[math-fun] surfaces of revolution & differential equations
Suppose I have a system of partial differential equations in 3D with rotational symmetry about an axis. Is it trivial to convert this system into an ordinary differential equation in 2D, such that the solution, considered as a surface of revolution, is the 3D solution? Ditto in the reverse direction -- from 2D to 3D.
Somehow this problem statement is confusing me. Is the solution of your system a function f(x,y,z), and then the "surface of revolution" of which you speak is the surface f(x,y,z)=0? If I have gotten this wrong, please clarify with the correct form of the solution. Intuitively, it would certainly seem as if the answer to your question should be "yes". Suppose the axis of rotational symmetry to be the z axis. Then new coordinates would be z and r = sqrt(x^2 + y^2). On Tue, May 31, 2011 at 9:24 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Suppose I have a system of partial differential equations in 3D with rotational symmetry about an axis. Is it trivial to convert this system into an ordinary differential equation in 2D, such that the solution, considered as a surface of revolution, is the 3D solution?
Ditto in the reverse direction -- from 2D to 3D.
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You are right; I didn't do a very good job of explaining. Here's an example: in 2D, a catenary models a hanging chain. What is the equivalent 3D structure (presumably some sort of hanging soap bubble) ? Is it a revolved catenary or something else? At 07:24 AM 5/31/2011, Allan Wechsler wrote:
Somehow this problem statement is confusing me. Is the solution of your system a function f(x,y,z), and then the "surface of revolution" of which you speak is the surface f(x,y,z)=0? If I have gotten this wrong, please clarify with the correct form of the solution.
Intuitively, it would certainly seem as if the answer to your question should be "yes". Suppose the axis of rotational symmetry to be the z axis. Then new coordinates would be z and r = sqrt(x^2 + y^2).
On Tue, May 31, 2011 at 9:24 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Suppose I have a system of partial differential equations in 3D with rotational symmetry about an axis. Is it trivial to convert this system into an ordinary differential equation in 2D, such that the solution, considered as a surface of revolution, is the 3D solution?
Ditto in the reverse direction -- from 2D to 3D.
participants (2)
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Allan Wechsler -
Henry Baker