Linear vs. Volume Expansion Coefficient

At first glance, we may interpret the coefficient of volume thermal expansion, 

Consider a cubic element within a material of volume V = lxlylz- Substituting into the definition of coefficient of volume expansion and using the product rule  

The sample lengths can be written in terms of initial length plus change in length  

If the change in length is small compared to the overall sample length, the last three terms may be considered insignificant  

Two assumptions have been made which will certainly not be true for all specimens  

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