Hi all. I am very puzzled by the following.(adsbygoogle = window.adsbygoogle || []).push({});

Let [itex]x_1[/itex] and [itex]x_2[/itex] be two coordinate systems related by [itex]x_1=1-x_2[/itex].

Now if [itex]y(x_1) = x_1[/itex] and [itex]z(x_2) = 1-x_2[/itex], then clearly [itex]y(x_1)=z(x_2)[/itex].

Now integrating the function in each coordinate system gives

[tex]Y(x_1) = \int y(x_1) dx_1 = \int x_1 dx_1 = \frac{x_1^2}{2} + C [/tex]

[tex]Z(x_2) = \int z(x_2) dx_2 = \int (1-x_2) dx_2 = -\frac{x_2^2}{2} + x_2 + D [/tex]

Now, however,

[tex]Y(x_1) = Y(1-x_2) = \frac{(1-x_2)^2}{2} + C = \frac{1}{2} - x_2 + \frac{x_2^2}{2} + C \neq Z(x_2) [/tex]

In words, [itex]Y(x_1) \neq Z(x_2)[/itex] regardless of the values of [itex]C[/itex] and [itex]D[/itex]. One would expect, however that [itex]Y(x_1) = Z(x_2)[/itex] - but then where was my mistake? Thanks!

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# Integration in two different coordinate systems

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