Vectors and coordinate transformations

[rockyshephear]

Can anyone explain why it's important to be able to take vectors in an x,y,z coordinate system and be able to transform them into other coordinate systems. Could not all vector considerations be grappled with in the standard x,y,z coordinate systems? How important is this ability to physicists and mathematicians wrt to how often this is done? All the time, occassionally, hardly ever?

 

[Dale]

A coordinate system is usually chosen because it matches some symmetry of the system that you want to analyze and so the math becomes simpler in that coordinate system. Most important problems are halfway solved once you figure out how to write the equations in the right coordinate system.

 

[rockyshephear]

Can you give a minimal example of the choice of a cs to match the symmetry of a system?
Thx

 

[Dale]

Sure, if you want to calculate the stress in a set of turbine blades (or even just a spinning disk) you would probably want to use cylindrical coordinates. If you want to calculate the fields from a dipole antenna you would probably use spherical coordinates. Etc.

 

[nicksauce]

Spherical/rotational symmetry appears EVERYWHERE in physics. I'm having a hard time coming up with examples that don't have spherical symmetry. Spherical coordinates are so very important for this reason.

 

[rockyshephear]

Thanks. Can you reference an actual problem so I can see how it's done. On the web possibly? Or a textbook?

 

[rockyshephear]

Spherical coordinates are very important

Yet I could define a sphere easily in (I guess what you call R^3 space, x,y,z coordinates. I guess it would be more difficult though now that I think about it.

 

[fluidistic]

Spherical coordinates are very important

Yet I could define a sphere easily in (I guess what you call R^3 space, x,y,z coordinates. I guess it would be more difficult though now that I think about it.

x^2+y^2+z^2=1 is a sphere of radius 1. In spherical coordinates it becomes \rho ^2 =1. Don't know if this helps or I misunderstood you.

 

[nicksauce]

Example: use the Maxwell equation \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} to find the Electric field everywhere inside a sphere of uniform charge density \rho.

Notice that there is a spherical symmetry in this problem. Therefore we know that \mathbf{E} will be entirely in the radial direction and depend on only the radial coordinate. Voila! We have already turned a 3 dimensional problem into a 1 dimensional problem simply by considering rotational symmetry. Now we apply the Maxwell equation using the divergence in spherical coordinates (see http://mathworld.wolfram.com/SphericalCoordinates.html)

This gives
<br /> \frac{d}{dr} E + \frac{2}{r}E = \frac{\rho}{\epsilon_0}.<br />

The solution to this ODE, with the condition that E(r=0) = 0, is
<br /> E(r) = \frac{\rho r}{3 \epsilon_0}

Now try doing the same thing in Cartesian coordinates. It would be excruciating!

 

[rockyshephear]

I get it! Thanks.

 

[rockyshephear]

Hey, to get the pretty mathematical symbols do you use a scripting language, keyboard shortcuts or a software you have to download? What is it and and do I do it? Dunka

 

[nicksauce]

The language is called LaTeX. You can find plenty of introductions on the subject with a google search, but this one looks pretty good http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/ .

Also you can click on any equation to see the code for it. I imagine you'll be able to pick up the very basics that way.

 

[rockyshephear]

\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}

 

[MATLABdude]

\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}

[itex ] fits stuff on one line, so that it's inline.

[tex ] lays out equations when you want them to stand alone.

Compare the use of \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} with:
\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}

/Nit-picky \LaTeX user.

 

[rockyshephear]

I'm not a big fan of plastics but I love LaTeX. So this forum's message boxes are LaTeX friendly. How can you tell if other text editors are as well? Is there a standalone editor you can have on your personal computer that allows this scripting? Maybe there's a LaTeX forum, sorry if this is in the wrong location.

 

[jtbell]

Try Googling for "latex for windows" (or whatever your favorite operating system is). It's actually a complete document-preparation system. Equation display is only part of it. It's the standard way to prepare and submit physics and math journal articles. When you submit an article to the Physical Review, for example, you write it up in LaTeX using PR's formatting macros, and it looks exactly like it will appear in print. After they've approved your article (usually after you've made some revisions), the PR people simply drop the LaTeX file into their publishing system. It saves a lot of work for them.

And if PR doesn't accept your article, it's easy to submit it to another journal because all you have to do is substitute the other journal's formatting macros.