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So we are considering a material line element parallel with the [itex]x_{1}[/itex] axis being deformed.

At a point in the proof we have

[itex]du_{1} = u_{1}(x_{1}+dL_{0}, x_{2}, x_{3})-u_{1}(x_{1}, x_{2}, x_{3})[/itex]

This is just saying we have a deformation happening along the[itex] x_{1}[/itex] axis.

where [itex]u[/itex] is a

__displacement vector__: [itex]u=R-r[/itex] where r is the intiial position of an element in the elastic material and R is the same element's position while deformed. [itex]u_{1}[/itex] is parallel to the x_{1} axis

[itex]L_{0}[/itex] is a material line element parallel to the [itex]x_{1}[/itex] axis. So considering that line element: [itex]dr = dL_{0}\hat{i}_{1}[/itex]

Anyway they have the following step:

[itex]du_{1} = u_{1}(x_{1}+dL_{0}, x_{2}, x_{3})-u_{1}(x_{1}, x_{2}, x_{3})[/itex]

[itex]=u_{1}(x_{1},x_{2}, x_{3})+\frac{\partial u_{1}}{\partial x_{1}}dL_{0}+O(dL_{0})^{2}-u_{1}(x_{1}, x_{2}, x_{3})[/itex]

I am stuck with what happened there. Any help with what happened there would be appreciated.