0:00 The Euclidian metric: ds^2=dx^2 + dy^2 + dz^2. I express it as an inner product of a 1x3 and 3x1 matrix.
1:00 The spherical-to-cartesian coordinate conversions, and calculating the Jacobian.
1:30 My new notation for Jacobian, shows little arrows to indicate the direction which a matrix should be written. I show how the arrow notation builds a matrix in an explicit form. (This is an improvement on Einstein Notation)
3:00 calculating a few of the partial derivatives in my head.
3:30 Better... Use Mathematica to find the Jacobian.
4:30 I've got things plugged in, by pasting 9 times, and then going across and down, making replacements with x,y,z, r, theta, phi.
6:00 Jacobian is protectes,so I name it "Jake"
6:30 I explain how the value of (dx, dy, dz) is equal to (dr, dtheta, dphi) times the Jacobian.
8:11 Going through what dx is as you multiply (dr dtheta dphi) by the first column of the Jacobian.
9:27 When finding the transpose of a product of two matrices, you have to swap the order of the factors, when you transpose them.
10:30 Take our Jake.Transpose[Jake]. It looks enormous... But there are hints of some things that should simplify.
11:18 Simplify boom, we have a very simple looking diagonal matrix. The huge mess from the jacobian
11:50 Writing out the equality, how the metric goes between the (dr dtheta dphi) written horizontally and vertically.
12:40 After the matrices are multiplied out, we obtain the first equation in the Wheeler article linked by George Dishman on Researchgate on May 19, 2018.
14:30 Wheeler sets dr = 0 and eliminates two of the columns of the three-dimensional metric.
15:40 What was the mapping for? What if we had x,y,z and wanted to convert to r,theta,phi. Instead of giving x,y,z as functions of r,theta,phi. We want r,theta,phi as functions of x,y,z. I determine that theta and phi can be expressed as arccosines of functions of x,y,z.
18:20 We've got the Jacobian, and multiplied by its transpose, to get the metric... What would we do for the inverse metric?
19:20 Test this: The inverse metric is done up by putting the r theta phi on top, and the x,y,z on bottom. I want to check if these are inverse metrics in the next video.