integrationem - Math 21a Supplement on Integration and...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 21a Supplement on Integration and Electro-Magnetism The Math 21a Supplement on Electricity and Magnetism presented the equations that describe all known electric and magnetic phenomena. In particular, recall that Maxwell’s equations describe electric fields by a vector valued function of space and time, E (t, x), while magnetic fields are described by a similar object, B (t, x). Then, Maxwell postulates that only such pairs ( E , B ) which obey the constraints div E = ρ , div B = 0, t E = curl B - j , t B = - curl E , (1) appear in nature. Here, the function ρ and the vector valued function j (both functions of time and space) are determined by the distribution and velocities of the various charged materials or particles present. The pair ( ρ , j ) must also satisfy a constraint, which is t ρ + div j = 0 . (2) The purpose of this supplement is illustrate various important ramifications of (1) and (2) which follow from applications of either Stokes’ Theorem or the Divergence Theorem. a) A short review Before getting to the purpose at hand, I will remind you of two definitions. For the first, suppose that v is a vector valued function on R 3 . (In the examples below the vector valued functions under consideration will be E , B and j , thus they may also depend on time.) Now, suppose that S is a surface in R 3 with a chosen normal direction, n . For example, S could be the whole surface of a sphere and n the outward pointing normal, or S could be something much more intricate. With regard to the normal vector n , keep in mind that most surfaces have two possible normal directions which point opposite. The first definition to review is that of the flux of v through S in the direction of the normal vector n . In particular, remember that this flux is defined to be the value of the surface integral S v n dA . (3) Why call (3) ‘flux’? Here is one reason: The value of (3) is evidently proportional to the average amount that v points outward from S along the normal n . Thus, if v is the velocity vector of a
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
moving fluid at each point, then the positivity of (3) indicates that there is a net flow of fluid through S in the direction n . On the other hand , if (3) is negative, then the net flow of fluid through S is in the direction of - n . So, when v is the velocity of a fluid, the integral in (3) acts just like flux should act. When v represents some other physical quantity, the corresponding integral in (3) is still called a flux integral. The second definition in this review requires the specification of a closed path
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

integrationem - Math 21a Supplement on Integration and...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online