Let us suppose that the transformation equations from a rectangular (x, y, z) system to the and systems are given by

and

and their inverses

and

There will exist a transformation directly from the system to the system defined by  

and conversely.

From 5) we obtain

By the definition of a gradient we have

and

Now the vector A is represented in the two coordinate systems as

The left members of 7) and 8) are thus equal. We now equate the coefficients of in 7) and 8) to obtain

Substituting equations 6) with p = 1, 2, 3 in any of the equations 10) and equating coefficients of

 on each side, we obtain

End of proof.