At the Wikipedia there are the differential formulation for [Euler-Bernoulli Beam](https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory) \eqref{1} and [Timoshenko Beam](https://en.wikipedia.org/wiki/Timoshenko%E2%80%93Ehrenfest_beam_theory) \eqref{2}
$$
\begin{align}
&\dfrac{d^2}{dx^2}\left(EI\dfrac{d^2w}{dx^2}\right) = q(x) \label{1}\tag{1}\\
&\begin{cases}\dfrac{d^2}{dx^2}\left(EI\dfrac{d\varphi}{dx}\right) = q(x) \\
\dfrac{dw}{dx} = \varphi - \dfrac{1}{\kappa AG} \cdot \dfrac{d}{dx}\left(EI\dfrac{d\varphi}{dx}\right)
\end{cases} \label{2}\tag{2}
\end{align}
$$

Both of formulations supposes that the undeformed beam is at $(\vec{e}_x)$ direction, the distributed charge $q$ is at $(\vec{e}_z)$ direction and $w$ is the displacement at $(\vec{e}_z)$ direction.

The deduction uses, for example that

$$
Q = \dfrac{dM}{dx}
$$

But the shear force $\vec{Q}$ and the momentum $\vec{M}$ are not colinear:

$$
\vec{Q} = Q \cdot \vec{e}_{z} = \dfrac{dM}{dx} \cdot \vec{e}_z  \ne \dfrac{dM}{dx} \cdot \vec{e}_{y} = \dfrac{d}{dx}\left(M \cdot \vec{e}_{y}\right) = \dfrac{d}{dx} \vec{M}
$$

**Question:** Then, is there a formulation $(1)$ and $(2)$ using vectorial notation? Like for example

$$
\vec{Q} = \nabla \times \vec{M}
$$

Cause

$$
\vec{Q} = 
\begin{bmatrix}
0 \\ 0 \\ Q
\end{bmatrix}=
\begin{bmatrix}
\dfrac{-dM}{dz} \\ 0 \\ \dfrac{dM}{dx}
\end{bmatrix} =
\det \begin{bmatrix}
\vec{e}_x & \vec{e}_y & \vec{e}_z \\
\dfrac{d}{dx} & \dfrac{d}{dy} & \dfrac{d}{dz} \\
0 & M & 0
\end{bmatrix}
= 
\begin{bmatrix}
\dfrac{d}{dx} \\ \dfrac{d}{dy} \\ \dfrac{d}{dz}
\end{bmatrix} \times 
\begin{bmatrix}
0 \\ M \\ 0
\end{bmatrix}
$$

**Motivation:** I want an analytic model for a 3D beam which neutral line follows an arbitrary path $p(t) \in \mathbb{R}^{3}$. When I tried to get it, I could not use the scalar rotations cause the vectors' directions were not the same and I should use rotations.