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## Main Question or Discussion Point

I'm studying linear elasticity and I came across an equation that I having problems figuring out the derivation. I want to understand this in case I may need to know it later in the course. The equation is as follows:

[itex]\frac{\partial^2e_{zz}}{\partial x \partial y} = \frac{\partial}{\partial z}\left(\frac{\partial e_{yz}}{\partial x} + \frac{\partial e_{zx}}{\partial y} - \frac{\partial e_{xy}}{\partial z}\right)[/itex]

I'm trying to understand this derivation with the given relationships below.

[itex]e_{xx} = \frac{\partial U_x}{\partial x}[/itex]

[itex]e_{yy} = \frac{\partial U_y}{\partial y}[/itex]

[itex]e_{zz} = \frac{\partial U_z}{\partial z}[/itex]

[itex]e_{xy} = 1/2\left(\frac{\partial U_y}{\partial x} + \frac{\partial U_x}{\partial y}\right)[/itex]

[itex]e_{yz} = 1/2\left(\frac{\partial U_z}{\partial y} + \frac{\partial U_y}{\partial z}\right)[/itex]

[itex]e_{zx} = 1/2\left(\frac{\partial U_x}{\partial z} + \frac{\partial U_z}{\partial x}\right)[/itex]

The only thing that makes sense is that the derivative with respect to x and y was taken on [itex]e_{zz}[/itex].

[itex]\frac{\partial^2e_{zz}}{\partial x \partial y} = \frac{\partial}{\partial z}\left(\frac{\partial e_{yz}}{\partial x} + \frac{\partial e_{zx}}{\partial y} - \frac{\partial e_{xy}}{\partial z}\right)[/itex]

I'm trying to understand this derivation with the given relationships below.

[itex]e_{xx} = \frac{\partial U_x}{\partial x}[/itex]

[itex]e_{yy} = \frac{\partial U_y}{\partial y}[/itex]

[itex]e_{zz} = \frac{\partial U_z}{\partial z}[/itex]

[itex]e_{xy} = 1/2\left(\frac{\partial U_y}{\partial x} + \frac{\partial U_x}{\partial y}\right)[/itex]

[itex]e_{yz} = 1/2\left(\frac{\partial U_z}{\partial y} + \frac{\partial U_y}{\partial z}\right)[/itex]

[itex]e_{zx} = 1/2\left(\frac{\partial U_x}{\partial z} + \frac{\partial U_z}{\partial x}\right)[/itex]

The only thing that makes sense is that the derivative with respect to x and y was taken on [itex]e_{zz}[/itex].