# Is there any direction of pressure i.e. according to force applied or something..?

I was reading about pressure in general and came to know that it is a scalar and being scalar it has no direction(by definition of scalars). But I think that there is some direction as when we push a wall we apply pressure on it in a particular direction,while cutting an object with a knife etc. I don't know if its correct.Please help. Any help and edits are appreciated

Pressure (in the sense of the hydrostatic pressure $P$, i.e., the normal stress on all sides) is a scalar. Pressure (in the sense of the stress $\sigma$ in the context of materials science) is not a scalar but is instead a 3×3 matrix.

The matrix $\sigma=\sigma_{ij}$ arises because of the variety in possible directions, as you intuit: there are three orthogonal directions (e.g., $1$, $2$, and $3$, corresponding to the $x$, $y$, and $z$ directions, respectively) that the surface might face, and there are three independent orthogonal directions along which a force might be applied to that surface.

Thus, the stress matrix (aka the Cauchy stress tensor), as a 3-D generalization of "pressure", is $$\sigma_{ij}=\begin{pmatrix}\sigma_{11} & \sigma_{12} & \sigma_{13}\\&\sigma_{22}&\sigma_{23}\\&&\sigma_{33}\\\end{pmatrix}$$ where the first index is the direction of a vector normal to the surface and the second index is the direction of the force vector.

The matrix is diagonally symmetric because static equilibrium tells us that certain elements must be identical; for example, the magnitude of the force in the $1$ direction on the surface pointing in the $2$ direction must be equal to the magnitude of the force in the $-1$ direction on the surface pointing in the $-2$ direction, or the object would start accelerating along the $2$-axis. Furthermore, it must also be equal to the magnitude of the force in the $2$ direction on the surface pointing in the $1$ direction, or the objective would start rotating around the $3$ axis. Consequently, there are six independent stress values.

Finally, the scalar pressure $P$ corresponds to a stress matrix of $$\sigma_{ij}=\begin{pmatrix}-P & 0 & 0\\0&-P&0\\0&0&-P\\\end{pmatrix}$$ because hydrostatic pressure is compressive.

Pressure does not have a direction. As your reading suggested, it is a scalar. The way we introduce pressure to students is we talk about spreading a force out over some area, and pressure is the force divided by the area over which it is spread. More exactly, the force is the pressure times the area, and the area is what defines the direction of the force.

What's making this complicated is that I'm implicitly assuming that the pressure on one side of that area has one value, the pressure on the other is zero, and the surface has no width to it. In other words, the pressure suddenly drops. If we make things more realistic, the pressure will drop smoothly over some distance. The rate of change of pressure (drop in pressure per unit distance of drop) is a force density: the force applied per unit volume to that infinitesimal volume along the direction the rate of change was evaluated. The components of the force come from looking at the rate of change of pressure in all three directions (ie the gradient).

The definition of pressure is $p=\frac{|\vec{F}_n|}{A}$ where A is the area of the surface of contact and $\vec{F}_n$ is the "part"(or component) of the force that is normal to the surface(or the projection of the force on the unit normal vector to the wall) and $|\vec{F}_n|$ is the magnitude of that part of the force. Thus, pressure is a scalar quantity but it contains information about the direction of the force that goes into its definition.
So, if you think of the pressure as having direction normal to the surface you consider you are really thinking about the direction of $\vec{F}_n$. So, keep this in mind in order to build your intuition while solving problems but always remember that pressure is strictly a scalar quantity.