# B Why are pressure and current scalar?

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1. Oct 27, 2016

### donaldparida

I know that a vector quantity or simply a vector is a physical quantity which has a magnitude and is associated with some definite direction. According to this definition should not pressure and current be vectors since both are associated with some definite direction?In some places they are treated as scalars while in other places as tensors. What types of quantities are they actually?Also is there any method we can use to prove that a physical quantity is a vector or not.

2. Oct 27, 2016

### vanhees71

Are you referring to relativistic or non-relativistic tensors? In Newtonian physics one is talking about tensors in the sense of 3D vector calculus, and there transformation properties refer to rotations. Now pressure is a special case of stress, and stress is described by a symmetric rank-two tensor. In the case of a static fluid the stress tensor reads $$\hat{\sigma}=-\mathrm{diag}(p,p,p).$$

The current density (not current!) is a vector (field). If $\rho$ is the density and $\vec{v}$ the flow field, then
$$\vec{j}=\rho \vec{v}.$$
The current through a surface $A$ is given by
$$I=\int_{A} \mathrm{d}^2 \vec{a} \cdot \vec{j}$$
is thus indeed a scalar. It gives the amount of mass of the fluid going through the surface $A$. The surface-normal vectors can be oriented as you like. The two different orientations of the surface just defines the sign of the current, i.e., the direction of the normal vectors tells you which direction of the flow counts as positive.

You can proof of any components $T_{ijk\ldots}$ whether they are tensor components by checking that they transform under rotations (orthogonal linear transformations) as (Einstein summation implies; all components wrt. to Cartesian bases)
$$T'_{abc\ldots} = O_{ai} O_{bj} O_{ck} \cdots T_{ijk\ldots}.$$