How do you prove Euler's Theorem d u = ( ∂ u ∂ x ) d x + ( ∂ u ∂ y ) d yif u = f ( x , y ).I also heard that Ramanujan developed another method can somebody know that?

Question

How do you prove Euler&#039;s Theorem  d  u  =      (                            ∂          u                          ∂          x                      )    d  x  +      (                            ∂          u                          ∂          y                      )    d  yif   u  =  f  (  x  ,  y  ).I also heard that Ramanujan developed another method can somebody know that?

phinny5608tt · Accepted Answer

I would propose an &quot;Analysis II based&quot; 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.

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?

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