OK, so the formulas
(ds)^2 = (dx)^2 + (dy)^2
= (dr)^2 + r^2 (dA)^2
Give us arc length formulas for the plane in rectangular and polar coordinates, respectively.
These very naturally generalize to three (or more) dimensions:
(ds)^2 = (dx)^2 + (dy)^2 + (dz)^2 <--- rectangular
= (dr)^2 + r^2 (dA)^2 + (dz)^2 <----cylindrical
= (dr)^2 + r^2 (dA)^2 + r^2 cos^2 (A) (dB)^2 <---spherical
In the last, A represents latitude and B represents longitude. This is slightly different than typical (usually, spherical uses the angle down from the z-axis instead of latitude, which is up from the equator)
But, suppose we live on a sphere? (well, we do, at least approximately). Just fix r in the spherical coordinates and so dr=0. So, on a sphere we have
(ds)^2 = r^2 (dA)^2 + r^2 cos^2 (A) (dB)^2.
Now, these formulas for the arc length are called 'line elements' and each surface has one that describes the arc length on that surface in whatever coordinates you want to use.
It turns out that these line elements contain a LOT of information about the surface. In particular, we can determine how to find areas on the surface from them. We can determine curvature of the surface.
For example, on the sphere, we can determine area by doing a *double integral* over a region of r^2 cos(A) dA dB. This is the square root of the product of the two terms in the line element.
For spherical coordinates in three dimensions, we can compute *volume* by doing something similar and getting a *triple integral* of r^2 cos(A) dr dA dB.
More later.
(ds)^2 = (dx)^2 + (dy)^2
= (dr)^2 + r^2 (dA)^2
Give us arc length formulas for the plane in rectangular and polar coordinates, respectively.
These very naturally generalize to three (or more) dimensions:
(ds)^2 = (dx)^2 + (dy)^2 + (dz)^2 <--- rectangular
= (dr)^2 + r^2 (dA)^2 + (dz)^2 <----cylindrical
= (dr)^2 + r^2 (dA)^2 + r^2 cos^2 (A) (dB)^2 <---spherical
In the last, A represents latitude and B represents longitude. This is slightly different than typical (usually, spherical uses the angle down from the z-axis instead of latitude, which is up from the equator)
But, suppose we live on a sphere? (well, we do, at least approximately). Just fix r in the spherical coordinates and so dr=0. So, on a sphere we have
(ds)^2 = r^2 (dA)^2 + r^2 cos^2 (A) (dB)^2.
Now, these formulas for the arc length are called 'line elements' and each surface has one that describes the arc length on that surface in whatever coordinates you want to use.
It turns out that these line elements contain a LOT of information about the surface. In particular, we can determine how to find areas on the surface from them. We can determine curvature of the surface.
For example, on the sphere, we can determine area by doing a *double integral* over a region of r^2 cos(A) dA dB. This is the square root of the product of the two terms in the line element.
For spherical coordinates in three dimensions, we can compute *volume* by doing something similar and getting a *triple integral* of r^2 cos(A) dr dA dB.
More later.
