Why does ∂Σ{Ni}/∂Nj equal 1 but ∂N/∂Nj equal 0 in Lagrange multipliers?

[kidsasd987]

Hi, I have a question about lagrange multiplier

Let's say we are given with the following constraints

Σ{Ni}=N and Σ{NiEi}=total energy. N and total energy are constants by definition.
if we take the derivative with respect to Nj,

∂Σ{Ni}/∂Nj=∂N/∂Nj
where i=j, ∂Σ{Ni}/∂Nj=1 and ∂N/∂Nj = 0 because N is constant.

On slide 14, it says ∂N/∂Nj = 0 while ∂Σ{Ni}/∂Nj=1 with the preceding constraint Σ{Ni}=N.
Then, we can conclude

∂Σ{Ni}/∂Nj= ∂N/∂Nj=0.

This is quite ambiguous to me.
if we assume we have a constraint x1+x2+x3..+xn=const.

partial of this constraint with respect to xj will be 1=0 therefore it is contradiction.
how shoud I interpret this?

 

[ShayanJ]

A constraint means that your variables are not independent from each other. So when I have $x_1+x_2+\dots+x_n=const$, it means when I decrease one of the $x$s, another increases to maintain the constant. But the way you differentiate it, you assume they're independent variables which is a wrong assumption.

 

[kidsasd987]

A constraint means that your variables are not independent from each other. So when I have $x_1+x_2+\dots+x_n=const$, it means when I decrease one of the $x$s, another increases to maintain the constant. But the way you differentiate it, you assume they're independent variables which is a wrong assumption.

hmm I get it thanks .