Let M be a manifold in **R**^{n} defined by a vector equation f(x)=0 for some differentiable function f:**R**^{n}→**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=(f_{1},...,f_{m}), with each f_{i}:**R**^{n}→**R**, the system of vectors is the perhaps more familiar

(∇ f_{1})(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) = a_{1} (∇ f_{1})(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.