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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.^[1]

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

$a\ dx+b\ dy=a'dx'+b'dy',$

(1)

$A\ dx+B\ dy=A'dx'+B'dy',$

(2)

then

$a=a'{\frac {\partial x'}{\partial x}}+b'{\frac {\partial y'}{\partial x}},\ b=a'{\frac {\partial x'}{\partial y}}+b'{\frac {\partial y'}{\partial y}}$ $A=A'{\frac {\partial x'}{\partial x}}+B'{\frac {\partial y'}{\partial x}},\ B=A'{\frac {\partial x'}{\partial y}}+B'{\frac {\partial y'}{\partial y}},$

and it is easy to see that

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

This implies that

$(aB-Ab)dx\ dy=(a'B'-A'b')dx'dy'$

(3)

Now this relation can be obtained from the previous pair by the process of multiplication used in Grassmann's calculus of extension.^[2] 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

$a={\frac {\partial }{\partial x}},\ b={\frac {\partial }{\partial y}},\ a'={\frac {\partial }{\partial x'}},\ b'={\frac {\partial }{\partial y'}}$ $\left({\frac {\partial B}{\partial x}}-{\frac {\partial A}{\partial y}}\right)dx\ dy=\left({\frac {\partial B'}{\partial x'}}-{\frac {\partial A'}{\partial y'}}\right)dx'dy',$

(4)

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

1. The theory of integral invariants has been developed by Poincaré, Mécanique céleste, t. 3 ; Méthodes nouvelles de la Mécanique céleste, t. 3, p. 33 ; Goursat, Liouville's Journal (6), t. 4, p. 331; Koenigs, Comptes rendus, t. 122 (1906), pp. 25-37.
2. Scott's Determinants, p. 16.