3  L01 — Mathematical Foundations

Vectors, Tensors, and Index Notation

📽 Slides: Open presentation

3.1 Vectors and Vector Spaces

A vector space \(V\) over \(\mathbb{R}\) is a set closed under addition and scalar multiplication satisfying the standard axioms: 1. Closure under addition: if \(\mathbf{u}, \mathbf{v} \in V\) then \(\mathbf{u} + \mathbf{v} \in V\) 2. Associativity and commutativity of addition 3. Closure under scalar multiplication: if \(\mathbf{u} \in V\) and \(\alpha \in \mathbb{R}\) then \(\alpha\mathbf{u} \in V\) 4. Distributive properties of scalar multiplication 5. Existence of additive identity (zero vector) and multiplicative identity

An \(n\)-dimensional space has exactly \(n\) linearly independent vectors.

Note

Physical interpretation: A vector is a mathematical object representing a direction and magnitude. In continuum mechanics, vectors describe positions (\(\mathbf{x}\)), displacements (\(\mathbf{u}\)), forces, and velocities. Unlike scalars (which depend only on magnitude), vectors transform in predictable ways under coordinate changes—a property essential for writing constitutive laws that do not depend on the observer’s coordinate system.

3.1.1 Inner Product and Norms

The inner product (or dot product) of two vectors \(\mathbf{u}, \mathbf{v} \in V\) is a scalar \(\mathbf{u} \cdot \mathbf{v} \in \mathbb{R}\) satisfying:

1. Symmetry: \(\mathbf{u} \cdot \mathbf{v} = \mathbf{v} \cdot \mathbf{u}\)
2. Bilinearity: \((\alpha\mathbf{u} + \beta\mathbf{v}) \cdot \mathbf{w} = \alpha(\mathbf{u} \cdot \mathbf{w}) + \beta(\mathbf{v} \cdot \mathbf{w})\) for \(\alpha, \beta \in \mathbb{R}\)
3. Positive-definiteness: \(\mathbf{v} \cdot \mathbf{v} \geq 0\), with equality iff \(\mathbf{v} = \mathbf{0}\)
4. Non-degeneracy: if \(\mathbf{v} \cdot \mathbf{w} = 0\) for all \(\mathbf{w} \in V\), then \(\mathbf{v} = \mathbf{0}\)

The Euclidean norm (or length) of a vector is: \[ |\mathbf{v}| = \|\mathbf{v}\| = \sqrt{\mathbf{v} \cdot \mathbf{v}} \]

Cauchy-Schwarz Inequality: For any \(\mathbf{u}, \mathbf{v} \in V\), \[ |\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}| \, |\mathbf{v}| \]

Proof sketch: Consider \(\mathbf{w}(\lambda) = \mathbf{u} - \lambda\mathbf{v}\) for \(\lambda \in \mathbb{R}\). By positive-definiteness, \(|\mathbf{w}(\lambda)|^2 = \mathbf{w}(\lambda) \cdot \mathbf{w}(\lambda) \geq 0\). Expanding and minimizing over \(\lambda\) yields the inequality. Equality holds iff \(\mathbf{u}\) and \(\mathbf{v}\) are parallel.

Triangle Inequality: \(|\mathbf{u} + \mathbf{v}| \leq |\mathbf{u}| + |\mathbf{v}|\) follows from Cauchy-Schwarz.

3.1.2 Angles, Orthogonality, and Orthonormal Bases

The angle \(\theta\) between non-zero vectors \(\mathbf{u}\) and \(\mathbf{v}\) is defined by: \[ \cos\theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| \, |\mathbf{v}|} \]

Vectors are orthogonal if \(\mathbf{u} \cdot \mathbf{v} = 0\).

An orthonormal basis \(\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\}\) for \(\mathbb{R}^3\) satisfies \(\mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij}\), where \(\delta_{ij}\) is the Kronecker delta. In an orthonormal basis, any vector can be uniquely written as: \[ \mathbf{v} = (v_1, v_2, v_3) = v_i \mathbf{e}_i \] with components \(v_i = \mathbf{v} \cdot \mathbf{e}_i\).

Worked example: Let \(\mathbf{u} = (1, 2, 0)\) and \(\mathbf{v} = (3, 0, 1)\) in the standard Cartesian basis \(\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\}\).

