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!

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- October 6th 2010, 09:53 PMrebghbIncrements of area dxdy
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! - October 6th 2010, 11:32 PMProve It
Area and volume elements in polar coordinate systems

By the way, IS true. - October 7th 2010, 04:44 AMHallsofIvy
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 .) - October 7th 2010, 10:21 PMrebghb
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,