Spherical coordinates - phi vs theta

[Calpalned]

Homework Statement

My textbook states that when ##\theta = c ## where c is the constant angle with respect to the x-axis, the graph is a "half-plane". However, when ##\phi = c ## it is a half-cone. The only difference I see is that ##\phi## is the angle with respect to the z-axis, rather than the x-axis. Why is it that one ends up being a cone and the other a plane?

Homework Equations

n/a

The Attempt at a Solution

That is the only part I don't understand. It is clear how ##\rho = c ## is a sphere, but why are ##\phi## and ##\theta## so different?

 

[LCKurtz]

Homework Statement

My textbook states that when ##\theta = c ## where c is the constant angle with respect to the x-axis, the graph is a "half-plane". However, when ##\phi = c ## it is a half-cone. The only difference I see is that ##\phi## is the angle with respect to the z-axis, rather than the x-axis. Why is it that one ends up being a cone and the other a plane?

Homework Equations

n/a

The Attempt at a Solution

That is the only part I don't understand. It is clear how ##\rho = c ## is a sphere, but why are ##\phi## and ##\theta## so different?

Look at the geometry. If you pick a point where ##\theta = c## and vary ##r>0## and ##\phi## do you see why you get a half plane? Now pick a point where ##\phi = c##. What happens to that point as you vary ##\theta##? When you very ##r##? It might help to be looking at http://en.wikipedia.org/wiki/Spherical_coordinate_system when you answer. Use the second picture there as it has the usual notation used in math.

 

[HallsofIvy]

Homework Statement

My textbook states that when ##\theta = c ## where c is the constant angle with respect to the x-axis, the graph is a "half-plane". However, when ##\phi = c ## it is a half-cone. The only difference I see is that ##\phi## is the angle with respect to the z-axis, rather than the x-axis.

You are mistaken. For one thing, [itex]\theta[/itex] can range from 0 to [itex]2\pi[/itex] while [itex]\phi[/itex] only ranges from 0 to [itex]\pi[/itex]. Do you see why that is true?
("Mathematics notation" and "physics notation" reverse [itex]\theta[/itex] and [itex]\phi[/itex]. I am assuming that you are using "mathematics notation".)

Why is it that one ends up being a cone and the other a plane?

Homework Equations

n/a

The Attempt at a Solution

That is the only part I don't understand. It is clear how ##\rho = c ## is a sphere, but why are ##\phi## and ##\theta## so different?

 

[BvU]

How about some physical experience ?

Stand up straight and lift one arm straight up. Let the direction your feet are pointing in represent ##\theta## and the angle your arm makes with the vertical when you lower it represent ##\phi## ("math notation" - I didn't know it existed). Fortunately they have r in common imagine a very long arm...

Moving your arm up and down (##[0,\pi]##) with feet fixed defines a half plane.
And rotating on your feet (##[0,2\pi]##) with arm angle fixed gives you the cone. (I take it the mistake Ivy refers to is that it's a cone surface, not half a cone).

And now I will revert to "physics notation" in order not to confuse myself -- after all this is PF and not MF !