Hello everyone!
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?
Thanks!
Area and volume elements in polar coordinate systems
By the way, IS true.
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 .)
Dear HallsofIvy,
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...
Best,