1. Inner product: \(\mathbf{u} \cdot \mathbf{v} = 1 \cdot 3 + 2 \cdot 0 + 0 \cdot 1 = 3\).
2. Norms: \(|\mathbf{u}| = \sqrt{1 + 4 + 0} = \sqrt{5}\), \(|\mathbf{v}| = \sqrt{9 + 0 + 1} = \sqrt{10}\).
3. Angle: \(\cos\theta = \frac{3}{\sqrt{5} \cdot \sqrt{10}} = \frac{3}{\sqrt{50}} = \frac{3}{5\sqrt{2}} \approx 0.424\), so \(\theta \approx 65^\circ\).
4. Unit vector: \(\hat{\mathbf{u}} = \frac{\mathbf{u}}{|\mathbf{u}|} = \frac{1}{\sqrt{5}}(1, 2, 0)\).

3.2 Euclidean Space and Curvilinear Coordinates

In Euclidean space, we use Cartesian coordinates \(X^i\) (with \(i = 1, 2, 3\)) with orthonormal basis \(\{\mathbf{E}_i\}\) where \(\mathbf{E}_i \cdot \mathbf{E}_j = \delta_{ij}\).

For general curvilinear coordinates \(\theta^i\), we require a smooth, invertible mapping (diffeomorphism): \[ X^i = f^i(\theta^j) \quad \Leftrightarrow \quad \theta^j = g^j(X^i) \] where \(g^j\) is the inverse of \(f^i\). Both mappings must be continuous and differentiable. The position vector in Euclidean space is: \[ \mathbf{r}(\theta^1, \theta^2, \theta^3) = X^i(\theta^j)\,\mathbf{E}_i \]

3.2.1 Covariant (Tangent) Basis Vectors

As we move along a coordinate curve (say, varying \(\theta^1\) while holding \(\theta^2, \theta^3\) fixed), the position vector changes. The covariant basis vector in direction \(i\) is the tangent vector to the \(\theta^i\) coordinate curve: \[ \mathbf{g}_i = \frac{\partial \mathbf{r}}{\partial \theta^i} = \frac{\partial X^k}{\partial \theta^i}\mathbf{E}_k \]

These basis vectors are not, in general, orthonormal and not unit vectors. They form a local frame at each point in the coordinate system.

Note

Physical interpretation: In a deforming material (finite-strain kinematics), curvilinear coordinates attached to material particles move and stretch. The covariant basis vectors at a material point, when transported by the deformation, become the current basis vectors. This is why the distinction between covariant and contravariant components is essential: the former scale with basis changes (stretch), the latter scale opposite to basis changes (contract) to preserve geometric meaning.

3.2.2 Contravariant (Reciprocal) Basis Vectors

The reciprocal (contravariant) basis vectors \(\mathbf{g}^i\) are defined by the biorthogonality condition: \[ \mathbf{g}^i \cdot \mathbf{g}_j = \delta^i_j \]

In three dimensions, the reciprocal basis can be constructed explicitly using the cross product: \[ \mathbf{g}^1 = \frac{\mathbf{g}_2 \times \mathbf{g}_3}{\mathbf{g}_1 \cdot (\mathbf{g}_2 \times \mathbf{g}_3)}, \quad \mathbf{g}^2 = \frac{\mathbf{g}_3 \times \mathbf{g}_1}{\mathbf{g}_1 \cdot (\mathbf{g}_2 \times \mathbf{g}_3)}, \quad \mathbf{g}^3 = \frac{\mathbf{g}_1 \times \mathbf{g}_2}{\mathbf{g}_1 \cdot (\mathbf{g}_2 \times \mathbf{g}_3)} \]

The denominator is the scalar triple product, which is non-zero if the basis is linearly independent.

3.3 The Metric Tensor

3.3.1 Definition and Geometric Meaning

The metric tensor (with covariant components) is defined by: \[ g_{ij} = \mathbf{g}_i \cdot \mathbf{g}_j \]

It is symmetric (\(g_{ij} = g_{ji}\)) and positive-definite (all eigenvalues are positive in a well-defined coordinate system).

