# Is pressure a scalar or a tensor

## If force is a vector then why is pressure a scalar? [Duplicate]

Pressure is a proportionality factor. The area gives you the direction. You need to remember that pressure is defined anywhere in the volume, not just on the surface. A gas volume has a defined pressure everywhere. And the direction of the force is up to you - the way you align your surface that you put into the gas.

F.⃗ = pA.⃗
Here you can see that the area is the vector.

Quote Wikipedia:

It is wrong (although more common) to say "the pressure is going one way or the other". The pressure as a scalar has no direction. The force given by the previous relationship to the crowd has a direction, but the pressure does not. If we change the orientation of the surfel, the direction of the normal force changes accordingly, but the pressure remains the same.

I should make this clear the through the pressure caused force calculated so that given the area vector, you get the force perpendicular to the surface. It is the defining Equation and the only one that records what pressure actually does. So it is always true, but it must be understood as a formula for calculating the force from pressure.

If F.⃗ is only caused by the pressure, it can nothing other than perpendicular to the area, otherwise other forces will be present in the system or the liquid will not be isotropic. Apart from that, assuming the F.⃗ is only caused by the pressure, one could calculate p by absolute values:

p = | F. || A |

Mathematically speaking, you've converted a vector equation into a scalar equation, the parallel vectors presupposes. Now you are not allowed to enter anything, only lengths (or projections - similar argument) of F and A, which were otherwise guaranteed to be parallel. You get nonsense. You also lose the pressure mark (this cannot happen with gases, but with elastic solids or liquids it can "pull" due to intermolecular forces).

Strictly speaking, pressure is a Tensor , but for gases it is isotropic so that it acts as a scalar. Furthermore, imagine, without going into details, what a tensor is pA.⃗, p can also change the direction, not just the size of EIN⃗ so the force does not have to point perpendicularly. This applies to elastic solids where you sideways Can transfer forces to the surface, and for viscous fluids where the viscous force is also just a tension (generalized pressure) transferred to the surface. In this situation, p has 6 independent components, so you they don't can only measure by measuring a force on a single surface. You would have to measure all force components on 3 surfaces that are arranged in different orientations. Only with gases can you be sure that the force will be the same regardless of the orientation.

https://en.wikipedia.org/wiki/Cauchy_stress_tensor

### Orion

The equation I have given is the basic definition of force defined by pressure and is always correct. It can't fail but when you get it invert want to calculate the pressure then it will be your mistake if you put the wrong forces or areas into it (remember, you cannot divide by a vector so the inversion loses information and induces guesswork). It's kind of like that y = x2 always gives you y when you enter a value of x, but inversion x = y√ allows you to enter nonsense values ​​(e.g. negative if we limit ourselves to real values) and lose that in the process Sign of the result.

### Orion

You can't define it that way, it's not general, and it's wrong if your conditions aren't met. Stress tensor and pressure are always defined as those the Powers cause , not the other way around. This is just a simplified explanation of elementary school pressure, but physically it doesn't go well with generalizing.

### Orion

Especially since there is pressure without forces. Forces appear when a surface is present, they are just edge effects. Pressure is more than that. It is a more fundamental quantity, and when you are describing the complete movement of materials (including deformation, sound, flow, thermodynamics) the definition "force" is not helpful and defined "backwards".