21 Appendix A — Skew Tensors, Permutation, and Tensor Algebra
Axial vectors, the Levi-Civita tensor, and applications
This appendix gathers the algebra and advanced properties of skew-symmetric 2nd-order tensors and the permutation tensor, which underpin many formulas in continuum mechanics and elasticity. References to L02 tensor calculus appear throughout.
21.1 A3.1 Skew-Symmetric Tensors and Axial Vectors
21.1.1 Definition and Structure
A second-order tensor \(\mathbf{W}\) is skew-symmetric (antisymmetric) if: \[ \mathbf{W}^T = -\mathbf{W}, \quad \text{or equivalently} \quad W_{ij} = -W_{ji} \]
In 3D, a skew-symmetric tensor has exactly three independent components. By contrast, a general 2nd-order tensor has 9 components, and a symmetric tensor has 6 components.
Example: The spin tensor (skew part of the velocity gradient): \[ \mathbf{W} = \operatorname{skew}(\mathbf{L}) = \frac{1}{2}(\mathbf{L} - \mathbf{L}^T) \]
21.1.2 The Axial Vector
Every skew-symmetric tensor \(\mathbf{W}\) in 3D corresponds to a unique axial vector \(\mathbf{w}\) (also called the dual vector or pseudovector) such that: \[ \mathbf{W}\mathbf{v} = \mathbf{w} \times \mathbf{v} \quad \forall \, \mathbf{v} \]
Extraction formula: Given \(\mathbf{W}\), extract \(\mathbf{w}\) by: \[ w_i = -\frac{1}{2}\varepsilon_{ijk}W_{jk} \]
where \(\varepsilon_{ijk}\) is the Levi-Civita symbol (defined in Section A3.2 below).
Inverse (construction formula): Given \(\mathbf{w}\), construct the skew tensor: \[ W_{ij} = -\varepsilon_{ijk}w_k = \varepsilon_{ikj}w_k \]
21.1.3 Component Relation
In component form, with \(\mathbf{w} = w_1\mathbf{e}_1 + w_2\mathbf{e}_2 + w_3\mathbf{e}_3\):
\[ \mathbf{W} = \begin{pmatrix} 0 & -w_3 & w_2 \\ w_3 & 0 & -w_1 \\ -w_2 & w_1 & 0 \end{pmatrix} \]
Check: \(\mathbf{W}\mathbf{v} = \begin{pmatrix} -w_3 v_2 + w_2 v_3 \\ w_3 v_1 - w_1 v_3 \\ -w_2 v_1 + w_1 v_2 \end{pmatrix} = \mathbf{w} \times \mathbf{v}\) ✓
21.1.4 Physical Interpretation: Angular Velocity
In kinematics, the spin tensor \(\mathbf{W} = \operatorname{skew}(\dot{\mathbf{F}}\mathbf{F}^{-1})\) has axial vector \(\mathbf{w} = \boldsymbol{\omega}\), the angular velocity of the material.
A material point undergoing rigid rotation with angular velocity \(\boldsymbol{\omega}\) has velocity: \[ \mathbf{v} = \boldsymbol{\omega} \times \mathbf{x} \]
Equivalently (in matrix form): \[ \mathbf{v} = \mathbf{W}\mathbf{x}, \quad \mathbf{W} = \begin{pmatrix} 0 & -\omega_3 & \omega_2 \\ \omega_3 & 0 & -\omega_1 \\ -\omega_2 & \omega_1 & 0 \end{pmatrix} \]
21.1.5 Worked Example: Extracting Angular Velocity
Suppose a rigid body has spin tensor: \[ \mathbf{W} = \begin{pmatrix} 0 & 0 & -2 \\ 0 & 0 & 0 \\ 2 & 0 & 0 \end{pmatrix} \]
Extract the axial vector using \(w_i = -\frac{1}{2}\varepsilon_{ijk}W_{jk}\):
- \(w_1 = -\frac{1}{2}(\varepsilon_{123}W_{23} + \varepsilon_{132}W_{32}) = -\frac{1}{2}(1 \cdot 0 + (-1) \cdot 0) = 0\)
- \(w_2 = -\frac{1}{2}(\varepsilon_{231}W_{31} + \varepsilon_{213}W_{13}) = -\frac{1}{2}(1 \cdot 2 + (-1) \cdot (-2)) = -\frac{1}{2}(2 + 2) = -2\)
- \(w_3 = -\frac{1}{2}(\varepsilon_{312}W_{12} + \varepsilon_{321}W_{21}) = -\frac{1}{2}(1 \cdot 0 + (-1) \cdot 0) = 0\)
So \(\mathbf{w} = (0, -2, 0)^T\), meaning the body rotates about the \(\mathbf{e}_2\)-axis with angular velocity magnitude 2 rad/s.