Geometric meaning: - Diagonal elements \(g_{ii}\) (no sum) represent the squared length of the \(i\)-th basis vector: \(|\mathbf{g}_i|^2 = g_{ii}\). - Off-diagonal elements \(g_{ij}\) (with \(i \neq j\)) encode the angle between \(\mathbf{g}_i\) and \(\mathbf{g}_j\): \(\cos\theta_{ij} = \frac{g_{ij}}{\sqrt{g_{ii}g_{jj}}}\).

The inverse metric tensor \(g^{ij}\) satisfies: \[ g_{ik} g^{kj} = \delta_i^j \]

The arc length (infinitesimal distance) between two nearby points at \(\theta^i\) and \(\theta^i + d\theta^i\) is: \[ (ds)^2 = d\mathbf{r} \cdot d\mathbf{r} = \frac{\partial \mathbf{r}}{\partial \theta^i}d\theta^i \cdot \frac{\partial \mathbf{r}}{\partial \theta^j}d\theta^j = g_{ij}\,d\theta^i\,d\theta^j \]

In Cartesian coordinates, \(g_{ij} = \delta_{ij}\) (unity metric).

Cylindrical coordinates \((r, \phi, z)\) with mapping \(X^1 = r\cos\phi\), \(X^2 = r\sin\phi\), \(X^3 = z\):

Covariant basis vectors: \[ \mathbf{g}_r = \frac{\partial \mathbf{r}}{\partial r} = \cos\phi\,\mathbf{E}_1 + \sin\phi\,\mathbf{E}_2 \] \[ \mathbf{g}_\phi = \frac{\partial \mathbf{r}}{\partial \phi} = -r\sin\phi\,\mathbf{E}_1 + r\cos\phi\,\mathbf{E}_2 \] \[ \mathbf{g}_z = \frac{\partial \mathbf{r}}{\partial z} = \mathbf{E}_3 \]

Metric tensor components: \[ g_{rr} = 1, \quad g_{\phi\phi} = r^2, \quad g_{zz} = 1, \quad g_{r\phi} = g_{r z} = g_{\phi z} = 0 \]

Inverse metric: \[ g^{rr} = 1, \quad g^{\phi\phi} = \frac{1}{r^2}, \quad g^{zz} = 1, \quad g^{r\phi} = g^{rz} = g^{\phi z} = 0 \]

Arc length element: \((ds)^2 = dr^2 + r^2 d\phi^2 + dz^2\) ✓

Volume element: \(dV = \sqrt{\det g_{ij}} \, dr\,d\phi\,dz = r \, dr\,d\phi\,dz\) (the Jacobian).

3.3.2 Spherical Coordinates (Optional Detail)

Spherical coordinates \((R, \theta, \varphi)\): \(X^1 = R\sin\theta\cos\varphi\), \(X^2 = R\sin\theta\sin\varphi\), \(X^3 = R\cos\theta\).

Metric tensor: \[ g_{RR} = 1, \quad g_{\theta\theta} = R^2, \quad g_{\varphi\varphi} = R^2\sin^2\theta, \quad g_{R\theta} = g_{R\varphi} = g_{\theta\varphi} = 0 \]

Volume element: \(dV = R^2 \sin\theta \, dR\,d\theta\,d\varphi\).

3.4 Second-Order Tensors

A second-order tensor \(\mathbf{T}\) is a linear map \(V \to V\). Component representation in a Cartesian basis: \[ \mathbf{T} = T_{ij}\,\mathbf{e}_i \otimes \mathbf{e}_j \]

Key operations:

Operation	Expression
Transpose	\((T^T)_{ij} = T_{ji}\)
Trace	\(\mathrm{tr}\,\mathbf{T} = T_{ii}\)
Determinant	\(\det\mathbf{T}\)
Double contraction	\(\mathbf{A}:\mathbf{B} = A_{ij}B_{ij}\)

Symmetric and Skew-symmetric decomposition: \[ \mathbf{A} = \frac{1}{2}(\mathbf{A} + \mathbf{A}^T) + \frac{1}{2}(\mathbf{A} - \mathbf{A}^T) = \mathbf{sym}(\mathbf{A}) + \mathbf{skew}(\mathbf{A}) \]

3.5 Index Notation and the Summation Convention

Einstein summation: repeated index implies sum over 1, 2, 3. \[ a_i b_i \equiv \sum_{i=1}^{3} a_i b_i \]

Free indices appear once; dummy (summation) indices appear twice.

