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Math Help - Show that an object is a tensor

  1. #1
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    Show that an object is a tensor

    Suppose that an object with two indices has the property that A(\mu,\nu)C^{\mu\nu} is a scalar for any arbitrary tensor C^{\mu\nu}. Show that A(\mu,\nu) is a tensor.


    Hint: start with the relationship A^{'\nu}_{\mu}x^{'\mu}x^{'\nu}=A^\beta_\alpha x^\alpha x^\beta
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  2. #2
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    Suppose that with respect to coordinate system (x^i) we have that <br />
A(\mu, \nu)C^{\mu \nu}=k, where k is a scalar. Now, in coordinate system (\bar x^i) we have that \bar A(\mu, \nu)\bar C^{\mu \nu}=\bar k =k.

    Since C^{\mu \nu} are the components of a contravariant tensor of order two, we have that

    \displaystyle{\bar C^{\mu \nu}=\frac{\partial \bar x^\mu}{\partial x^i}\frac{\partial \bar x^\nu}{\partial x^j}C^{i j}}.

    Hence,

    \displaystyle{<br />
[\bar A(\mu, \nu)\frac{\partial \bar x^\mu}{\partial x^i}\frac{\partial \bar x^\nu}{\partial x^j}-A(i, j)]C^{i j}=0.

    Since this holds for arbitrary components C^{i j}, we have that

    \displaystyle{<br />
\bar A(\mu, \nu)\frac{\partial \bar x^\mu}{\partial x^i}\frac{\partial \bar x^\nu}{\partial x^j}-A(i, j)=0}.<br />

    Multiplying both sides by \displaystyle{\frac{\partial x^i}{\partial \bar x^r}\frac{\partial x^j}{\partial \bar x^s}} we get

    \displaystyle{<br />
\bar A(\mu, \nu)\delta_r^\mu\delta_s^\nu=A(i,j)\frac{\partial x^i}{\partial \bar x^r}\frac{\partial x^j}{\partial \bar x^s}}.<br />

    Hence,

    \displaystyle{<br />
\bar A(r,s)=A(i,j)\frac{\partial x^i}{\partial \bar x^r}\frac{\partial x^j}{\partial \bar x^s}}.<br />

    That is, A(\mu ,\nu) are the components of a covariant tensor of order two.
    Last edited by ojones; March 26th 2011 at 01:47 AM.
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