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is an approximation
The force <math> \vec F = -k \vec x</math> can be interpreted as a vector field depending on the displacement vector <math>\vec x</math>. Since this field is the gradient of the function <math>\frac{1}{2} k|x|^2</math>, this means that <math>\vec F</math> is a [[conservative field]]. As a result, <math>E = \frac{1}{2} kx^2</math> is conserved, and is the potential energy stored in the string. Moreover, <math>E</math> depends only on <math>\vec x</math>, and not the path that the particle takes to get to <math>\vec x</math>, and so it is said to be a [[path independent]] quantity.
The force <math> \vec F = -k \vec x</math> can be interpreted as a vector field depending on the displacement vector <math>\vec x</math>. Since this field is the gradient of the function <math>\frac{1}{2} k|x|^2</math>, this means that <math>\vec F</math> is a [[conservative field]]. As a result, <math>E = \frac{1}{2} kx^2</math> is conserved, and is the potential energy stored in the string. Moreover, <math>E</math> depends only on <math>\vec x</math>, and not the path that the particle takes to get to <math>\vec x</math>, and so it is said to be a [[path independent]] quantity.
It should be noted that in reality, Hooke's law is merely an approximation, and no physical spring actually has precisely this behavior. However, for most materials a version of Hooke's law holds for reasonable ranges of <math>\vec x</math>. This is called the ''elastic range'' of the material.
[[Category:Physics]]
[[Category:Physics]]