Examples: \[ (\mathbf{A}\mathbf{b})_i = A_{ij}b_j, \qquad (\mathbf{A}\mathbf{B})_{ij} = A_{ik}B_{kj}, \qquad \mathbf{A}:\mathbf{B} = A_{ij}B_{ij}. \]

The Kronecker delta \(\delta_{ij}\) and Levi-Civita symbol \(\varepsilon_{ijk}\): \[ \delta_{ij} = \begin{cases}1 & i=j \\ 0 & i\neq j\end{cases}, \qquad \varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km}. \]

3.6 Principal Values and Directions

For a symmetric tensor \(\mathbf{A}\), the eigenvalue problem: \[ \mathbf{A}\mathbf{n}^{(\alpha)} = \lambda^{(\alpha)}\mathbf{n}^{(\alpha)}, \qquad \alpha = 1,2,3. \]

The characteristic equation is \[ \det(\mathbf{A} - \lambda\mathbf{I}) = -\lambda^3 + I_1\lambda^2 - I_2\lambda + I_3 = 0, \] where \(I_1, I_2, I_3\) are the principal invariants: \[ I_1 = \mathrm{tr}\,\mathbf{A}, \qquad I_2 = \tfrac{1}{2}[(I_1)^2 - \mathrm{tr}\,\mathbf{A}^2], \qquad I_3 = \det\mathbf{A}. \]

The spectral decomposition: \[ \mathbf{A} = \sum_{\alpha=1}^{3} \lambda^{(\alpha)}\,\mathbf{n}^{(\alpha)}\otimes\mathbf{n}^{(\alpha)}. \]

3.7 Coordinate Transformations

Under an orthogonal transformation \(\mathbf{Q}\) (\(\mathbf{Q}\mathbf{Q}^T = \mathbf{I}\)): \[ \mathbf{v}^* = \mathbf{Q}\mathbf{v}, \qquad \mathbf{T}^* = \mathbf{Q}\mathbf{T}\mathbf{Q}^T. \]

Invariants are unaffected by rotations.

3.8 Covariant and Contravariant Components

Any vector \(\mathbf{u}\) can be written in two equivalent forms using the covariant and reciprocal bases: \[ \mathbf{u} = u^i \mathbf{g}_i = u_i \mathbf{g}^i \]

Contravariant components \(u^i\) (upper indices) are the coefficients when \(\mathbf{u}\) is expanded in the covariant basis \(\{\mathbf{g}_i\}\).

Covariant components \(u_i\) (lower indices) are the coefficients when \(\mathbf{u}\) is expanded in the reciprocal basis \(\{\mathbf{g}^i\}\).

Alternatively, covariant components can be obtained by projection: \[ u_i = \mathbf{u} \cdot \mathbf{g}_i \]

The two representations are related via the metric tensor (raising and lowering indices): \[ u_i = g_{ij} u^j \quad \text{(lower an index)} \quad \text{and} \quad u^i = g^{ij} u_j \quad \text{(raise an index)} \]

3.8.1 Geometric Intuition

In an oblique basis (non-orthonormal), the distinction is clear: - Contravariant components \(u^i\) answer the question: “How much of basis vector \(\mathbf{g}_i\) do I need?” They scale opposite to how basis vectors scale. If we stretch the basis, contravariant components shrink. - Covariant components \(u_i\) are the projections of \(\mathbf{u}\) onto the reciprocal vectors. They scale with the basis vectors.

In Cartesian coordinates where \(\mathbf{g}_i = \mathbf{E}_i\) (orthonormal), the covariant and reciprocal bases are identical, so numerically \(u_i = u^i\) and the distinction disappears. This is why introductory mechanics courses often ignore index placement!

Worked example (2D oblique basis): Let \(\mathbf{g}_1 = (1, 0)\) and \(\mathbf{g}_2 = (\cos\alpha, \sin\alpha)\) (two basis vectors at angle \(\alpha\)).

Metric tensor: \(g_{11} = 1\), \(g_{22} = 1\), \(g_{12} = \cos\alpha\).

