What is necessary to memorize for coordinate systems?

[JVanUW]

I've been told that for upper level physics classes it's imperative to know how to switch between coordinate systems, however I'm unsure of what is exactly necessary to know. For example, today I was reading up on divergence and I noticed that there are formulas for
divergence in spherical and cylindrical coordinates. Same with the laplacian etc. What does one actually need to know by the time they are taking mechanics with differential equations or a little later quantum mechanics? Should these just be memorized? It's not that I'm opposed to learning how to derive them, but I'm a little more ahead in my physics than my math. Thanks

 

[espen180]

Yes, you will need to be able to switch between different coordinate systems. You don't have to memorize them, but you should be able to derive the transformations from one to another.

Some systems can be either utterly trivial or very difficult depending on the coordinate system you use. As an example, try working out the trajectories of uniform linear motion in spherical coordinates, using Newton's 2nd law.

 

[Lavabug]

Converting between cartesian and polar/cylindrical & spherical coordinates is one of the first things you'll learn how to do in a multivariable calculus course if not earlier, its very important in EM and classical mechanics.

But I've never been expected to know the divergence, curl and gradient operators in spherical and cylindrical coordinates by heart on an exam, and I'm taking my last EM and optics courses. We used it in QM but its never been a necessity in problem solving.

 

[Jorriss]

Just yesterday I needed the laplacian in polar coordinates and I couldn't remember the exact form of the 'r' component so I... looked it up!

In practice, it's good to be able to derive the formulas so you can convince yourself that they are right but you don't need to memorize them once convinced. Also, in practice, it's conveinent to have things memorized so I try to memorize most things but you always have reference books if you forget.

 

[lisab]

A bit of advice (a.k.a. learn from my fail ): coordinate transformations are ugly, but don't fight it, just learn it.

 

[Polymathiah]

Learn polar and cylindrical as well as general curvilinear coordinates and you should be set, I think.

 

[mal4mac]

I've been told that for upper level physics classes it's imperative to know how to switch between coordinate systems, however I'm unsure of what is exactly necessary to know. For example, today I was reading up on divergence and I noticed that there are formulas for
divergence in spherical and cylindrical coordinates. Same with the laplacian etc. What does one actually need to know by the time they are taking mechanics with differential equations or a little later quantum mechanics? Should these just be memorized? It's not that I'm opposed to learning how to derive them, but I'm a little more ahead in my physics than my math. Thanks

It's only imperative if you need to do it! The particular situation you are faced with will determine what you need to know.

Arfken lists thirteen different coordinate systems - if you find it fun you might want to learn how to convert between them all. Me, I'd wait until you actually need to do it.

Why on Earth would you memorise such things? Good luck with equations 2.145 of Arfken. Look 'em up when you need them!

You can learn to derive them if you've a summer to spare. Me I'd go to the beach and accept the results - someone has already done the "undue amount of algebra", no need for you to do it again. (Unless the exam demands it if course, then you need to practice it...)

 

[Leveret]

I definitely wouldn't dedicate time specifically to memorizing them. Most classes that make heavy use of them will have you practice with them a lot, and you'll end up memorizing some of them as a byproduct of the work you're doing anyway. You'll probably learn things like the polar surface element or the spherical volume element unconsciously in that manner, and you don't really need the rest memorized.

 

[mathwonk]

just learn how the coordinates are assigned to the points. e.g. spherical coordinates work like aiming a telescope. you have three coordinates: the angle of inclination from the horizon (or the declination from vertical), and the angle the shadow of the telescope on the ground makes with a ray from the origin (the base of the telescope), and the distance of the star from the origin.

i.e. you rotate the telescope until it points directly under the star, then you rotate it up until it points directly at the star, and then you focus it for the distance to the star.then you try to relate these numbers to the x,y,z coordinates of the star. e.g. the square d^2 of the distance to the star is d^2 = x^2+y^2+z^2. and if r^2 = x^2 + y^2, where r is the distance of the shadow of the star in the x,y plane, from the origin, then z/d = the tangent of the angle of inclination. etc...once you understand how to derive these relations, just look up the resulting formulas when you need them. its more important to be able to recognize whether the formulas are correct than to try to memorize them.

 

[AlephZero]

its more important to be able to recognize whether the formulas are correct than to try to memorize them.

Especially since different books may use different conventions. For example the "axis" of cylindrical polars may be the cartesian x or z axis. and the polar coordiate system may be written as (r, theta, z) or (x, r, theta).

If you haven't seen those sort of variations yet, you've led a sheltered life!