Let $\;f:\mathbb R^n\rightarrow \mathbb R^m\;$. The gradient matrix of $\;f\;$ is :
$\;\nabla f=\begin{pmatrix} \frac{\partial f_1}{\partial x_1} \dots \frac{\partial f_1}{\partial x_n}\\ \;\dots\;\\ \frac{\partial f_m}{\partial x_1} \dots \frac{\partial f_m}{\partial x_n}\\ \end{pmatrix}\;$
And $\;v=(v^1,\dots,v^n)\;$ is the unit normal vector field.
I know $\;\frac{\partial f}{\partial v}= \nabla f \cdot v\;$.
Question:
If I take the transpose matrix $\;(\nabla f)^T\;$, then would it be true to claim $\;\frac{\partial f}{\partial v}=(\nabla f)^T \cdot v\;$?
Thanks in advance!