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5. The Transformation of Integral Forms.
The general theory of the transformation of physical problems by means of a change of coordinates can be developed in a convenient manner by studying a number of integral forms, examining the way in which they are related to one another, and obtaining the formulae by means of which they can be transformed.

We shall commence by studying the simple case of a transformation in two variables from (x, y) to (x', y'). Suppose that

then

and it is easy to see that

$aB-Ab=(a'B'-A'b')\frac{\partial(x',y')}{\partial(x,y)}.$

This implies that

Now this relation can be obtained from the previous pair by the process of multiplication used in Grassmann's calculus of extension. In this calculus, the sign of a product depends on the order of the terms; thus

$dx\ dy=-dy\ dx$ and $dx\ dx=0$

This rule can also be applied to the case in which the quantities a and b are differential operators; thus, if

a relation which may be obtained directly by means of Green's theorem.