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In the third edition of the book Classical Mechanics, by H. Goldstein et al., on chapter 13 it is stated that:

For an elastic rod obeying Hooke’s law, it will be remembered that the extension of the rod per unit length is directly proportional to the force or tension exerted on the rod, a relation that can be written as $$F = Y \xi ,$$ where $\xi$ is the elongation per unit length and $Y$ is Young’s modulus.

Now, if I am not mistaken, the correct formula should be: $$F = Y A \xi\,,$$ where $A$ equals the area of the cross-section perpendicular to the applied force. If this were not the case, in the first formula $Y$ would then have the dimension of a force, and not that of a pressure.

Am I correct?

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    $\begingroup$ Correct. Goldstein et al. have conflated the stress-strain and force-displacement versions of Hooke’s Law. $\endgroup$ Commented Apr 30, 2018 at 15:17

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Yes you are correct. You can state Hooke's law in this form as well. Stress is proportional to strain. Modulus of elasticity can be defined as ratio of stress to strain.

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