- #1
TubbaBlubba
This has been in the back of my head during a course in Mathematical Physics. In for example cylindrical or spherical coordinates, we have one or two unit vectors describing position (er, for example), their directions in a Cartesian system the function of three variables. But we also have e.g. eφ. I've vaguely understood these unit vectors as relating to differential and integral operations, i.e. describing the manner in which the direction of position vectors change as the relevant angles change (very obvious in cylindrical or 2D polar coordinates), as well as with regard to the intuitive notion that you somehow need n vectors in n dimensions.
But in physics, we might encounter fields of the form, in for example spherical coordinates, A = (Ar, Aθ, Aφ), and here I'm not quite sure what the latter two components are trying to say. I do know that if a force field is given by F = ∇×A, then, say, an inverse square force can be given by A = eφ(-cot θ)/r. So, is it the case that field components pertaining to these unit vectors are only interesting in the case of vector potentials? Any other subtleties to it?
Essentially, when I see a field with components like that, what does that tell me about the field?
Thanks.
EDIT: Wow, after pondering this for a few days, minutes after posting, I just had an epiphany and realized what a hare-brained question this is. If I have a vector field depending on position, then a position unit vector can only be used to describe a radial force. If I want to describe forces with components orthogonal to that, I need to have linearly independent unit vectors. And then, that relates to the "describing the change of direction of the position vector with respect to angle" notion by virtue of the fact that a force in such a direction will indeed change the angle.
Well, good thing I got that cleared up...
But in physics, we might encounter fields of the form, in for example spherical coordinates, A = (Ar, Aθ, Aφ), and here I'm not quite sure what the latter two components are trying to say. I do know that if a force field is given by F = ∇×A, then, say, an inverse square force can be given by A = eφ(-cot θ)/r. So, is it the case that field components pertaining to these unit vectors are only interesting in the case of vector potentials? Any other subtleties to it?
Essentially, when I see a field with components like that, what does that tell me about the field?
Thanks.
EDIT: Wow, after pondering this for a few days, minutes after posting, I just had an epiphany and realized what a hare-brained question this is. If I have a vector field depending on position, then a position unit vector can only be used to describe a radial force. If I want to describe forces with components orthogonal to that, I need to have linearly independent unit vectors. And then, that relates to the "describing the change of direction of the position vector with respect to angle" notion by virtue of the fact that a force in such a direction will indeed change the angle.
Well, good thing I got that cleared up...
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