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'''Darboux theorem may''' may refer to one of the following assertions:
* Darboux theorem on local canonical coordinates for symplectic structure;
* Darboux theorem on intermediate values of the derivative of a function of one variable.

== Darboux theorems for symplectic structure ===
{{MSC|37Jxx,53Dxx}}

Recall that a [[symplectic structure]] on an even-dimensional manifold $M^{2n}$ is a closed nondegenerate 2-form $\omega$:
$$
\omega\in\varLambda^2(M),\qquad \rd \omega=0,\qquad \forall v\in T_p M\quad \exists w\in T_p M:\ \omega_p(v,w)\ne0.
$$

The matrix $S(z)$ of a symplectic structure, $S_{ij}(z)=\omega(\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_i})=-S_{ji}(z)$  in any local coordinate system $(z_1,\dots,z_{2n})$ is antisymmetric and nondegenerate: $\omega=\frac12\sum_{1}^{2n} S_{ij}(z)\,\rd z_i\land \rd z_j$. 

The ''standard symplectic structure'' on $\R^{2n}$ in the ''standard canonical coordinates'' $(x_1,\dots,x_n,p_1,\dots,p_n)$ is given by the form
$$
\omega=\sum_{i=1}^n \rd x_i\land \rd p_i.
$$

'''Theorem''' (Darboux theorem<ref name=AG>Arnold V. I., Givental A. B.</ref>, sometimes also referred<ref name=AG/> to as the Darboux-Weinstein theorem<ref>Guillemin V., Sternberg S., </ref>).

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== Darboux therem for intermediate values of differentiable functions ==