Hi sjastro,
Assuming that direct product is the same as outer product?
I tried to apply what you've said to the text in the book where I got stuck.
The book gives the simple example of a 2nd order tensor Proj
u that operates on a vector v to give the projection of v onto u:
i.e. Proj
uv = (
v.
u)
u
Easy enough to prove and understand.
The author then goes on to make the breathtaking leap of stating
"
a generalisation of this is that the direct product
uv of two vectors
u and
v is a tensor that sends any vector
w into a new vector according to the rule
uv(
w) =
u(
v.
w)
"
The text is not using the (x) symbol - i.e. outer product - , so I'm lost as to how to interpret this, since the result is supposed to be a vector, not a second order tensor...
Does it mean
[ u (x)
v ] (x)
w or
u (x)
v . [
w] or some thing else?
