Hey guys, just a quick question on vectors. I am no stranger to vectors but have never considered them in any co-ordinate system but cartesian.
when a vector is given in spherical co-ordinates for example, is the phi vector the magnitude along an imaginary "ring" in the "sphere"ical co-ordinate space? or is the phi vector always tangential to its ρ(radius) vector and if this is the case, what does the magnitude mean?

forgive my ignorance but i have always only considered phi simply as the angle that a point in space's projection on the x-y plane makes with the positive x axis...

 

Cylindrical coordinates are used very extensively in antenna modeling programs, where you need to describe relative field strength in any given direction from the origin. This notatiion is convenient because you can describe any direction with just two figures, azimuth and elevation (or zenith). For absolute field strength at any given point in space, you need a distance coordinate along a vector as well, and this may be more convenient to describe with spherical coordinates, since each coordinate has the same "weight".

One should always attempt to be "fluent" in different notation systems...it gives you a much more coherent view of the universe!

Eric

 

There are no universal standards for coordinate systems, so you're better off drawing a diagram to show unambiguously what your terminology is.

The unit vectors in spherical coordinates are mutually orthogonal. There's a good picture on wikipedia's page http://en.wikipedia.org/wiki/Spherical_coordinates (the picture is http://en.wikipedia.org/wiki/File:Spherical_coordinate_surfaces.png) that shows the three coordinates. The unit vectors will be perpendicular to the red, blue, and yellow surfaces. Unlike cartesian unit vectors, the θ and phi (missing from the symbols...) unit vectors change their directions in space for different values of θ and phi.

You can use an orthogonal transformation (i.e., the matrix given at http://en.wikipedia.org/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates) to convert back and forth between the coordinate systems. These orthogonal matrices have a determinant of 1 and their transpose is their inverse (nice, if you've ever had to invert matrices by hand without a calculator). They form a representation of the SO(3) group.

 

Thanks for the replies guys,
perhaps I did a crappy job in explaining what i'm on about. Spherical coordinates I understand and have no issue with. Its depicting a vector in this system that i do not fully understand just yet. As I said, i do not know how to interpret the phi vector for example (and the same applies to the θ vector)

**edit** well i think i finally understand the maths of it now so its really helped me grasp it. thanks for help

 

This might help...

Consider what happens when, for some arbitrary point, you increase one of the coordinates while holding the others constant. Say you increase r by one unit. Just for sake of this example, suppose we're using centimeters. Wherever the point was, you'll make it one centimeter farther from the origin. Draw an arrow in space from the point's old position to the new position; this is the "basis vector" for r.

Now suppose, for that arbitrary point, we increase theta instead of r. Suppose we're measuring all our angles in degrees. We scoot that point along, increasing its theta by one degree. It stays the same distance from the origin, and phi doesn't change either. Things get interesting here, because if the point was somewhere close to the origin, it moves only a little ways, but if it's kilometers away, it'll move far. Whatever the case, draw an arrow from its old position to its new, and that's the theta basis vector at that location in space.

Of course, we should be doing proper calculus, and moving our points by infinitesimal (extremely small) amount, not by one unit of measure.

Imagine going through this process for every point in space. Anywhere you go, there's a well defined radial basis vector, theta basis vector, and phi basis vector defined by the coordinate system. Now suppose you want to describe how the wind is blowing at some place. That's just a vector (imagine an arrow drawn in space) which exists just fine on its own apart from any coordinate system.

To describe this vector mathematically in spherical coordinates, find out how much of each of the three basis vectors you need to add to sum to the wind's vector there. It's just like adding multiples of unit X, Y, and Z vectors in rectangular coordinates to make a given vector, but now the "unit" vectors are at crazy angles and oddball lengths. Still, we can state for example "at the point (r=75, theta=40, phi=15) add 1/2 of the radial basis vector, 5 times the theta basis and -2.1 times the phi basis vector. This is the wind velocity at that point."

To make things easier for us simple-minded physicists and engineers, to relate our work to the laws of physics, we may normalize our basis vectors. The radial vector needs no modification; it is already unit length. For the theta basis vector, divide it by r to create a unit-length vector. (Beware of dragons! Stay away from the origin.) For phi, it's hard to explain well in just words - you may want to sketch and contemplate - but imagine what a one degree increase in longitude means for someone on Earth at the equator, and what it means for someone in northern Alaska. The unit phi vector is the coordinate-based phi basis vector divided by r*sin(theta). For this to work stay away from the origin and also stay away from the Z axis (north and south pole axes). [note: sine? cosine? check the definition of your particular coordinate system; scientists and engineers in different realms may use different conventions, especially in older books and software.]

Now we can use multiples of these _unit_ vectors to form sums to describe any given physical vector, just like unit X, Y, and Z basis vector except that the triad of unit vectors will be cocked at some funny angle depending on where in space it is.

So, for every point of space, we imagine increasing each coordinate in turn while holding the others steady, defining the "coordinate basis vectors," and use a linear combination of these to describe any given physical vector. We can take an extra step and define the "unit basis vectors" by normalizing and use those to describe the physical vector.

Dealing with gradients, divergence and so forth becomes nearly easy if you use the unit basis vectors, pretending they're X,Y,Z although oriented at crazy angles. However, since gradients and such involve comparing physical fields between nearby points in space, we must account for the fact that our unit r,theta,phi basis vectors change in direction from point to point. This involves "connection coefficients" aka just "connections," as they're called in some areas of science, and goes beyond the scope of this little essay, so I'll stop here.


DSW