How Do Lagrange Multipliers Extend Beyond Two Variables?

[okkvlt]

How does lagrange multipliers work?
i was able to work out this proof of the idea, but its only true for a function with two independent variables and one dependent variable.

Rn=the space that is the independent variables.

x[Rn]=x
C[Rn]=C=constant.


dx/d[Rn]=grad(x)*v; v is a unit vector
dC/d[Rn]=grad(c)*v

because C is held constant, dC/d[Rn]=0 everywhere.
because cos(pi/2)=0, **grad(C) is perpendicular to v.**

In order for extrema to exist, dx/d[Rn]=0. grad(x)*v is zero meaning **grad(x) is perpendicular to v.**

in the case Rn=R2:
both grad(x) and grad(c) are perpendiular to v, it means grad(x) must be parallel to grad(c).

That is the requirement given by the system
grad(x)=L*grad(c)
C[Rn]=C
where L is the scalar multiplier (upside down y).

but it seems as though this is only true for the R2 case. in 3 dimnensions, if both grad(x) and grad(c) are perpendicular to v, it doesn't necessarily mean grad(x) is parallel to grad(c). It seems like I am missing something.


How do i extend this to more than two independent variables?

 

[HallsofIvy]

What do you mean by "dx/d[Rn]" where Rn is n dimensional Euclidean space? I don't believe that is standard notation.

 

[okkvlt]

i know it isnt
[Rn]=x,y,z, etc
basically i meant the directional derivative by dx/d[Rn]"