integrationem

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

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

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

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

View Full Document
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
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
Ask a homework question - tutors are online