Ok, sorry about the disjointedness, everyone.  I started this thread for a discussion a friend and I were having about vectors in a polar frame, but the actual discussion ended up happening over IM instead of the forum.  So, anyway, switching to didactic mode, I'll try to rehash what we talked about for general review.

Hmm, I'm trying to figure out how to upload a picture.  All this would make much more sense with a few diagrams.

Anway, normally when we represent a vector in 2 dimensions in elementary physics, we write it as the sum of components along some x- and y-axes:



We can write our 2-dimensional vector as a linear combination of any two linearly independent vectors in the plane , and our choice of and indicate our selection of a particular basis for our vectors, namely the Cartesian basis.

For some problems, however, the Cartesian basis becomes rather cumbersome to use.  Say, for example, that we want to write out the centripetal accelleration and the velocity of a particle in centripetal motion at constant speed.  We know that the accelleration is pointing inward along a radial direction, and that the velocity is tangential.  For this problem, it becomes more convenient to choose a non-Cartesian basis, one that uses a radial unit vector and a tangential unit vector instead of and

The polar unit vectors at a point in specified by are given by:
     
     

If I had a picture, this would be easier to see and verify by derivation.

Note that points tangentially in the counter-clockwise (increasing ) direction.

So, with this choice of basis, we can simplify the vectors in our centripetal motion problem to:





Here I have assumed that the particle is moving counter-clockwise and so v is positive, but it need not be so.

Now navigating in a new basis can be weird, but there are several things that carry over from the way you're probably used to working with vectors.  The first of these things is the mechanism of the dot product.  The dot product is a geometrical object, a property having to do with the space , not with your choice of basis, so switching from the Cartesian frame to the polar frame does not affect how the dot product works.

The dot product is a bilinear operator (linear on both vectors involved) that is fully defined in a space by knowing how any particular basis dots among itself.  For example, in an arbitary basis , the dot product between two arbitrary vectors and is given by



       

In other words, the dot product obeys the distributive property with addition, and you can pull constants out front.  Because of these two properties, all you have to know about the space to be able to calculate the dot product is how the basis vectors of any particular basis for that space dot with each other. 

Now the norm, or length, of a vector is defined in terms of the dot product.  For a given vector , the norm or length is given by



Anyway, with that out of the way, we can get to what is probably the most important property of both the Cartesian basis and the polar basis: orthonormality.  What this means is that all the vectors in either basis are unit vectors, and they are perpendicular to one another.  Observe that


     
     
     
     
     
     

So is a unit vector.



     
     
     
     
     
     

So is a unit vector.


     
     
     
     

So and are orthonormal [unit vectors all mutually orthogonal (perpendicular)].

The important simplifying result of all this is that, now that this has been proved, you can treat and like and when you dot them, just multiplying the coefficients of the same basis vector and ignoring any cross-terms.  Thus, if you have two arbitrary vectors and , then



because, when you expand the dot product, the dot products and .

And the last property I'll metion here is the one I raced through last post.  You can calculate the length of a vector in the polar basis the same way you would in a Cartesian basis - by taking the square root of the sum of the squares of the components, because, applying the last result,


     
     


Phew.  Anyway, that should give a basic explanation of what the polar coordinate vector basis is and how to use its basic properties.  There is one other very important property I haven't mentioned yet, which is that, unlike the Cartesian basis, the polar basis is not constant - it rotates as you increase .  This has some important consequences, such as when you are taking derivatives.  I'll have to cover that one in a later post, though.

Last edited by M@Man (2005-09-13 16:10:44)

is ket vectors the solution for subparticle experimenTAL ANALYSIS, to control or limit the lost of energy displayed and error calculations, IF not so, then what noncommutative operator would be