The Student Room Group
Reply 1
Goldenratio
Can anyone explain what exactly a local inverse is and how do i go about calculating where/if F_xy=(x^2-y^2,-2xy) has them?

as far as i can see i have to use the Jacobian but i'm not sure about it.

thanks:smile:
Not exactly sure about the terminology you're using, but if you consider Taylor's theorem, you can see that "locally", any function will behave like an affine function (constant + linear map). If the linear map is invertible, then you can form an inverse map that also works locally.

(The above is not rigourous; if memory serves, it can be made so, but I don't remember all the fine detail).
Reply 2
hmm and how could I apply this theorem to my example?

i think the terminology is non-standard thats why i am having trouble finding any information about it:frown:
Reply 3
Almost certainly, you are just supposed to evaluate the Jacobian and find where it's invertible.

If you want more information, try a google on "Implicit Function Theorem".
Reply 4
so for example:

W(x,y,z)=(x+y,2xy^2)

J=2y(2x-y)

has no inverse where y=0 and y=2x, but has an inverse everywhere else?

So if i wanted to show that these vector fields are differentiable what do I have to do?

lim(h->0) (f(x+h)-f(h))/h

and then the same for y? but since it's a vector i cannot devide by this...