Lagrange multiplier rule

Let M be a manifold in Rⁿ defined by a vector equation f(x)=0 for some differentiable function f:Rⁿ→R^m. (A solution of a smooth vector equation is generally some manifold.) Let g:M→R be a differentiable function, and let v be an extremum point of M.

The Lagrange multiplier rule says that at v, the system of m+1 vectors

(∇ f)(v)
(∇ g)(v)

has less than full degree (i.e. is linearly dependent).

If we write f=(f₁,...,f_m), with each f_i:Rⁿ→R, the system of vectors is the perhaps more familiar

(∇ f₁)(v)
...
(∇ f_m)(v)
(∇ g) (v),

and one possible linear dependence is given by (∇ g)(v) being a linear combination of the (∇ f_i)(v)'s; writing

(∇ g)(v) = a₁ (∇ f₁)(v) + ... + a_m (∇ f_m)(v)

in this case shows you why the a_i's are called Lagrange multipliers.

If you think about it, what the Lagrange multiplier rule is telling you is merely that if you constrain an end of a rubber band to a curved surface and pull the other end in some direction, the constrained end will come to a stop when the rubber band is perpendicular to the surface.