# Tensor rank of pressure

1. Jul 13, 2010

### ovais

Hello everybody, yesterday I stand to teach vectors and scalars to 12th standard students in a coaching.While giving examples of scalars I named mass , work , pressure etc.Then a student argued me that pressure should be a vector quantity since when you apply a push on wall that is force then the pressure would also acts in the direction where you are applying the force.So according to him pressure is a vector quantity.While in books I always read pressure a scalar.Actually I my self used to wonder how is pressure a scalar quantity while it seems associated with direction just as a force do, there I answer that since we do talk of pressure(while on solid surfaces) as pressure on a surface, which always acts normal to the surface no matter be the surface plane or curved thus it is immaterial to say the direction of the pressure since it is always calculated on the surface that is perpendicular to it or in opposite direction of area vector( just as we take are in guss law, which however has nothing to do with pressure but is helpful here to explain about area vector).Though I explain this, neither me nor he was fully satisfied, since still it has a direction though fixed.I search the net and get many useful points but still it is not getting clear as to how pressure a scalar quantity.Then I come across with tensors, I want to know-1: what is the tensor rank of pressure? 2:If its rank is zero (that is if is scalar) then please explain what physical quantities can be taken with non-zero rank. Thanks in advance

2. Jul 13, 2010

### DrDu

I am also not a specialist on this, but, at least in a solid, pressure is the trace of the stress tensor (or minus one third of it, to be precise) or, the isotropic part of it. Hence it transforms as a scalar. Liquids are special in so far as all other components of strain are zero.

3. Jul 13, 2010

### Phrak

If I may restate the question

Force is a vector. Force over an area is a pressure. How can pressure be a scalar?

I might add that area is properly expressed as a 2-form rather than a tensor of two upper indices such that the area is oriented; there is a unique choice for which direction the surface normal points.

Last edited: Jul 13, 2010
4. Jul 13, 2010

### DrDu

Well, so to speak, pressure is the scalar product of normal surface vector (of unit area) times force averaged over all orientations of the surface.

5. Jul 13, 2010

### ovais

thanks for your replies DrDu, I will say I am not clear when you say pressure is the trace of the stress tensor or the isotropic part of it. And how a part of something(stress tensor) non -scalar could become scalar. Thankx again

6. Jul 13, 2010

### DrDu

Well, the trace of a tensor T_ij is the sum over all T_ii, which is clearly a scalar.
E.g., I can define the tensor of the tensor product of the vectors r with itself $$T_{ij}=r_i r_j$$. It is important e.g. in the definition of the moment of inertia or of the electric multipole moments. Now the trace of this tensor $$tr(T)=\sum_i r_i r_i$$ is simply the squared length of the vector r.
What does that mean in the case of pressure:
Consider a cube made out of an elastic material in a bench vice. The force on one of the two forces in the vice is F and the area of the surface is A. The force acting on the other surfaces is 0. Then the pressure is 1/3 (F/A+0+0).

7. Jul 13, 2010

### ovais

ok the trace of tensor T_ij is the sum of all t_ii, can you please state the physical relationship between T_ij and T_ii using words other than trace?

8. Jul 13, 2010

### DrDu

T_ii are the diagonal elements of the tensor T_ij, i.e., the elements T_ij for which i=j.
Hence, if you think of the tensor as a 3x3 matrix, the trace (or spur) is the sum over all diagonal elements of the matrix.

9. Jul 13, 2010

### ovais

ah i got it u explained very well.Now I will like to know what is that tensor whose trace you defined as pressure, i know it is stress tensor as you said i want to know how i visulaize the stress tensor?where can i feel its effect?I am familiar with pressure but stress tensor is new to me?

10. Jul 13, 2010

### Q_Goest

When pressure is referred to as a scalar quantity, what is meant is that pressure at any point within a fluid has no intrinsic direction to it. Consider the pressure of a fluid NOT acting against a surface. For example, consider what pressure water has at a depth of 10 feet in the middle of the ocean. If the water isn't acting against anything, there is no directional force, so we can't call it a vector. But there is certainly pressure at that depth in the ocean, just as there is pressure in a fluid at all locations within the fluid. So the pressure in a fluid is a scaler quantity. We call pressure a scalar quantity because without considering what surface the pressure is acting against, there is no force and no vector. (Note that "fluid" here means any liquid or gas.)

In contrast, pressure acting on a surface becomes a vector quantity because the interaction of the fluid against the solid surface creates a normal force. That normal force is just the pressure times the area (integrated of course) so that FORCE is a vector quantity.

11. Jul 13, 2010

### ovais

did you mean the concept of pressure applies only to fluid?and the thing which acts on solid surfaces is pressure force(a vector) not pressure.Is that you mean?

12. Jul 13, 2010

### Q_Goest

When refering to pressure as a scalar quantity, the underlying assumption is that we're talking about a fluid. For a fluid acting on a solid, there is a force produced which is a vector quantity, but the concept of pressure in a fluid shouldn't be confused with the concept of that pressure acting at a surface.

13. Jul 13, 2010

### ovais

so pressure in fluid is scalar and pressure acting on solid surface has direction normal to it?and in the bulk fluids have pressure in scalar form?

14. Jul 13, 2010

### dulrich

Here is a quote from the http://en.wikipedia.org/wiki/Pressure#Definition" article:
This way of saying it works for both fluids and solids. The difference is that in a fluid the pressure is the same in all directions.

Last edited by a moderator: Apr 25, 2017
15. Jul 13, 2010

### ovais

the statement that pressure is the scalar proportionality constant that relates two normal vectors.please tell me what are these two vectors and how is pressure(as a constant) relates these two vectors. thanks a ton

16. Jul 13, 2010

### Studiot

If you want a description to convince them, forget tensors and try this.

How do you combine (add) vectors?
How do you combine (add) scalars?
How do you combine (add) pressures?

Think of a sealed vessel half full of water. At any point in the water there is a pressure.
Now pump up the air pressure in the other half.

How much does the pressure increase at any point in the water?

Another thing to bear in mind.

Only scalars can affect every point in this way. Vectors can only affect things in their line of action.

17. Jul 13, 2010

### dulrich

(1) The vector normal to the surface, and (2) the force that creates the pressure. One way to think of it is that P = F/A implies F = PA. But we can write the second using both scalars and vectors so that it applies in both cases.

18. Jul 13, 2010

### Studiot

I have to observe that pressure can never be a vector, simply because it does not combine according to the laws of vector addition.

19. Jul 13, 2010

### ovais

dear studiot your explanation works well when dealing with fluid in containers but what should I call the quantity F/A.Suppose i apply a force of 10 N normal to a plate .1 m2 in area.then we say we are applying a pressure of 100 N/m2 on the wall.Now if we say our pressure has nothing to do with direction, dosn't it seem wrong?

20. Jul 13, 2010

### ovais

The vector normal to the surface-it that you write A and the force vector F?If this the case then P=F/A should be meaningless, since vectors do not follow division.we can give a meaning to it by your second relation F=PA.