Differentiation with respect vector

[Jhenrique]

Helow!

For a long time I aks me if exist differentiation/integration with respect to vector and I think that today I discovered the answer! Given:
$$f(\vec{r}(t))$$
So, df/dt is:
$$\bigtriangledown f\cdot D\vec{r}$$
But, df/dt is:
$$\frac{df}{d\vec{r}}\cdot \frac{d\vec{r}}{dt}$$
This means that:
$$\frac{d\vec{r}}{dt}=D\vec{r}$$
and:
$$\frac{df}{d\vec{r}}=\bigtriangledown f$$

So, analogously, the integration with respect to vector is:
$$\int f\;d\vec{r}=\bigtriangledown ^{-1}f$$

What make I think that:
$$\frac{\mathrm{d} }{\mathrm{d} \vec{r}}=\bigtriangledown$$

Then, would be possible to formalize the differintegration of vector with respect to vector too by:
$$\bigtriangledown \cdot \vec{f},\;\;\;\bigtriangledown \times \vec{f},\;\;\;\bigtriangledown^{-1} \cdot \vec{f}\;\;\;and\;\;\;\bigtriangledown^{-1} \times \vec{f}$$

What you think about all this?

 

[Simon Bridge]

It may make more sense of you expand \vec r in a basis - say: Cartesian.
See which ideas make sense.

But you can differentiate and integrate vectors - look up vector-valued functions for instance.