Confusion about some basics of spherical coordinates

[random_soldier1337]

If the radius unit vector is giving us some direction in spherical coordinates, why do we need the angle vectors or vice versa?

[timo]

Using unit vectors to define directions indeed contains the same information as using appropriate angles. And either can be used in conjunction with a radial coordinate to define location (but only the version with the angles is called spherical coordinates). Either version can be more appropriate for a practical problem, I think. In my experience, direction vectors tend to be more useful for trigonometric/geometric questions, and spherical coordinates tend to be more useful for integrals.

[random_soldier1337]

Let me clarify. I am confused because one of my subjects has a variable that is a function of various things, position and direction being a few of them. Position is already specified in terms of vector r. However, direction is the radius unit vector. I don't quite understand since it seems redundant and doesn't make sense at the same time.

[timo]

It is hard to tell without knowing your "subject" or the variable. For spherical coordinates of a single location, the unit direction vector indeed supplies no additional information to the location vector. Maybe the direction refers to something else than the location? Like the state of a small magnet, which (ignoring momenta) is defined by its location and its orientation at this location. Or much simpler: Direction refers to the direction of travel. Another idea I could think of is that if your position and direction are data in a data set processed on a computer. Then, they could just be in there for convenience of the user or some (possibly overambitious) performance optimization of calculations that only need the direction and want to skip the normalization step.

Just semi-random ideas.

[random_soldier1337]

Thanks. Even though those were semi random ideas, I think they actually make sense with what I am referring to. They are analogous if you will.

Though I think I may now just be confused by what exactly the radius unit vector may be pointing towards. May I please know?

[timo]

I assume you refer to my magnet example: The force between two magnets depends not only on their location, but also on their orientation. Take two rod magnets NS in one dimension which are one space (here I literally mean the space character) apart. In the case NS NS their attract. If one is oriented the other way round, e.g. NS SN, they repel each other. On other words: Their force does not only depend on their location, but also on their orientation (the equations in 3D are readily found via Google, but the common choice of coordinates may not obviously relate to what you describe). So for calculating forces or energies of magnets, you need their orientation as an additional parameter. This orientation can be expressed as a unit vector (and to relate to my first post: since this is a geometric and not an integration topic, unit vectors are better suited than angles).

[studiot]

4 hours ago, random_soldier1337 said:

Let me clarify. I am confused because one of my subjects has a variable that is a function of various things, position and direction being a few of them. Position is already specified in terms of vector r. However, direction is the radius unit vector. I don't quite understand since it seems redundant and doesn't make sense at the same time.

I am not quite sure what you mean by an 'angle vector' can you provide an  example?

Meanwhile consider this.

 

Much use is made in field and potential theory of 'shells' of the quantity of interest.

The three variables,      [math]{\rm{r,}}\;{\rm{\theta }}\;{\rm{and}}\;{\rm{\varphi }}\;[/math]   specify a point on such a shell.

The theta and phi specify the direction of the outward normal at that point.

As Timo has already pointed out integrating over all space will give a flux through that shell so we can applu Gauss' theorem

 

The thing is that we often wish to go further than this and place an active (test) particle at the specified point.
By active I mean that it interacts with the flux passing through the shell so it may have momentum, a magnetic moment or whatever.
This interaction needs to be described with another vector, different from the outward normal.
(This interaction includes the case of zero net action for a second vector orthogonal to the outward normal.)

Differential equations in the three variables   [math]{\rm{r,}}\;{\rm{\theta }}\;{\rm{and}}\;{\rm{\varphi }}\;[/math]   can often be solved by the method of separation of variables, leading to Legendre's equation for instance. This applies most particularly to Laplace's equation which leads to spherical harmonics when the separation is applied.

 

 

 

Edited October 9, 2018 by studiot

[random_soldier1337]

18 hours ago, studiot said:

I am not quite sure what you mean by an 'angle vector' can you provide an  example?

I was referring to the theta and phi directions, their vectors I mean.

18 hours ago, studiot said:

The theta and phi specify the direction of the outward normal at that point.

Wait don't you mean tangent? I thought the radius unit vector would be the normal.

[MigL]

In spherical polar co-ordinates, r specifies a sphere about the origin, of radius r.
The theta and phi angles specify the position on that sphere analogous to latitude and longitude

As Studiot has mentioned, the example that comes to mind is the Hydrogen atom solution, using separation of variables, of the wave equation in spherical polar co-ordinates.

[studiot]

4 hours ago, random_soldier1337 said:

22 hours ago, studiot said:

The theta and phi specify the direction of the outward normal at that point.

Wait don't you mean tangent? I thought the radius unit vector would be the normal.

I said normal, you said normal he she or it said normal, what's the problem?

Yes the tangent is orthogonal to the normal and is another vector.