Editing Lagrange multiplier (section)

===Multiple constraints===
Let <math>\ M\ </math> and <math>\ f\ </math> be as in the above section regarding the case of a single constraint. Rather than the function <math>g</math> described there, now consider a smooth function <math>\ G:M\to \R^p (p>1)\ ,</math> with component functions <math>\ g_i: M \to \R\ ,</math> for which <math>0\in\R^p</math> is a [[regular value]]. Let <math>N</math> be the submanifold of <math>\ M\ </math> defined by <math>\ G(x)=0 ~.</math>

<math>\ x\ </math> is a stationary point of <math>f|_{N}</math> if and only if <math>\ \ker( \operatorname{d}f_x )\ </math> contains <math>\ \ker( \operatorname{d}G_x ) ~.</math> For convenience let <math>\ L_x = \operatorname{d}f_x\ </math> and <math>\ K_x = \operatorname{d}G_x\ ,</math> where <math>\ \operatorname{d}G</math> denotes the tangent map or Jacobian <math>\ TM \to T\R^p ~</math> (<math>\ T_x\R^p</math> can be canonically identified with <math>\ \R^p</math>). The subspace <math>\ker(K_x)</math> has dimension smaller than that of <math>\ker(L_x)</math>, namely <math>\ \dim(\ker(L_x)) = n-1\ </math> and <math>\ \dim(\ker(K_x)) = n-p ~.</math> <math>\ker(K_x)</math> belongs to <math>\ \ker(L_x)\ </math> if and only if <math>L_x \in T^{\ast}_x M</math> belongs to the image of <math>\ K^{\ast}_x: \R^{p\ast}\to T^{\ast}_x M ~.</math> Computationally speaking, the condition is that <math>L_x</math> belongs to the row space of the matrix of <math>\ K_x\ ,</math> or equivalently the column space of the matrix of <math>K^{\ast}_x</math> (the transpose). If <math>\ \omega_x \in \Lambda^{p}(T^{\ast}_x M)\ </math> denotes the exterior product of the columns of the matrix of <math>\ K^{\ast}_x\ ,</math> the stationary condition for <math>\ f|_{N}\ </math> at <math>\ x\ </math> becomes
<math display="block"> L_x \wedge \omega_x = 0 \in \Lambda^{p+1} \left (T^{\ast}_x M \right ) </math>
Once again, in this formulation it is not necessary to explicitly find the Lagrange multipliers, the numbers <math>\ \lambda_1, \ldots, \lambda_p\ </math> such that
<math display="block">\ \operatorname{d}f_x = \sum_{i=1}^p \lambda_i \operatorname{d}(g_i)_x ~.</math>