I've lately been thinking about this: why isn't equal to .
Okay now, and .
So using the derivative rule, we say that and that ...
But when we carry out we get something very long...
How is that?
You cannot just multiply differentials like that: If x= u(s, t) and y= v(s, t) then where J(x,y;u,v) is the Jacobian determinant: .
In this case, u and v are r and , respectively, and so .
(In more advanced mathematics, differential geometry, we use that to define the "algebra of differentials" in such a way that it is anti-commutative- that is, that ab= -ba. From that so all squares are 0.
If then and if then .
Then . The terms we would get by multiplying dr and dr together or and together are 0 leaving which, since multiplication is anti-commutative, is the same as .)
Thank you for commenting, I am aware of the Jacobian (stretching factor)... But never considered this case.
Regarding differential Geometry, can you recommend me a book (an intro) from calculus III to differential geometry? I would appreciate it...