PSI is a relative scalar field of weight w in the sense(adsbygoogle = window.adsbygoogle || []).push({});

that it transforms between the hatted and unhatted coordinate systems

according to,

[tex]

\bar{\psi}=J^{w}\psi[/tex]

Where J is the jacobian. According to the book, the ordinary partial

derivative of a relative scalar field is not a relative tensor field

because,

[tex]

\frac{\partial\bar{\psi}}{\partial\bar{x}^{j}}=J^{w}\frac{\partial x^{h}}{\partial\bar{x}^{j}}\frac{\partial\psi}{\partial\bar{x}^{h}}+wJ^{w-1}\frac{\partial J}{\partial\bar{x}^{j}}\psi[/tex]

(this transformation law is not the tranformation law of any sort

of relative scalar field unless the second term on the right it zero)

Later the book asserts the following relation relation (note the LHS

below is a factor in the second term on the RHS above):

[tex]\frac{\partial J}{\partial\bar{x}^{j}}=J\frac{\partial^{2}x^{h}}{\partial\bar{x}^{j}\partial\bar{x}^{l}}\frac{\partial\bar{x}^{l}}{\partial x^{h}}[/tex]

... Why isn't this zero always??? Invoking the chain rule [tex]\sum\frac{\partial a}{\partial x^{h}}\frac{\partial x^{h}}{\partial b}=\frac{\partial a}{\partial b}[/tex]

gives (einstein notation is being used - sums are implicit),

[tex]

\frac{\partial^{2}x^{h}}{\partial\bar{x}^{j}\partial\bar{x}^{l}}\frac{\partial\bar{x}^{l}}{\partial x^{h}}=\frac{\partial^{2}x^{h}}{\partial\bar{x}^{j}\partial x^{h}}=\frac{\partial}{\partial\bar{x}^{j}}\left(\frac{\partial x^{h}}{\partial x^{h}}\right)=\frac{\partial}{\partial\bar{x}^{j}}\left(n\right)=0[/tex]

This there something wrong in what is written immediatly above?

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# Why is this term not zero?

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