Reciprocal basis (by formula, or by solving \(\mathbf{g}^i \cdot \mathbf{g}_j = \delta^i_j\)): \[ \mathbf{g}^1 = (1, -\cot\alpha), \quad \mathbf{g}^2 = (0, \csc\alpha) \]

Consider the vector \(\mathbf{u} = (1, 0)\) (horizontal unit vector in Cartesian space).

Contravariant components: Solve \(\mathbf{u} = u^1 \mathbf{g}_1 + u^2 \mathbf{g}_2 = u^1(1,0) + u^2(\cos\alpha, \sin\alpha)\). - From \(1 = u^1 + u^2\cos\alpha\) and \(0 = u^2\sin\alpha\), we get \(u^2 = 0\) and \(u^1 = 1\).

Covariant components: \(u_i = \mathbf{u} \cdot \mathbf{g}_i\). - \(u_1 = (1,0) \cdot (1,0) = 1\). - \(u_2 = (1,0) \cdot (\cos\alpha, \sin\alpha) = \cos\alpha\).

Verify via metric: \(u_i = g_{ij}u^j\) gives \(u_1 = g_{11} \cdot 1 = 1\) ✓ and \(u_2 = g_{21} \cdot 1 = \cos\alpha\) ✓.

In this oblique system, the covariant representation \(\mathbf{u} = 1 \cdot \mathbf{g}^1 + \cos\alpha \cdot \mathbf{g}^2\) is not as intuitive as the contravariant one; both are valid and equivalent.

3.9 Tensor Contractions

Outer Product (⊗): \[ (\mathbf{a} \otimes \mathbf{b})_{ij} = a_i b_j \] produces a second-order tensor (no summation).

Single Contraction (·): \[ (\mathbf{A} \cdot \mathbf{v})_i = A_{ij} v^j \] contracts one upper and one lower index, reducing order by 1.

Double Contraction (:): \[ \mathbf{A} : \mathbf{B} = A_{ij} B^{ij} \] contracts two pairs of indices, producing a scalar result.

In curvilinear coordinates, use metric to align indices: \[ \mathbf{u} \cdot \mathbf{v} = u^i v_i = u_i v^i = g_{ij} u^i v^j \]

3.10 The Levi-Civita Symbol and Cross Product

3.10.1 Levi-Civita Permutation Symbol

The Levi-Civita symbol (or permutation symbol) \(\varepsilon_{ijk}\) in 3D is defined as: \[ \varepsilon_{ijk} = \begin{cases} +1 & \text{if } (i,j,k) \text{ is an even permutation of } (1,2,3) \\ -1 & \text{if } (i,j,k) \text{ is an odd permutation of } (1,2,3) \\ 0 & \text{if any two indices are equal} \end{cases} \]

Examples: \(\varepsilon_{123} = +1\), \(\varepsilon_{231} = +1\), \(\varepsilon_{312} = +1\) (even), \(\varepsilon_{213} = -1\), \(\varepsilon_{132} = -1\), \(\varepsilon_{321} = -1\) (odd), \(\varepsilon_{112} = 0\).

Relation to the determinant: For a \(3 \times 3\) matrix with entries \(A_{ij}\), \[ \det \mathbf{A} = \varepsilon_{ijk} A_{1i} A_{2j} A_{3k} \] (summing over all repeated indices).

3.10.2 Cross Product in Index and Direct Notation

The cross product \(\mathbf{u} \times \mathbf{v}\) is a vector perpendicular to both \(\mathbf{u}\) and \(\mathbf{v}\).

Index notation: \[ (\mathbf{u} \times \mathbf{v})_k = \varepsilon_{kij} u_i v_j \]

Geometric interpretation: - Magnitude: \(|\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}| \sin\theta\), where \(\theta \in [0, \pi]\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\). - Direction: By the right-hand rule: point fingers along \(\mathbf{u}\), curl them towards \(\mathbf{v}\), thumb points in direction of \(\mathbf{u} \times \mathbf{v}\). - Geometric meaning: The magnitude equals the area of the parallelogram spanned by \(\mathbf{u}\) and \(\mathbf{v}\). (The area of the triangle is half this.)

