The Unapologetic Mathematician

Mathematics for the interested outsider

Identifying Tensor Fields

Just as for vector fields, we need a good condition to identify tensor fields in the wild. And the condition we will use is similar: if T is a smooth tensor field of type (r,s), then for any coordinate patch (U,x) in the domain of T, we should be able to write out

\displaystyle T\vert_U=\sum\limits_{i_1,\dots,i_r,j_1,\dots,j_s=1}^nT^{i_1\dots i_r}{}_{j_1\dots j_s}\frac{\partial}{\partial x^{i_1}}\otimes\dots\otimes\frac{\partial}{\partial x^{i_r}}\otimes dx^{j_1}\otimes\dots\otimes dx^{j_s}

for some smooth functions T^{i_1\dots i_r}{}_{j_1\dots j_s} on U. Conversely, this formula defines a smooth tensor field on U.

Indeed, we can find these coefficient functions by evaluation:

\displaystyle T^{i_1\dots i_r}{}_{j_1\dots j_s}=T\left(dx^{i_1},\dots,dx^{i_r},\frac{\partial}{\partial x^{j_1}},\dots,\frac{\partial}{\partial x^{j_s}}\right)

for using this definition, if we plug these coordinate vector fields and coordinate covector fields into either the left or the right side of the expression above we will get the same answer. Any vector or covector fields on U can be written as a linear combination of these coordinate fields with smooth functions as coefficients, and the multilinear properties of tensors will ensure that both sides get the same value no matter what fields we evaluate them on.

Similarly, if \alpha is a differential k-form and (U,x) is a coordinate patch within its domain, then we can write

\displaystyle\alpha\vert_U=\sum\limits_{1\leq i_1<\dots<i_k\leq n}\alpha_{i_1\dots i_k}dx^{i_1}\wedge\dots\wedge dx^{i_k}

for some smooth functions \alpha_{i_1\dots i_k} on U. The proof in this case is similar, following from the definition

\displaystyle\alpha_{i_1\dots i_k}=\alpha\left(\frac{\partial}{\partial x^{i_1}},\dots,\frac{\partial}{\partial x^{i_k}}\right)

In this case we can pick the indices to be strictly increasing because of the antisymmetry of the tensors.

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July 7, 2011 - Posted by | Differential Topology, Topology

5 Comments »

  1. [...] seen that given a local coordinate patch we can decompose tensor fields in terms of the coordinate bases and on and , respectively. But what happens if we want to pass [...]

    Pingback by Change of Variables for Tensor Fields « The Unapologetic Mathematician | July 8, 2011 | Reply

  2. [...] for exterior multiplication, we will use the fact that we can write any -form as a linear combination of -fold products of -forms. Thus we only have to check [...]

    Pingback by Pulling Back Forms « The Unapologetic Mathematician | July 13, 2011 | Reply

  3. [...] show that the result doesn’t actually depend on this choice. Picking a coordinate patch gives us a canonical basis of the space of -forms over , indexed by “multisets” . Any -form over can be written [...]

    Pingback by The Uniqueness of the Exterior Derivative « The Unapologetic Mathematician | July 19, 2011 | Reply

  4. [...] check this, we will use our trick: let be a coordinate patch around , giving us the basic coordinate vector fields in the patch. If [...]

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  5. [...] we know that any -form on can be written out as a sum of functions times -fold [...]

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