Scalar, vector and tensor calculus

[Jhenrique]

I noticed that sometimes exist a parallel between scalar and vector calculus, for example:

$v=at+v_0$

$s=\int v dt = \frac{1}{2}at^2 + v_0 t + s_0$

in terms of vector calculus

$\vec{v}=\vec{a}t+\vec{v}_0$

$\vec{s}=\int \vec{v} dt = \frac{1}{2}\vec{a}t^2 + \vec{v}_0 t + \vec{s}_0$

So, this same equation could be written in terms of tensor calculus? Or exist some equation that can assume a scalar, vector and tensor form?

 

[HallsofIvy]

I'm not sure what you intend here. The definitions in Calculus as extended to vectors and tensors are done in imitation of scalar Calculus so of course you have the same formulas.

(I would not consider your equation "of vector calculus" to actually be "vector Calculus". Your coefficients are vectors but your variables are not.)

 

[Jhenrique]

The definitions in Calculus as extended to vectors and tensors are done in imitation of scalar Calculus so of course you have the same formulas.

You can give me an example of some formule that have a scalar(rank0), a vector(rank1) and a tensor(rank2) version?