# Vector identity for $\dfrac{\mathrm d}{\mathrm dt}\nabla \varphi(\vec r(t),t)$

Suppose I have a scalar field $$\varphi\left(\vec r(t), t\right)$$ where $$\vec r$$ is a vector in $$\mathbb R^3$$ that depends on the parameter $$t$$.

I'd like to figure out what $$\dfrac{\mathrm d}{\mathrm dt}\nabla \varphi$$ is.

I start as follows, $$\nabla \varphi = \partial_i \varphi \hat{e}^i$$ (which I attempt to write in Einstein notation)

From there, I take the time derivative component-wise and using the directional derivative I get $$\dfrac{\mathrm d}{\mathrm dt}\nabla \varphi = \left( v_j \partial _j \partial_i \varphi +\partial _t \partial _i \varphi \right)\hat{e}^i$$ where $$v^\nu = \dfrac{\mathrm d r^\nu }{\mathrm dt}$$.

I can rewrite this as (where $$^T$$ denotes the transpose)$$\dfrac{\mathrm d\nabla\varphi }{\mathrm dt} = \pmatrix{\nabla \partial _x\varphi^T \\ \nabla \partial _y \varphi ^T \\ \nabla \partial_z \varphi ^T }\vec v+\dfrac{\partial}{\partial t}\nabla \varphi$$ and am wondering if there is a way for me to simplify this further, in particular if there is a notation or name for the $$\pmatrix{\nabla \partial _x\varphi^T \\ \nabla \partial _y \varphi ^T \\ \nabla \partial_z \varphi ^T }$$ matrix.

• I think you mean e.g. $\nabla\varphi=\partial_\mu\hat{e}^\mu$ or, even better, $\nabla\varphi=\partial_i\hat{e}^i$.
– J.G.
Jun 11, 2022 at 22:08
• @J.G. yes, thank you Jun 11, 2022 at 22:21
• Sorry, I meant $\partial_i\varphi\hat{e}^i$. You still don't want an index on $\varphi$.
– J.G.
Jun 11, 2022 at 22:23
• Hi! Let me know if you want more details/ more notation help. Jun 11, 2022 at 22:31
• @J.G. ah yes, it's a scalar -- for some reason I was thinking of $\varphi^i$s as components of the gradient :/ Jun 11, 2022 at 22:32

$$\frac{d}{dt} \nabla \phi = \left[ \nabla_v + \partial_t \right] \nabla \phi$$
It maybe confusing to see $$\nabla_v$$ acting on a vector here. The meaning of the notation is that you act the operator on all the component of the vector.
• Btw you can actually think about $\nabla$ acting twice on a vector. This is known as the vector gradient. Jun 11, 2022 at 22:31