21.2 A3.2 The Permutation Tensor and Levi-Civita Symbol
21.2.1 Levi-Civita Symbol (Notation)
The Levi-Civita symbol \(\varepsilon_{ijk}\) is a coordinate-dependent symbol with values:
\[ \varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is an even permutation of } (1,2,3) \\ -1 & \text{if } (i,j,k) \text{ is an odd permutation of } (1,2,3) \\ 0 & \text{if any two indices are equal} \end{cases} \]
Explicit values (in 3D): \[ \varepsilon_{123} = \varepsilon_{231} = \varepsilon_{312} = 1, \quad \varepsilon_{132} = \varepsilon_{213} = \varepsilon_{321} = -1, \quad \text{all others } = 0 \]
21.2.2 Levi-Civita Tensor (True Tensor)
The Levi-Civita tensor \(\boldsymbol{\varepsilon}\) is a genuine third-order tensor that transforms properly under coordinate changes. In Cartesian coordinates, its components coincide with the symbol:
\[ \varepsilon_{ijk}^{\text{Cartesian}} = \varepsilon_{ijk}^{\text{symbol}} \]
However, in curvilinear coordinates with metric tensor \(g_{ij}\), the tensor components are:
\[ \varepsilon_{ijk}^{\text{tensor}} = \sqrt{g}\,\varepsilon_{ijk}^{\text{symbol}}, \quad g = \det(g_{ij}) \]
This course uses Cartesian coordinates exclusively, so we work with the symbol and tensor interchangeably.
21.2.3 Epsilon-Delta Identity
The most useful formula relates the permutation symbol to the Kronecker delta:
\[ \boxed{\varepsilon_{ijk}\varepsilon_{lmk} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}} \]
Proof sketch: Expand both sides for all values of \(i, j, l, m\) and verify that both equal 1 when \(\{i,j\} = \{l,m\}\) (in the same or opposite order) and 0 otherwise. ✓
Special cases: - \(\varepsilon_{ijk}\varepsilon_{ijk} = 6\) (contracting all three pairs) - \(\varepsilon_{ijk}\varepsilon_{ljk} = 2\delta_{il}\) (contracting the last two pairs)
21.2.4 Determinant Formula
The determinant of a 3×3 matrix \(\mathbf{A}\) can be written as:
\[ \det\mathbf{A} = \varepsilon_{ijk}A_{i1}A_{j2}A_{k3} = \varepsilon_{ijk}A_{1i}A_{2j}A_{3k} \]
More generally, any column can be used: \[ \det\mathbf{A} = \varepsilon_{ijk}A_{im}A_{jn}A_{kp}\varepsilon_{mnp} \]
Application: In continuum mechanics, the volume change is \(J = \det\mathbf{F} = \frac{1}{6}\varepsilon_{ijk}\varepsilon_{lmn}F_{il}F_{jm}F_{kn}\).
21.2.5 Vector Triple Product
The vector triple product \(\mathbf{a} \times (\mathbf{b} \times \mathbf{c})\) can be expanded using permutation notation:
\[ [\mathbf{a} \times (\mathbf{b} \times \mathbf{c})]_i = \varepsilon_{ijk}a_j(\mathbf{b} \times \mathbf{c})_k = \varepsilon_{ijk}a_j\varepsilon_{klm}b_l c_m \]
Using the epsilon-delta identity with \(\varepsilon_{ijk}\varepsilon_{klm} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}\):
\[ [\mathbf{a} \times (\mathbf{b} \times \mathbf{c})]_i = a_j b_i c_j - a_j c_i b_j = (\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{a}\cdot\mathbf{b})\mathbf{c} \]
\[ \boxed{\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{a}\cdot\mathbf{b})\mathbf{c}} \]
This formula is invaluable in deriving vector identities and checking consistency in mechanics equations.
21.2.6 Curl (Rotation) Operator
In index notation, the curl of a vector field \(\mathbf{u}(\mathbf{x})\) is:
\[ (\nabla \times \mathbf{u})_i = \varepsilon_{ijk}\frac{\partial u_k}{\partial x_j} \]
For example, the vorticity in fluid mechanics is: \[ \boldsymbol{\omega} = \nabla \times \mathbf{v} = \text{axial vector of skew}(\nabla\mathbf{v}) \]
The curl is automatically skew because \(\varepsilon_{ijk}\) is skew in its last two indices.
