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

• MHB
• kishan
In summary: e2 0] = w1 * [0 -e3 e2; e3 0 -e1; -e2 e1 0] - w2 * [0 -e3 e1; e3 0 -e2; -e1 e2 0] = w1 * [0 -e3 e2; e3 0 -e1; -e2 e1 0] + w2 * [0 e2 e1; -e2 0 e3; -e1 -e3
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

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 -

## 1. What is a skew-symmetric tensor?

A skew-symmetric tensor is a type of tensor that is antisymmetric, meaning that it is equal to the negative of its transpose. In other words, if a tensor A is skew-symmetric, then AT = -A. This property makes it useful for representing physical quantities such as rotation and angular velocity.

## 2. What is the difference between a skew-symmetric tensor and a symmetric tensor?

A symmetric tensor is equal to its transpose, while a skew-symmetric tensor is equal to the negative of its transpose. In other words, if a tensor A is symmetric, then AT = A, while if A is skew-symmetric, then AT = -A. This means that a symmetric tensor has all its components symmetrically distributed about the diagonal, while a skew-symmetric tensor has all its components antisymmetrically distributed about the diagonal.

## 3. How do you compute the skew-symmetric tensor [W]e?

To compute the skew-symmetric tensor [W]e, you can use the cross product of two vectors e and w, where e is a unit vector representing the direction of the axis of rotation and w is the angular velocity vector. The resulting tensor will be [W]e = e x w, where x denotes the cross product.

## 4. What is an axial vector w?

An axial vector w represents the direction and magnitude of rotation in three-dimensional space. It is often used in conjunction with the skew-symmetric tensor [W]e to represent rotational motion and calculate physical quantities such as torque and angular momentum.

## 5. How is the skew-symmetric tensor [W]e used in physics?

The skew-symmetric tensor [W]e is used in physics to represent rotational motion and calculate various physical quantities related to rotation, such as torque and angular momentum. It is also used in mechanics and engineering for applications involving rigid body dynamics and kinematics.

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