Changes

The negative sign here is to indicate that the force is in the direction opposite the displacement.  For example, if <math>\vec x</math> is to the right, then <math>-k \vec x</math> is to the left, indicating that the mass is pulled back towards the spring.

The negative sign here is to indicate that the force is in the direction opposite the displacement.  For example, if <math>\vec x</math> is to the right, then <math>-k \vec x</math> is to the left, indicating that the mass is pulled back towards the spring.

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|\vec x|^2</math>, this means that <math>\vec F</math> is a [[conservative field]].  As a result, <math>E = \frac{1}{2} k|\vec x|^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.

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.