On the topic of changing variables in multi-dimensional Integration; can anybody explain why the Jacobian is needed? I have been using 'Pauls Online Notes' to read up on it but i don't feel it explains why it is used, instead just stating that it is necessary.

This is the exact equation i am having difficulty understanding: http://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables_files/eq0033MP.gif

Question

On the topic of changing variables in multi-dimensional Integration; can anybody explain why the Jacobian is needed? I have been using 'Pauls Online Notes' to read up on it but i don't feel it explains why it is used, instead just stating that it is necessary.

This is the exact equation i am having difficulty understanding: http://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables_files/eq0033MP.gif

experimentx · Answer

explained here http://www.youtube.com/watch?v=UZb9hZIAvL4

experimentx · Answer

from this point http://www.youtube.com/watch?v=UZb9hZIAvL4

experimentx · Answer

19 m

anonymous · Answer

Great, watching now thanks!

anonymous · Answer

fantastic, explained perfectly! thanks a lot!

experimentx · Answer

all breaks down into liner approximation of multivariable function and simple geometry.

anonymous · Answer

My lecturer explained it in terms of a first-order Taylor approximation. It threw me off a little I guess, thanks for clearing it up for me!

experimentx · Answer

first order Taylor approximation is linear approximation

anonymous · Answer

And if that was made clear during the proof it would have probably made more sense