Vector Calculus - Feynman's treatment

         This is just a shout out for the treatment of "Differential Calculus of Vector Fields" and "Vector Integral Calculus" given in Chapters 2 and 3 of Richard Feynman's "Lectures on Physics", Vol II.
         In my opinion, when these topics are presented by math professors they are often separated from their real-world origins and applications. Separated like so, the defined entities have no physical life so to speak and just exist as analytical tools (however useful they might be). It always bothered me that this topic, usually encountered near the end of the Calculus sequence, never came to life the way that most of the rest of Calculus did. Of course, what I did not realize, was that that life was just over the Physics horizon (Electromagnetism, Fluid Dynamics, etc) when I turned to study more abstract mathematics.
         The theorems are quite beautiful in that they establish properties which hold for general surfaces curves. That is, it might be pretty easy to analytically establish a result for a given surface (like a section of a sphere) where known properties of that surface can be used. But to establish a result for any surface is a pretty nice analytical accomplishment. On the other hand, if there is a physical interpretation involved, picturing the situation might bring the abstract result to life and make it more memorable!!
     In any case, I am pointing the work out for anyone who might be interested and for anyone who might like to be reminded, here are the four summary results of the two chapters.

1) The operators δ/δx, δ/δy, δ/δz, can be considered as the components of a vector operator ∇ and the formulas that result from vector algebra by treating this operator as a vector are correct.

2) The difference of the values of a scalar field η at two points equals the line integral of the tangential component of the gradient of that scalar along any curve between the first and second point.
   η(2) - η(1) = line integral of ∇η ･ ds along any curve connecting (1) to (2)  , ∫  ∇η ･ ds, where ds is the differential segment in the direction from (1) to (2), that is ds  = t ds where t is the unit tangent vector along the curve. So η(2) - η(1) = ∫  ∇η ･ t ds.

3) The surface integral of the normal component of any vector field C over a closed surface S equals the integral of the divergence of the vector field over the interior volume V. That is
∫  C･n da = ∫ ∇･C dV where the left integral is taken over the closed surface S and the right integral is taken over the volume V.

4) The line integral of the tangential component of any vector field C around a closed loop Γ equals the surface integral of the normal component of the curl of that vector field over any surface S bounded by Γ. That is ∫ C ･ ds = ∫ (∇xC) ･ n da, where the left integral is taken over Γ and the right integral is taken over the surface S.

   And that's what vector calculus is all about.