Properties: - Anti-commutativity: \(\mathbf{u} \times \mathbf{v} = -\mathbf{v} \times \mathbf{u}\) - Linearity: \((\alpha\mathbf{u} + \beta\mathbf{v}) \times \mathbf{w} = \alpha(\mathbf{u} \times \mathbf{w}) + \beta(\mathbf{v} \times \mathbf{w})\) - Orthogonality: \(\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v}) = 0\) and \(\mathbf{v} \cdot (\mathbf{u} \times \mathbf{v}) = 0\) - Magnitude identity: \(|\mathbf{u} \times \mathbf{v}|^2 = |\mathbf{u}|^2 |\mathbf{v}|^2 - (\mathbf{u} \cdot \mathbf{v})^2\) (follows from Cauchy-Schwarz)

3.10.3 Scalar Triple Product

The scalar triple product is the scalar: \[ [\mathbf{u}, \mathbf{v}, \mathbf{w}] = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) \]

Index notation: \[ [\mathbf{u}, \mathbf{v}, \mathbf{w}] = \varepsilon_{ijk} u_i v_j w_k \]

Geometric meaning: The absolute value \(|[\mathbf{u}, \mathbf{v}, \mathbf{w}]|\) equals the signed volume of the parallelepiped (a 3D “slanted box”) spanned by the three vectors. It is positive if \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) form a right-handed system, negative if left-handed, and zero if they are coplanar.

3.10.4 The Epsilon-Delta Identity and BAC-CAB Rule

A fundamental identity relating the Levi-Civita symbol to the Kronecker delta is: \[ \varepsilon_{ijk}\varepsilon_{imn} = \delta_{jm}\delta_{kn} - \delta_{jn}\delta_{km} \]

This identity is used to prove the BAC-CAB rule (also called vector triple product rule): \[ \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = \mathbf{v}(\mathbf{u} \cdot \mathbf{w}) - \mathbf{w}(\mathbf{u} \cdot \mathbf{v}) \]

Proof (index notation): Component \(i\) of LHS is \[ [\mathbf{u} \times (\mathbf{v} \times \mathbf{w})]_i = \varepsilon_{ijk} u_j (\mathbf{v} \times \mathbf{w})_k = \varepsilon_{ijk} u_j (\varepsilon_{klm} v_l w_m) = \varepsilon_{ijk}\varepsilon_{klm} u_j v_l w_m. \] Using the epsilon-delta identity and relabeling, this simplifies to the RHS.

Worked example: Let \(\mathbf{u} = (1, 0, 0)\), \(\mathbf{v} = (0, 1, 0)\), \(\mathbf{w} = (0, 0, 1)\) (standard basis).

Cross product: \(\mathbf{v} \times \mathbf{w} = (1, 0, 0) = \mathbf{u}\).

Scalar triple product: \([\mathbf{u}, \mathbf{v}, \mathbf{w}] = \mathbf{u} \cdot \mathbf{u} = 1\) (volume of unit cube).

BAC-CAB check: \(\mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = \mathbf{u} \times \mathbf{u} = \mathbf{0}\). By BAC-CAB, \(\mathbf{v}(\mathbf{u} \cdot \mathbf{w}) - \mathbf{w}(\mathbf{u} \cdot \mathbf{v}) = \mathbf{v} \cdot 0 - \mathbf{w} \cdot 0 = \mathbf{0}\) ✓.

3.10.5 General Permutation Symbol in Curvilinear Coordinates

In curvilinear coordinates with metric tensor \(g_{ij}\), the permutation symbol must be scaled by \(\sqrt{\det g_{ij}}\) to account for the geometry: \[ \varepsilon_{ijk} = \sqrt{g} \, e_{ijk} \] where \(g = \det g_{ij}\) and \(e_{ijk}\) is the standard Cartesian Levi-Civita symbol. This ensures that the cross product formula and scalar triple product remain invariant (geometrically meaningful) under coordinate transformations.

