Page 178 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 15: Tensors in Cartesian Coordinates

Exercise 15.27: Likewise, raise index J in (1). Notes
Another well known object is the Levi-Civita symbol. This is a three dimensional array
determined by the following formula:

 0, if among j,k,q,there at least two equal numbers;
jkq 
    1, if (j k q) is even permutation of numbers (1 2 3); ...(3)
jkq
  1, if (jkq) is odd permutation of numbers (1 2 3).

The Levi-Civita symbol (3) is not a tensor. However, we can produce two tensors by means of
Levi-Civita symbol. The first of them

  det(g )  ijk ...(4)
ijk
ij
is known as the volume tensor. Another one is the dual volume tensor:
ijk
ij
  det(g )  ijk . ...(5)
Let’s take two vectors x and y. Then using (4) we can produce covector a:

3 3
j
k
i 
a   ijk x y . ...(6)
j 1 k 1


Then we can apply index raising procedure and produce vector a:
3 3 3
ri
j
r
k
a   g  ijk x y . ...(7)
i 1 j 1 k 1



Formula (7) is known as formula for the vectorial product (cross product) in skew-angular basis.
Exercise 15.28: Prove that the vector a with components (7) coincides with cross product of
vectors x and y, i.e. a = [x, y].
15.12 Summary
Suppose we have a vector x and a covector a. Upon choosing some basis e , e , e , both of
 1 2 3
them have three coordinates: x , x , x for vector x, and a , a , a for covector a. Let’s denote
2
3
1
1
2
3
by a,x the following sum:
3
a,x =  a x .
i
i
i 1

The sum is written in agreement with Einstein’s tensorial notation. It is a number depending
on the vector x and on the covector a. This number is called the scalar product of the vector
x and the covector a. We use angular brackets for this scalar product in order to distinguish
it from the scalar product of two vectors in E, which is also known as the dot product.
Defining the scalar product a,x by means of sum we used the coordinates of vector x and
of covector a, which are basis-dependent.

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