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Unit 15: Tensors in Cartesian Coordinates
Notes
3
X ......... ... S ... X ...h ... ...(3)
...i ...
....
...
i
...........
h 1 h
In a similar way, each lower index is served by inverse transition matrix T and also produces one
summation in formula (1):
3
....
.........
X ...ja... ... T ... X .......... ...(4)
...
k
...k ...
k 1 j
Formulas (3) and (4) are the same as (1) and used to highlight how (1) is written. So tensors are
defined. Further we shall consider more examples showing that many well-known objects
undergo the definition.
Task What are the valencies of vectors, covectors, linear operators, and bilinear
forms when they are considered as tensors.
Exercise 15.19: Let a be the matrix of some bilinear form a. Lets denote by b components of
ij
ij
inverse matrix for a . Prove that matrix b under a change of basis transforms like matrix of
ij
ij
twice-contravariant tensor. Hence, it determines tensor b of valency (2, 0). Tensor b is called a
dual bilinear form for a.
15.6 Dot Product and Metric Tensor
The covectors, linear operators, and bilinear forms that we considered above were artificially
constructed tensors. However there are some tensors of natural origin. Lets remember that we
live in a space with measure. We can measure distance between points (hence, we can measure
length of vectors) and we can measure angles between two directions in our space. Therefore,
for any two vectors x and y we can define their natural scalar product (or dot product):
(x, y) = |x| |y| cos() ...(1)
where is the angle between vectors x and y.
Task Remember the following properties of the scalar product (1):
(1) (x + y, z) = (x, z) + (y, z);
(2) (a x, y) = a (x, y);
(3) (x, y + z) = (x, y) + (x, z);
(4) (x, a y) = a (x, y);
(5) (x, y) = (y, x);
(6) (x, x) 0 and (x, x) = 0 implies x = 0.
These properties are usually considered in courses on analytic geometry or vector algebra, see
Vector Lessons on the Web.
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