An expression for the gradient in cylindrical coordinates was obtained. Divergence and curl were also shown in cylindrical coordinates [Transparancy 4]. These expressions are derived in section 3.10 of the textbook, using the definition of divergence in terms of net outflux per unit volume, and the definition of curl in terms of the swirl per unit area.
Another way to derive these expressions is to calculate from the coordinate transformation equations and do all the algebra. (This could take a long time, unless you use Maple!)
As an aside, I also mentioned that Maple can be useful for calculating gradient, divergence, curl, etc. These commands are in the "linalg" package, which must first be loaded using the "with(linalg)" command. Here are some examples.
phi:=ln(sqrt(((x-1)^2+y^2+z^2+a^2)/((x+1)^2+y^2+z^2+a^2))); with(linalg): F:=grad(phi,[x,y,z]); F:=simplify(F); G:=curl(F,[x,y,z]); simplify(G);
[0, 0, 0]
Notice the use of the "simplify" command. This is because F and
G as first calculated by Maple are ugly messes, which Maple doesn't
automatically simplify. You can see that the curl of F has a
particularly nice simplification! (Hopefully you got this nice answer on
HW#05.)
with(linalg): laplacian(phi,[x,y,z]); simplify(%);Notice the use of '%' in order to refer to the previous result. This only works in Maple V Release 5. For all previous releases of Maple, you must use the '"' (ie. the double-quote) to refer to the previous result.
with(linalg): map(laplacian,F,[x,y,z]); simplify(%);Maple's "laplacian" command can't handle vectors as arguments, so we have to manually apply the "laplacian" function to each of the components using Maple's "map" command.
You can optionally specify cylindrical or spherical coordinates in Maple's grad, diverge, curl, and laplacian commands. Check the on-line help for more details.
(You will find two versions -- a "screen" version, which is 400 pixels wide and a corresponding number of pixels long, at a resolution of 100 by 100 dots per inch, and a "print" version, which is generally between 8 and 8.5 inches wide and up to 11 inches long, also at a resolution of 100 by 100 dots per inch. The screen versions generally have a file size of between 60 and 75 K, which downloads reltively quickly. The print versions generally have a file size of between 170 and 225 K, which takes a bit longer to download.)