21.3 A3.3 Lie Algebra of Skew-Symmetric Tensors: \(\mathfrak{so}(3)\)
21.3.1 The Lie Algebra \(\mathfrak{so}(3)\)
The set of all 3×3 skew-symmetric tensors forms a Lie algebra denoted \(\mathfrak{so}(3)\), with the commutator as the Lie bracket:
\[ [\mathbf{W}_1, \mathbf{W}_2] = \mathbf{W}_1\mathbf{W}_2 - \mathbf{W}_2\mathbf{W}_1 \]
Properties: - Closure: The commutator of two skew tensors is skew. - Dimension: 3 (one basis element per axial-vector component). - Basis: Three generators corresponding to rotations about each axis:
\[ \mathbf{E}_1 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}, \quad \mathbf{E}_2 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{pmatrix}, \quad \mathbf{E}_3 = \begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} \]
A general skew tensor \(\mathbf{W}\) with axial vector \(\mathbf{w} = (w_1, w_2, w_3)\) is: \[ \mathbf{W} = w_1\mathbf{E}_1 + w_2\mathbf{E}_2 + w_3\mathbf{E}_3 \]
21.3.2 The Exponential Map: From Skew to Rotation
The matrix exponential of a skew tensor \(\mathbf{W}\) gives a rotation:
\[ \mathbf{R}(t) = \exp(t\mathbf{W}) = \mathbf{I} + t\mathbf{W} + \frac{t^2}{2!}\mathbf{W}^2 + \frac{t^3}{3!}\mathbf{W}^3 + \cdots \]
is an orthogonal matrix with \(\det\mathbf{R} = 1\) (a rotation).
21.3.3 Rodrigues’ Formula
For a unit axial vector \(\hat{\mathbf{w}}\) (with magnitude 1) and a scalar angle \(\theta\), let: \[ \mathbf{W}_\theta = \theta\text{skew}(\hat{\mathbf{w}}) \]
Then the exponential closes in closed form (Rodrigues’ rotation formula):
\[ \boxed{\mathbf{R} = \exp(\mathbf{W}_\theta) = \mathbf{I} + \sin\theta\,\mathbf{W}_\theta + (1-\cos\theta)\mathbf{W}_\theta^2} \]
This describes a rotation by angle \(\theta\) about the axis \(\hat{\mathbf{w}}\).
Key property: For any skew tensor \(\mathbf{W}\), the exponential \(\mathbf{R} = \exp(\mathbf{W})\) is a rotation tensor. Conversely, any rotation can be written as the exponential of a (unique) skew tensor.
21.3.4 Connection to Kinematics
When the spin tensor \(\mathbf{W} = \operatorname{skew}(\mathbf{L})\) is given, the rotation part of the deformation gradient evolves as:
\[ \frac{d\mathbf{R}}{dt} = \mathbf{W}\mathbf{R} \]
Integrating formally (if \(\mathbf{W}\) is constant over a time interval \([0, t]\)):
\[ \mathbf{R}(t) = \exp(\mathbf{W}t)\mathbf{R}(0) \]
In a finite-element or finite-difference code, this is often integrated using Rodrigues’ formula or similar exponential-map algorithms to maintain orthogonality of \(\mathbf{R}\) exactly.
21.4 A3.4 Summary of Key Formulas
Axial Vector: \[w_i = -\frac{1}{2}\varepsilon_{ijk}W_{jk}, \quad W_{ij} = -\varepsilon_{ijk}w_k\]
Epsilon-Delta: \[\varepsilon_{ijk}\varepsilon_{lmk} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}\]
Cross Product (Component Form): \[(\mathbf{a} \times \mathbf{b})_i = \varepsilon_{ijk}a_j b_k\]
Vector Triple Product: \[\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a}\cdot\mathbf{c})\mathbf{b} - (\mathbf{a}\cdot\mathbf{b})\mathbf{c}\]
Determinant (Column 1): \[\det\mathbf{A} = \varepsilon_{ijk}A_{i1}A_{j2}A_{k3}\]
Rodrigues’ Formula: \[\mathbf{R} = \mathbf{I} + \sin\theta\,\mathbf{W} + (1-\cos\theta)\mathbf{W}^2, \quad \mathbf{W} = \operatorname{skew}(\hat{\mathbf{w}})\]
All formulas assume Cartesian coordinates and 3D Euclidean space.