Suppose that an object with two indices has the property that is a scalar for any arbitrary tensor . Show that is a tensor.

Hint: start with the relationship

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- March 23rd 2011, 07:12 PMdavesfaceShow that an object is a tensor
Suppose that an object with two indices has the property that is a scalar for any arbitrary tensor . Show that is a tensor.

Hint: start with the relationship - March 26th 2011, 12:36 AMojones
Suppose that with respect to coordinate system we have that , where is a scalar. Now, in coordinate system we have that

Since are the components of a contravariant tensor of order two, we have that

Hence,

Since this holds for arbitrary components , we have that

Multiplying both sides by we get

Hence,

That is, are the components of a covariant tensor of order two.