# Lagrangian multipliers

1. Mar 6, 2009

### 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

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:

That is the requirement given by the system
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 doesnt necessarily mean grad(x) is parallel to grad(c). It seems like im missing something.

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

2. Mar 6, 2009

### HallsofIvy

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

3. Mar 6, 2009

### okkvlt

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