How do you prove Euler's Theorem du=(du/dx)dx+(du/dy)dy if u=f(x,y). I also heard that Ramanujan developed another method can somebody know that?

valtricotinevh 2022-07-18 Answered
How do you prove Euler's Theorem
d u = ( u x ) d x + ( u y ) d y
if u = f ( x , y ).
I also heard that Ramanujan developed another method can somebody know that?
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Answers (1)

phinny5608tt
Answered 2022-07-19 Author has 17 answers
I would propose an "Analysis II based" approach to start with.
Let p = ( x , y ) and f : Ω R 2 R be a function defined on the open set Ω p.
We say that f is differentiable at p if for every h := ( d x , d y ) s.t. p + h Ω there exists a vector a R 2 s.t. d f ( p ) : R 2 R s.t.
f ( p + h ) f ( p ) = a , h + O ( h 2 ) .
The linear map a a , h from R 2 to R is called differential of f at p and it is denoted by d f ( p ). The vector h can be consideblack as a small increment around p.
If f is differentiable it follows that (this is shown on every textbook of Analysis)
d f ( p ) ( a ) := a , h = f ( p ) , h ,
where f ( p ) is the gradient of f at p. In summary, recalling that p = ( x , y ) and h = ( d x , d y ) we arrive at
d f ( p ) = f ( p ) , h = f x d x + f y d y ,
with the partial derivatives computed at p.
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