- #1

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- Thread starter arpon
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- #1

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- #2

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- #3

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Rewrite the expression as ##f=p~a##. In that expression force is a vector and pressure is a scalar, so what kind of quantity is area?

- #4

Stephen Tashi

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How about the idea that "pressure is a scalar part of a tensor"?

- #5

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The simple explanation is in post #2, the complicated one (in terms of the Cauchy stress-energy tensor) is hinted to in post#4. As soon as one goes to study continuum (classical) dynamics, then the true nature of 'p' results. For high-school physics, post#2 will suffice.

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- #6

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$$F_j=\int_{\partial V} \mathrm{d}^2 f_k \sigma_{kj},$$

where ##\sigma_{kj}## is the stress tensor of the fluid and ##\mathrm{d}^2 \vec{f}## the surface-normal vectors along the boundary of the volume pointing by convention outward of the volume.

Pressure is one part of the stress tensor of a fluid

$$\sigma_{jk}=s_{jk}-P \delta_{jk}.$$

By definition ##s_{jk}##, the "stress deviator", is traceless and thus

$$\sigma_{jj}=-3 P$$

Since ##\sigma_{jk}## is a 2nd rank tensor, the pressure is a scalar.

The geometrical meaning is that the pressure tries to change the magnitude of the volume, while the traceless stress deviator tends to deform it.

The Wikipedia article on this topic is very nice with very good figures, making the thing pretty intuitive:

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

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