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.



Mechanics: Model beam using differential vectorial formulation
The beam equations \eqref{1} and \eqref{2} "admit a vector formulation" in the sense that they can be rigorously deduced from the 3D theory of (nonlinear) elasticity. I'm not aware of a precise reference dealing specifically with those two examples, but the *Encyclopedia of Physics* entry [1] by Antman (which also alludes briefly to the deduction of \eqref{1} in §26, example $\delta$, page 698) presents two different classes of methods for solving the problem, namely projection methods ([1], §11, pp. 660-663) and the asymptotic method ([1], §13, pp. 664-665). Antman gives a fairly complete (even if somewhat dated) survey of the theories of beams, rods and all continua with 1D behavior, considering also the history of the subject. Well, my two cents.


**Edit after the downvote**. After I saw the OP and my answer downvoted, I decided to add a few words of explanation. According to Stuart Antman ([1], §1, p. 641),



> 
> A *theory of rods*2 or, equivalently a *one-dimensional theory of solids* is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a curve and the time as the only independent variables.  
>   
> 2We use "rod" as a generic name for "arch", "bar", "beam", "column", "ring", "shaft", etc. We employ rod both in the intuitive sense of a slender body and in several precise mathematical senses. The meaning will be clearer from the context.
> 
> 
> 


Said that, by carefully reading the question and its *motivation*, one sees that the techniques shown in [1] exactly suit the needs of the Asker: his search for a *particular* 3D equation for the models \eqref{1} and \eqref{2} is probably useless since it would be unsuitable for the study of a non-rectilinear beam, while the general procedures described in [1] where the equations describing a 1D medium are deduced from the standard, general 3D equations by applying them to *particular* constitutive equations, are applicable for beams described by a general spatial curve.


**Reference**


[1] Stuart S. Antman, "The Theory of Rods", in Flügge, S. & Truesdell, C. (Eds.) *Festkörpermechanik/Mechanic of Solids*, *Handbuch der Physik/Encyclopedia of Physics*, Vol. VI part 2, Springer-Verlag, 1972, 641-703.


