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Math Help - Pure contravariant tensor

  1. #1
    Super Member Showcase_22's Avatar
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    Pure contravariant tensor

    A pure contravariant tensor of order (p,0) (the 0 shows that a vector space V involved) has the form:

    \underbrace{ V^* \times \cdots \times V^* }_{p \ times} \rightarrow \mathbb{R}

    where V^* is the dual space of V.

    I'm having a lot of trouble trying to think of an example! I figure the easiest example is to take a vector space \mathbb{R}^3 and any 3 \times 3 matrix. These types of matrices will be in the dual space and they will represent a linear map.

    To map a linear map to a scalar without using any vectors in V, I figure the easiest way is to just take the determinant of the matrix.

    I think this example is okay, but I wondered if there was a better one I could use.

    Anyway, my question is: Can anyone think of any other good examples of a pure contravariant tensor?
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  2. #2
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    Quote Originally Posted by Showcase_22 View Post
    A pure contravariant tensor of order (p,0) (the 0 shows that a vector space V involved) has the form:

    \underbrace{ V^* \times \cdots \times V^* }_{p \ times} \rightarrow \mathbb{R}

    where V^* is the dual space of V.

    I'm having a lot of trouble trying to think of an example! I figure the easiest example is to take a vector space \mathbb{R}^3 and any 3 \times 3 matrix. These types of matrices will be in the dual space and they will represent a linear map.

    To map a linear map to a scalar without using any vectors in V, I figure the easiest way is to just take the determinant of the matrix.

    I think this example is okay, but I wondered if there was a better one I could use.

    Anyway, my question is: Can anyone think of any other good examples of a pure contravariant tensor?
    fix an element v \in V and define \varphi: V^* \times \cdots \times V^* \rightarrow \mathbb{R} by \varphi(f_1, \cdots , f_p)=\prod_{j=1}^p f_j(v). it's clear that \varphi is mutilinear (and thus it induces a linear map \tilde{\varphi}: V^* \otimes \cdots \otimes V^* \rightarrow \mathbb{R} defined by \tilde{\varphi}(f_1 \otimes \cdots \otimes f_p)=\prod_{j=1}^p f_j(v)).
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