How do you express unit vectors in spherical coordinates?

In summary, the conversation discusses difficulties with expressing unit vectors in spherical coordinates and suggests using trigonometry and projections onto the xy plane to convert between spherical and cartesian coordinates. Other methods are also mentioned in a physics forum thread.
  • #1
ak416
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Hi, I am having trouble with spherical coordinates. For example, how do you express the unit vectors x hat, y hat, z hat in terms of the spherical unit vectors r hat, theta hat, phi hat. I was able to go from spherical in terms of cartesian (with the help of mathworld.wolfram.com) but I can't get the other way. All i can think of is dR/dx / [dR/dx] = x hat where R is the position vectors in terms of spherical coords. I don't fully understand this but i saw this done as a derivation of theta hat in terms of x,y,z hat. So any suggestions?
 
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  • #2
Do you visually know what they are? If you draw a picture you can use trigonometry. Phi is the angle down from the z axis, theta is the angle counterclockwise from the x axis, and r is the length of the vector. I'm sure you know how to change from polar coordinates to rectangular coordinates. Try this approach: project the vector straight down onto the xy plane so that you have theta and s, where s is the length of that projection, and don't worry about what s is exactly yet. You can find x and y in terms of theta and s by conversion from polar coordinates. Then if you can use trigonometry to find an expression for s in terms of r and phi, you can substitute that expression into your intermediate expressions for x and y. And you can find z easily straight from r and phi.

There are other ways of doing it, probably some of them are simpler.
 
  • #3
Is it just r-hat cos theta? I am not sure. Please tell me if anyone knows.
 
  • #4

1. What are spherical coordinates and how are they used in physics?

Spherical coordinates are a coordinate system used to describe a point in three-dimensional space. They consist of three values: radius (r), inclination (θ), and azimuth (φ). In physics, they are commonly used to describe the position, velocity, and acceleration of objects moving in a spherical or circular path.

2. How do spherical coordinates differ from Cartesian coordinates?

Spherical coordinates are based on a spherical surface, whereas Cartesian coordinates are based on a flat plane. In spherical coordinates, the origin is located at the center of the sphere, while in Cartesian coordinates, the origin is at the intersection of the x, y, and z axes. Spherical coordinates also use different units, with radius measured in distance units and inclination and azimuth measured in angles.

3. What is the conversion formula between spherical and Cartesian coordinates?

The conversion from spherical coordinates to Cartesian coordinates is given by the formulas:

x = r sin(θ) cos(φ)

y = r sin(θ) sin(φ)

z = r cos(θ)

Conversely, the conversion from Cartesian coordinates to spherical coordinates is given by:

r = √(x² + y² + z²)

θ = arccos(z/r)

φ = arctan(y/x)

4. How are spherical coordinates used in solving problems in electromagnetism?

In electromagnetism, spherical coordinates are used to describe the electric and magnetic fields around a charged or magnetized object. They are particularly useful in solving problems involving symmetrical spherical distributions of charge or current, as the equations for electric and magnetic fields in these situations are simplified in spherical coordinates.

5. Can spherical coordinates be used in other areas of physics?

Yes, spherical coordinates are used in many other areas of physics, including fluid mechanics, quantum mechanics, and general relativity. They are especially useful for describing systems with spherical symmetry, such as planets, stars, and galaxies. Spherical coordinates can also be extended to higher dimensions, making them applicable in a wide range of physics problems.

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