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Re: How to plot divergence of gradient as contour plot

  • To: mathgroup at smc.vnet.net
  • Subject: [mg124035] Re: How to plot divergence of gradient as contour plot
  • From: Murray Eisenberg <murray at math.umass.edu>
  • Date: Fri, 6 Jan 2012 04:19:07 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <201201051100.GAA14785@smc.vnet.net>
  • Reply-to: murray at math.umass.edu

You need to use the VectorAnalysis package (carefully!), remembering that:

   (i) the default coordinate system is Cartesian[Xx, Yy, Zz]; and
   (ii) the result of using that package's Grad, Div, Curl is _not_ 
really a "function" of the variables (more about this later).

In your example:


   Div[Grad[Sin[Xx + Yy]]]
-2 Sin[Xx + Yy]


   ContourPlot[
        Div[Grad[Sin[Xx + Yy]]] /. {Xx -> x, Yy -> y},
         {x, -5, 5}, {y, -5, 5}]

   (* or first define a function *)

   f[x_, y_] := Div[Grad[Sin[Xx + Yy]]] /. {Xx -> x, Yy -> y}
   ContourPlot[f[x, y], {x, -5, 5}, {y, -5, 5}]

I find the whole paradigm of the VectorAnalysis package to be 
aggravatingly annoying. After all, from a mathematical perspective, if 
you're given a scalar-valued function with vector inputs, then its 
gradient is just a well-defined vector field -- and that has nothing 
whatsoever to do with so-called "coordinate systems". That is, one ought 
to be able to do something like this -- just like any standard modern 
text on multivariable calculus would show:

   phi[x_, y_] := Sin[x + y]
   Div[Grad[phi[x, y]]]

In Mathematica, with the VectorAnalysis defaults, you'd get 0 as result 
there, since you didn't override the default variable names Xx, Yy, Zz.

Then there's the whole nonsense of the gradient other coordinate 
systems. That, to my mind, is a very odd way to think about things. 
Mathematically, given a scalar field, it has an associated gradient 
vector field. Period. If you want to "express the gradient in terms of" 
another coordinate system, you're no longer finding that gradient; 
you're finding a composite of functions where you go first from that 
other coordinate system to cartesian coordinates, next take the 
gradient, and finally go back from cartesian coordinates in the result 
to the other coordinate system.

(I trust others will disagree with the preceding argument!)


On 1/5/12 6:00 AM, Szymon Roziewski wrote:
> Hello there,
>
> I am concerning to plot something like that.
>
> ContourPlot[Divergence[Gradient[Sin[x + y]]], {x, -5, 5}, {y, -5, 5}]
>
> Plotting divergence of gradient scalar field.
> How can I manage to do that?

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305



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