Hello!
"METRIC TENSOR IN POLAR COORDINATES"
We move from cartesian coordinates to polar coordinates
Cartesian basis vectors are static
Polar basis vectors change as the angle changes
We remember the coordinate change equations
"lambda" matrices were given by derivatives
We remember the equations for polar coordinates "r" and "theta"
"b" is contracted
"e_r" can be expanded following the Einstein notation
That also happens for "e_theta"
We substitute
We have "e_r"
We have "e_theta"
A matrix can be employed
We write "lambda"
We obtain the matrix for coordinate change
However remember that we must calculate the metric tensor
Remember that the metric tensor is defined as a basis vector product
We write "g" for cartesian coordinates
The basis vectors were defined through derivatives
We have an orthonormal basis
Remember the chain rule or Leibnitz rule
We may write "e_r" and "e_theta"
Now we substitute
We are derivating the equations for coordinate change
We can write the result in matrix form
And the "lambda" matrix is again obtained
The chain rule is equivalent to calculate the "lambda" matrix
Now we remember the "g" definition
It is a basis vector product
Now we substitute
Remember the cartesian basis is orthonormal
Now "e_theta"
We obtain "r^2"
Remember "e_x" times "e_y" is equal to zero
Now we write "e_r" times "e_theta"
Finally the metric tensor can be written
Bye Bye
