By P. Hammond
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In order to make progress with this programme we shall need to develop a mathematical language able to deal with distributed vector quantities. This language is known as vector analysis and we shall discuss it in general terms in the rest of this chapter. The results obtained in this study will then be applied to specific electromagnetic problems in the rest of the book. 2. 1. Source Coordinates and Field Coordinates In the last section we discussed the great advantage of field theory in being able to deal with effects distributed in space.
2 we were occupied with the formulation of electromagnetism in terms of the vector calculus. The power and elegance of this formula tion presents a great temptation to the writer of a textbook to present the whole subject as a branch of applied mathematics. It will become easier to resist the temptation when in later chapters we deal with applications of the theory. But even at this early stage it is worth while to take a de tached look at the mathematics and ask what physical information there may be in the various symbols and equations.
14. Explain what is meant by the curl of a vector field. 15. Derive the following relationships: div (au) = a div u + u . grad a curl (au) = a curl u — u x grad a div (u x v) = v . curl u — u . curl v. E. 60 APPLIED ELECTROMAGNETISM 3c 16. 17. Show that and curl (u x v) = (v. V) u — (u . V) v + u div v — v div u grad (u. v) = (v. V) u + (u . V) v + v x curl u + u x curl v. 18. Use the divergence theorem to prove that jjjgradVdv= § Vnds, where n is the unit vector normal to the surface. [Hint: In the divergence theorem divide the outflow into three components along the three rectangular coordinate direc tions.
Applied Electromagnetism by P. Hammond