Skew-Symmetric Tensor: Computing [W]e and Axial Vector w

[kishan]

If W is a skew-symmetric tensor,
(i) Write down the most general form of [W]e.
(ii) Show that there exist a vector w in R3 such that Wx = w*x for each x in R3. Such a vector w is
called the axial vector of W.
(iii) Use part (ii) to deduce that x.Wx = 0 for any x in R3.
(iv) If W has axial vector w = wi ei, write down [W]e

 

[jvicens]

in terms of w.

(i) The most general form of [W]e for a skew-symmetric tensor W is:
[W]e = [0 -w3 w2;
w3 0 -w1;
-w2 w1 0]

(ii) To show that there exists a vector w in R3 such that Wx = w*x for each x in R3, we can choose w = (w1, w2, w3) and x = (x1, x2, x3).
Then, we have:
Wx = [0 -w3 w2;
w3 0 -w1;
-w2 w1 0] * (x1, x2, x3)
= (w2x3 - w3x2, w3x1 - w1x3, w1x2 - w2x1)
= (w1, w2, w3) * (x2, -x1, 0)
= w * x

(iii) Using the result from part (ii), we can see that:
x.Wx = x * (w * x)
= w * (x * x)
= w * 0
= 0
Therefore, x.Wx = 0 for any x in R3.

(iv) If W has axial vector w = wi ei, then we can write [W]e in terms of w as:
[W]e = [0 -w3 w2;
w3 0 -w1;
-w2 w1 0]
= [0 -w3 w2;
w3 0 -w1;
-w2 w1 0] * (e1, e2, e3)
= (w2e3 - w3e2, w3e1 - w1e3, w1e2 - w2e1)
= w1 * [0 -e3 e2;
e3 0 -e1;
-e2 e1 0] + w2 * [0 e3 -e1;
-e3 0 e2;
e1 -