Problem With Explanation of Inner Product of Vector and Dyad

[marschmellow]

I've been trying to learn more about tensors with the help of this website, http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf, but its explanation on one little part about vectors has me puzzled.

It states that an inner product of a vector S and a dyad expressed as the product of vectors U and V, UV, is equal to S*UV=(S*U)V=kV where k is a scalar k=S*U. That makes perfect sense. But then it says that the result is a vector with magnitude k and direction determined by V. There was no requirement that any of these vectors were unit vectors, so wouldn't the magnitude be k|V|?

Also, when discussing tensors, is it assumed that the "product" of two tensors is the dyad product and not the inner or cross product unless so specified?

 

[llarsen]

(S*U) is a dot product which results in a scalar k=S*U. The scalar is them multiplied by the vector V. If you want to define a unit vector that points in the direction of V and call it Uv, then you can say that the magnitude is k|V| as you said and the result is k|V|Uv, but this is the same as kV, which is the result that they gave.

I am not a big fan of the notation that they use. Index notation that uses upper and lower indexes makes it much more clear whether you are using an inner product (the index is the same Ui*Vi - scalar result) or a dyadic product Ui*Vj = Aij dyadic result. The upper and lower indices indicate whether you are working with a vector or a covector. It seems to me that the notation used in the "introduction" raises a lot of questions because the type of object you re working with is implied rather than explicit.

 

[marschmellow]

I get that the result is kV and that k|V|Uv is the same thing as kV, but the site specified that the magnitude of kV, or |kV|, is equal to k, which just isn't true. I may find another source for this information, because of this little problem here and the stuff you just mentioned about lack of clarity in notation. I also find that the author sometimes skips steps that are very necessary to see for understanding but will then write out every little detail for something dead obvious (like the noncommutativity of dyadic multiplication). Does anyone know any better online sources for learning about tensors and their application in physics (classical or modern, but as non-esoteric as possible)?

 

[llarsen]

I found a short references you might try http://faculty.gg.uwyo.edu/dueker/tensor%20curvilinear%20relativity/tensor%20analysis%20intro.pdf" . It seems to do a good job explaining some of the tensor concepts.

There is a longer introduction http://arxiv.org/PS_cache/math/pdf/0403/0403252v1.pdf that you might look at for more details.

If you want to get a graphical feel for what tensors represent, I like "Applied Differential Geometry" by William Burke. He gives lots of pictures which I think help give a better sense of what you are working with. You can see the book on books.google.com if you want a quick preview. This book is more useful I think after you have some of the notation down. It gives some graphical sense of how to interpret the vector and covector building blocks that make up all tensors.

 

[adriank]

but the site specified that the magnitude of kV, or |kV|, is equal to k, which just isn't true.

The site is wrong.