Bo. Then which we write The integral on the right gives the total net charge Qo inside So;applying the divergence theorem to the integral on the left, we get which shows that (8') is also false, since it fails for So. Faraday's Law A changing magnetic field B(x,y, z; t) produces an electric field E. The relation between the two fields is given by Faraday's law, which can be stated (in suitable units) in two equivalent forms (c is the velocity of light): (9) 1 dBVxE = c dt differential form E . dr = ---B .dS integrated formails, As before, it is the integrated form which is more intuitive, though harder to calculate. The line integral on the left is called the electromotive force around the closed loop C;Faraday's law (9') relates it to the magnetic flux through any surface S spanning the loop C. A few comments on the two forms. The derivative in (9) is taken by just differentiating each component of B with respect to the time t. It is a partial derivative, since the com-ponents of B are also functions of x, y, z. In (9') on the other hand we have'an ordinary derivative, since after the integration, the flux is a function of t alone. It is understood in physics that on S the positive direction for flux and the positive direction on C must be compatibly chosen. The magnetic flux through S is the same for all surfaces S spanning the loop C. (This is a consequence of the physical law V . B = 0.) As a result, one speaks simply of "the flux4 V. VECTOR INTEGRAL CALCLUS through the loop C", meaning the flux through any surface spanning C, i.e. having C as its boundary. Once again, though (9) and (9') both express the same physical law, the equivalence between them is a mathematical statement; to prove it we use Stokes' theorem. Proof that (9) + (9'), by Stokes' theorem, if B has a continuous derivative and S is smooth, and finite in extent and in area. (This last equality is fairly subtle, and is taken up in theoretical advanced calculus courses.) Proof that (9') + (9). We show that if (9) is false, then (9') is false: 1
View Full Document
Unlocking...