- #1

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## Main Question or Discussion Point

All but one of the tensor operations can be defined without reference to either coordinates or a basis. This can be done for instance by defining a ##(^m_n)## tensor over vector space ##V## as a multi-linear function from ##V^m(V^*)^n## to the background field ##F##.

This allows us to define tensor addition and multiplication, and the application of a ##(_n^m)## tensor to a sequence of ##r## vectors and ##s## one-forms, where ##0\leq r\leq m## and ##0\leq s\leq n##, without referring to coordinates or using any basis of ##V##. For instance, for ##(_n^m)## tensors ##T1## and ##T2##, the ##(_n^m)## tensor ##(T1+T2):V^m(V^*)^n\rightarrow F## is defined by

##(T1+T2)(\vec{v_1},\vec{v_2}, ....,\vec{v_m}, \tilde{p_1},\tilde{p_2},...,\tilde{p_n}) =

T1(\vec{v_1},\vec{v_2}, ....,\vec{v_m}, \tilde{p_1},\tilde{p_2},...,\tilde{p_n})

+

T2(\vec{v_1},\vec{v_2}, ....,\vec{v_m}, \tilde{p_1},\tilde{p_2},...,\tilde{p_n})

##

and, for ##(_{n1}^{m1})## and ##(_{n2}^{m2})## tensors ##T1## and ##T2##, the tensor ##(T1\otimes T2):V^{m1+m2} (V^*)^{n1+n2}\rightarrow F## is defined by

##(T1\otimes T2)(\vec{v_1},\vec{v_2}, ....,\vec{v_{m1+m2}}, \tilde{p_1},\tilde{p_2},...,\tilde{p_{n1+n2}}) =

T1(\vec{v_1},\vec{v_2}, ....,\vec{v_{m1}}, \tilde{p_1},\tilde{p_2},...,\tilde{p_{n1}})

\times

T2(\vec{v_1},\vec{v_2}, ....,\vec{v_{m2}}, \tilde{p_1},\tilde{p_2},...,\tilde{p_{n2}}) ##

The other important tensor operation is

I tried a few strategies to define contraction in a coordinate/basis-independent way, but got nowhere.

Can anybody think of a way to define contraction that does not refer to either a coordinate system or a basis?

This allows us to define tensor addition and multiplication, and the application of a ##(_n^m)## tensor to a sequence of ##r## vectors and ##s## one-forms, where ##0\leq r\leq m## and ##0\leq s\leq n##, without referring to coordinates or using any basis of ##V##. For instance, for ##(_n^m)## tensors ##T1## and ##T2##, the ##(_n^m)## tensor ##(T1+T2):V^m(V^*)^n\rightarrow F## is defined by

##(T1+T2)(\vec{v_1},\vec{v_2}, ....,\vec{v_m}, \tilde{p_1},\tilde{p_2},...,\tilde{p_n}) =

T1(\vec{v_1},\vec{v_2}, ....,\vec{v_m}, \tilde{p_1},\tilde{p_2},...,\tilde{p_n})

+

T2(\vec{v_1},\vec{v_2}, ....,\vec{v_m}, \tilde{p_1},\tilde{p_2},...,\tilde{p_n})

##

and, for ##(_{n1}^{m1})## and ##(_{n2}^{m2})## tensors ##T1## and ##T2##, the tensor ##(T1\otimes T2):V^{m1+m2} (V^*)^{n1+n2}\rightarrow F## is defined by

##(T1\otimes T2)(\vec{v_1},\vec{v_2}, ....,\vec{v_{m1+m2}}, \tilde{p_1},\tilde{p_2},...,\tilde{p_{n1+n2}}) =

T1(\vec{v_1},\vec{v_2}, ....,\vec{v_{m1}}, \tilde{p_1},\tilde{p_2},...,\tilde{p_{n1}})

\times

T2(\vec{v_1},\vec{v_2}, ....,\vec{v_{m2}}, \tilde{p_1},\tilde{p_2},...,\tilde{p_{n2}}) ##

The other important tensor operation is

**contraction**. But it seems to always be defined using coordinates. It is possible to define it not explicitly referring to coordinates but instead referring to basis vectors, but that is still basis-dependent. The proof that the multilinear operator thus defined is independent of the coordinate system or basis used to define it is short and simple, but I still find it unsatisfying that an operation that does not depend on coordinates or bases should need to be defined in terms of them.I tried a few strategies to define contraction in a coordinate/basis-independent way, but got nowhere.

Can anybody think of a way to define contraction that does not refer to either a coordinate system or a basis?