3.11 Tensor Calculus: Gradient, Divergence, Curl

Gradient of a scalar \(\phi(\mathbf{x})\): \[ (\nabla\phi)_i = \frac{\partial\phi}{\partial x^i} \quad \text{(covariant vector)} \]

Gradient of a vector \(\mathbf{v}(\mathbf{x})\): \[ (\nabla\mathbf{v})_{ij} = \frac{\partial v_i}{\partial x^j} \quad \text{(second-order tensor)} \]

Divergence of a vector: \[ \nabla \cdot \mathbf{v} = \frac{\partial v^i}{\partial x^i} \quad \text{(scalar)} \]

Divergence of a tensor: \[ (\nabla \cdot \mathbf{T})_i = \frac{\partial T_{ij}}{\partial x^j} \quad \text{(vector)} \]

Curl of a vector: \[ (\nabla \times \mathbf{v})_k = \varepsilon_{kij} \frac{\partial v_i}{\partial x^j} \quad \text{(axial vector/pseudovector)} \]

In Cartesian coordinates, these simplifications hold. In curvilinear coordinates, Christoffel symbols modify the derivatives for covariance.

3.12 Tensor Invariants

For a second-order tensor \(\mathbf{A}\), the principal invariants are:

\[ I_1(\mathbf{A}) = \text{tr}(\mathbf{A}) = A_{ii} \]

\[ I_2(\mathbf{A}) = \frac{1}{2}\left[\text{tr}(\mathbf{A})^2 - \text{tr}(\mathbf{A}^2)\right] = \frac{1}{2}(A_{ii}A_{jj} - A_{ij}A_{ji}) \]

\[ I_3(\mathbf{A}) = \det(\mathbf{A}) \]

These invariants remain unchanged under orthogonal coordinate transformations.

Deviatoric part: \(\mathbf{A}' = \mathbf{A} - \frac{1}{3}\text{tr}(\mathbf{A})\mathbf{I}\)

Deviatoric invariants (e.g., \(J_2 = \frac{1}{2}\mathbf{A}' : \mathbf{A}'\)) are also invariant and commonly used in yield criteria.

3.13 Tensor Functions and Eigenvalue Decomposition

For a symmetric tensor \(\mathbf{A}\) with spectral decomposition: \[ \mathbf{A} = \sum_{\alpha=1}^{3} \lambda_\alpha \mathbf{n}_\alpha \otimes \mathbf{n}_\alpha \]

Any analytic tensor function \(f(\mathbf{A})\) can be computed as: \[ f(\mathbf{A}) = \sum_{\alpha=1}^{3} f(\lambda_\alpha) \mathbf{n}_\alpha \otimes \mathbf{n}_\alpha \]

Examples: \[ \mathbf{A}^2 = \sum_{\alpha=1}^{3} \lambda_\alpha^2 \mathbf{n}_\alpha \otimes \mathbf{n}_\alpha, \quad \sqrt{\mathbf{A}} = \sum_{\alpha=1}^{3} \sqrt{\lambda_\alpha} \mathbf{n}_\alpha \otimes \mathbf{n}_\alpha, \quad \ln(\mathbf{A}) = \sum_{\alpha=1}^{3} \ln(\lambda_\alpha) \mathbf{n}_\alpha \otimes \mathbf{n}_\alpha \]

This spectral approach is widely used in constitutive models for hyperelastic materials.

3.14 Christoffel Symbols and Covariant Differentiation

3.14.1 Motivation: Why Naive Derivatives Fail in Curvilinear Coordinates

In Cartesian coordinates, the partial derivative of a vector is straightforward: only the components change, not the basis vectors. But in curvilinear coordinates, when we move from one point to a nearby point in the coordinate grid, the basis vectors themselves change. A naive partial derivative thus mixes changes in components with changes in basis, and the result is not a tensor.

Consider a vector field \(\mathbf{u}(\theta^i) = u^j(\theta^i) \mathbf{g}_j(\theta^i)\). Taking the derivative: \[ \frac{\partial \mathbf{u}}{\partial \theta^i} = \frac{\partial u^j}{\partial \theta^i} \mathbf{g}_j + u^j \frac{\partial \mathbf{g}_j}{\partial \theta^i} \]

The second term accounts for basis change. The Christoffel symbol quantifies this change.

