V15 Relation t o 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 theorem 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 derived 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 aM Recall that the product of and the function M x y z is understood to be Then ax ax we have af af grad f Vf i j ax ay af k az N j Pk The divergence is sort of a symbolic scalar product if F M i while the curl as we have noted as a symbolic cross product i a curlF V x F M j N k a P 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 respectively V VECTOR INTEGRAL CALCLUS 2 Two important relations involving the symbolic operator are div curl F 0 V VxF 0 curl grad f 0 VxVf 0 7 7 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 A x A 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 equivalent 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 8 L V E 4np E dS 4x4 p charge density differential form 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 8 Proof that 8 Let D be the interior of the closed surface S Then L E d S L 4nQ V E d V by the divergence theorem by definition of p and Q V15 RELATION T O PHYSICS Proof that 8 3 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 a t Po we write this inequality as Say the quantity on the left is positive a t 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 VxE E dr 1dB c dt differential form ails B dS integrated form 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 components 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 flux V VECTOR INTEGRAL CALCLUS 4 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 If 9 is false this means that at some point Po V x E 10 VxE 1d B c dt 1d B we write this c dt 0 This means that …
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