I was studying from H.C. Verma - Concepts of Physics 2 (if you have the book, page 6 - topic 23.10). If you don't have the book, this is the concerned matter.
Note: If you know this stuff, you can jump to the end.
Consider a rod at temperature and suppose its length at this temperature is . As the temperature is changed to , the length is changed to . We define average coefficient of linear expansion in the temperature range as The coefficient of linear expansion at temperature is limit of average coefficient as , i.e., Suppose the length of a rod is at and at temperature measured in Celsius. If is small and constant over the given temperature interval, or The coefficient of volume expansion is defined in a similar way. If is the volume of a body at temperature , the coefficient of volume expansion at temperature is It is also known as coefficient of cubical expansion.
If and denote the volumes at and (measured in Celsius) respectively and is small and constant over the given temperature range, we have
It is easy to show that .
My question is: How do you prove that ?
Note for those who jumped: is the coefficient of cubical expansion, and is the coefficient of linear expansion.