3.14.2 Christoffel Symbol (Symbol of the Second Kind)

The Christoffel symbol of the second kind \(\Gamma^k_{ij}\) is defined by: \[ \frac{\partial \mathbf{g}_j}{\partial \theta^i} = \Gamma^k_{ij} \mathbf{g}_k \]

Explicit formula (in terms of the metric tensor): \[ \Gamma^k_{ij} = \frac{1}{2} g^{km} \left( \frac{\partial g_{mj}}{\partial \theta^i} + \frac{\partial g_{im}}{\partial \theta^j} - \frac{\partial g_{ij}}{\partial \theta^m} \right) \]

Key property: The Christoffel symbol is symmetric in the lower two indices: \[ \Gamma^k_{ij} = \Gamma^k_{ji} \]

Important note: The Christoffel symbol is not a tensor. It vanishes in Cartesian coordinates (where basis vectors are constant), but is non-zero in general curvilinear systems.

Cylindrical coordinates \((r, \phi, z)\): Metric is \(g_{rr} = 1\), \(g_{\phi\phi} = r^2\), \(g_{zz} = 1\), all others zero.

Non-zero Christoffel symbols: \[ \Gamma^r_{\phi\phi} = -r, \quad \Gamma^\phi_{r\phi} = \Gamma^\phi_{\phi r} = \frac{1}{r}, \quad \text{all others} = 0 \]

Physical meaning: \(\Gamma^r_{\phi\phi} = -r\) says that when you move in the \(\phi\) direction twice, the radial basis vector shrinks, which makes sense geometrically (the coordinate grid circles get smaller as you go outward).

3.14.3 Covariant Derivative of a Vector

To obtain a tensor derivative, we define the covariant derivative of a contravariant vector \(u^i\): \[ u^i_{;j} = \frac{\partial u^i}{\partial \theta^j} + \Gamma^i_{jk} u^k \]

Similarly, for a covariant vector \(u_i\): \[ u_{i;j} = \frac{\partial u_i}{\partial \theta^j} - \Gamma^k_{ij} u_k \]

The semi-colon notation \((;\,)\) denotes covariant differentiation. The result is a tensor (it transforms like a tensor under coordinate changes), unlike the naive partial derivative.

3.14.4 Covariant Derivative of a Tensor

For a general second-order tensor \(\mathbf{A} = A^{kl} \mathbf{g}_k \otimes \mathbf{g}_l\): \[ A^{kl}_{;m} = \frac{\partial A^{kl}}{\partial \theta^m} + \Gamma^k_{mn} A^{nl} + \Gamma^l_{mn} A^{kn} \]

The pattern: add Christoffel-symbol corrections for each upper index, subtract for each lower index.

3.14.5 Divergence in Curvilinear Coordinates

A direct application: the divergence of a vector field is: \[ \nabla \cdot \mathbf{u} = u^i_{;i} = \frac{\partial u^i}{\partial \theta^i} + \Gamma^i_{ik} u^k = \frac{1}{\sqrt{g}}\frac{\partial}{\partial \theta^i}\left(\sqrt{g}\,u^i\right) \]

where \(g = \det g_{ij}\) is the metric determinant.

Note

Physical interpretation: Covariant derivatives encode the fact that in a curved or non-orthonormal coordinate system, “parallel transport” of a vector (moving it while keeping it fixed in the local frame) requires a compensation term. In finite-strain mechanics, material coordinates deform with the material, and these coordinate curves become non-orthogonal. The Christoffel symbols then represent the geometric effect of the deformation on how we measure spatial derivatives.

3.14.6 Tensor Character

The covariant derivatives are components of tensors under coordinate transformations. This is why Grad, Div, and Curl are valid tensor operations: they produce results that transform as tensors, making them coordinate-independent and physically meaningful.

3.15 Derivative of a Tensor with Respect to a Tensor

For \(\mathbf{F} = \mathbf{F}(\mathbf{G})\) where both are second-order tensors:

\[ \mathbb{H} = \frac{\partial \mathbf{F}}{\partial \mathbf{G}}, \quad H_{klij} = \frac{\partial F_{kl}}{\partial G_{ij}} \]

This is a fourth-order tensor with 81 components in 3D.

Useful identities: \[ \frac{\partial \mathbf{A}}{\partial \mathbf{A}} = \mathbb{I}, \quad \frac{\partial \text{tr}(\mathbf{A})}{\partial \mathbf{A}} = \mathbf{I}, \quad \frac{\partial \det(\mathbf{A})}{\partial \mathbf{A}} = (\det \mathbf{A})\mathbf{A}^{-T} \]

These derivatives are essential when deriving stress-strain relationships in constitutive theory.