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## Main Question or Discussion Point

I think that many of us have had to endure working with Lagrange multipliers in the past, but it seems to me that it has always been taught incorrectly.

So the statement (if you will allow me to use differential forms) is

[tex] \lambda_0 df(p) = \lambda_1 dg_1(p) + \cdots + \lambda_k dg_k(p) [/tex]

Now normally of course, this wouldn't matter. We could just "normalize by [itex]\lambda_0[/itex]" to get rid of it, unless of course [itex] \lambda_0 = 0 [/itex]. I've seen most of the proofs of this theorem and the [itex] \lambda_0 [/itex] never arises. I can't convince myself why it should indeed arise, yet I can give a great deal of evidence that it is necessary.

Am I confusing this with another LM theorem? Does the [itex]\lambda_0 [/itex] only arise when we consider problems on sub-Riemannian and symplectic manifolds? In those cases, the [itex] \lambda_i [/itex] are not elements of [itex] \mathbb R [/itex] but instead are covectors in [itex] T^*_p M [/itex].

So the statement (if you will allow me to use differential forms) is

Now my issue is that it's well-known that this should beLet M be a smooth manifold. Suppose [itex] dg_i [/itex] are linearly independent at each point [itex] p \in M [/itex]. If p is a local extremum of f restricted to M then [itex] \exists \lambda_1,\ldots,\lambda_k \in \mathbb R [/itex] such that

[tex] df(p) = \lambda_1 dg_1(p) + \cdots + \lambda_k dg_k(p) [/tex]

[tex] \lambda_0 df(p) = \lambda_1 dg_1(p) + \cdots + \lambda_k dg_k(p) [/tex]

Now normally of course, this wouldn't matter. We could just "normalize by [itex]\lambda_0[/itex]" to get rid of it, unless of course [itex] \lambda_0 = 0 [/itex]. I've seen most of the proofs of this theorem and the [itex] \lambda_0 [/itex] never arises. I can't convince myself why it should indeed arise, yet I can give a great deal of evidence that it is necessary.

Am I confusing this with another LM theorem? Does the [itex]\lambda_0 [/itex] only arise when we consider problems on sub-Riemannian and symplectic manifolds? In those cases, the [itex] \lambda_i [/itex] are not elements of [itex] \mathbb R [/itex] but instead are covectors in [itex] T^*_p M [/itex].