Chapter 1 Preliminary Concepts

Last updated: 26 Apr 2026

1.2 Vector and Tensor calculus

1.2.1 Vector and tensor

Cartesian vectors: ~

Cartesian tensors: \(\boldsymbol{T} = T_{ij} \boldsymbol{e}_i \otimes \boldsymbol{e}_j\), where \(\otimes\) is the dyadic product that increases the order of the tensor by one. It should be noted that \(\boldsymbol{u}\otimes\boldsymbol{v} \neq \boldsymbol{v}\otimes\boldsymbol{u}\).

Several properties of the dyadic product are as follows: 1. \(a\boldsymbol{u}\otimes\boldsymbol{v} = \boldsymbol{u}\otimes a\boldsymbol{v} = a(\boldsymbol{u}\otimes\boldsymbol{v})\), where \(a\) is a scalar. 2. \(\boldsymbol{u}\otimes(\boldsymbol{v}+\boldsymbol{w}) = \boldsymbol{u}\otimes\boldsymbol{v} + \boldsymbol{u}\otimes\boldsymbol{w}\) 3. \((\boldsymbol{u}\otimes\boldsymbol{v})\cdot\boldsymbol{w} = \boldsymbol{u}(\boldsymbol{v}\cdot\boldsymbol{w})\), which means the inner product is applied to the adjacent vectors. For rank-\(m\) and rank-\(n\) tensors, the inner product yields a rank-\((m+n-2)\) tensor.

The transpose of a tensor is defined as \(\boldsymbol{T}^\top = T_{ji} \boldsymbol{e}_i \otimes \boldsymbol{e}_j\).

Symmetric tensors: \(\boldsymbol{S} = \boldsymbol{S}^\top\) and Skew tensors: \(\boldsymbol{W} = -\boldsymbol{W}^\top\). Every rank-2 tensor can be decomposed into a symmetric part and a skew part, i.e., \(\boldsymbol{T} = \boldsymbol{S} + \boldsymbol{W}\), where \(\boldsymbol{S} = (\boldsymbol{T} + \boldsymbol{T}^\top)/2\) and \(\boldsymbol{W} = (\boldsymbol{T} - \boldsymbol{T}^\top)/2\).

The symmetric part of displacement gradient\(\nabla\boldsymbol{u}=\partial\boldsymbol{u}/\partial\boldsymbol{x}\) is the strain tensor, and the skew part is the spin tensor.

The contraction operator or double inner product of two rank-2 tensors is defined as \(\boldsymbol{A}:\boldsymbol{B} = A_{ij}B_{ij}\), yielding a scalar. It can also be used to define the norm of a rank-2 tensor as \(\|\boldsymbol{A}\| = \sqrt{\boldsymbol{A}:\boldsymbol{A}}\).

Several properties of the contraction operator are as follows: 1. \(\boldsymbol{A}:\boldsymbol{B} = \boldsymbol{B}:\boldsymbol{A}\) 2. \(\boldsymbol{A}:(\boldsymbol{B}+\boldsymbol{C}) = \boldsymbol{A}:\boldsymbol{B} + \boldsymbol{A}:\boldsymbol{C}\) 3. \(\boldsymbol{A}:(\boldsymbol{B}\boldsymbol{C}) = (\boldsymbol{B}^\top\boldsymbol{A}):\boldsymbol{C} = (\boldsymbol{A}\boldsymbol{C}^\top):\boldsymbol{B}\)

The trace of a rank-2 tensor is defined as \(\text{tr}(\boldsymbol{A}) = \boldsymbol{A}:\boldsymbol{I} = \boldsymbol{I}:\boldsymbol{A} = A_{ii}\).

The orthogonal tensor represents the rotational relation between two coordinate systems: \(\boldsymbol{u}^* = \boldsymbol{\beta}\cdot\boldsymbol{u}\) and \(\boldsymbol{u} = \boldsymbol{\beta}^\top\cdot\boldsymbol{u}^*\), where the component of \(\boldsymbol{\beta}\) is \(\beta_{ij} = \boldsymbol{e}_i^* \cdot \boldsymbol{e}_j\). We also have \(\boldsymbol{\beta}^\top\cdot\boldsymbol{\beta} = \boldsymbol{\beta}\cdot\boldsymbol{\beta}^\top = \boldsymbol{I}\). The coordinate transformation of a rank-2 tensor is given by \(\boldsymbol{T}^* = \boldsymbol{\beta}\boldsymbol{T}\boldsymbol{\beta}^\top\), with \(T_{ij}^* = \beta_{ik}T_{kl}\beta_{jl}\).

The permutation symbol \(e_{ijk}\) is defined as follows: \(e_{ijk} = \begin{cases} 1 & \text{if } (i,j,k) \text{ is an even permutation of } (1,2,3) \\ -1 & \text{if } (i,j,k) \text{ is an odd permutation of } (1,2,3) \\ 0 & \text{otherwise} \end{cases}\) which can define the vector product as \(\boldsymbol{u}\times\boldsymbol{v} = e_{ijk}u_jv_k\boldsymbol{e}_i\).

An important property: \(e_{ijk}e_{lmk} = \delta_{il}\delta_{jm} - \delta_{im}\delta_{jl}\).

The dual vector of a skew tensor \(\boldsymbol{W}\) is defined as \(\boldsymbol{w} = -\frac{1}{2}e_{ijk}W_{jk}\boldsymbol{e}_i\), and the skew tensor can be expressed in terms of its dual vector as \(W_{ij} = e_{ijk}w_k\). Then we can express the vector product as \(\boldsymbol{W}\boldsymbol{u} = \boldsymbol{w}\times\boldsymbol{u}\).

1.2.2 Vector and tensor calculus

The gradient operator is defined as a vector

\[ \nabla = \frac{\partial}{\partial \boldsymbol{x}}=\boldsymbol{e}_i\frac{\partial}{\partial x_i} \]

For a vector field \(\boldsymbol{u}(\boldsymbol{x})\), the gradient is a rank-2 tensor defined as

\[ \nabla\boldsymbol{u} = \boldsymbol{u}\otimes\nabla = \frac{\partial u_i}{\partial x_j}\boldsymbol{e}_i\otimes\boldsymbol{e}_j = u_{i,j}\boldsymbol{e}_i\otimes\boldsymbol{e}_j \]

The divergence is given by

\[ \nabla\cdot\boldsymbol{u} = \text{tr}(\nabla\boldsymbol{u}) = \frac{\partial u_i}{\partial x_i} \]

The Laplace operator is defined as

\[ \nabla^2 = \nabla\cdot\nabla = \frac{\partial^2}{\partial x_i \partial x_i} \]

1.2.3 Integral theorems

The divergence theorem states that if a tensor \(\boldsymbol{A}\) is continuously differentiable in \(\Omega\), then:

\[ \iint_\Omega \nabla\cdot\boldsymbol{A} \, d\Omega = \int_{\Gamma} \boldsymbol{n}\cdot\boldsymbol{A} \, d\Gamma \]

where \(\Omega\) is a domain bounded by \(\Gamma\) with outward normal \(\boldsymbol{n}\). A variant is the gradient theorem, which states:

\[ \iint_\Omega \nabla\boldsymbol{A} \, d\Omega = \int_{\Gamma} \boldsymbol{n}\otimes\boldsymbol{A} \, d\Gamma \]

The Reynolds transport theorem states that for \(\boldsymbol{f}(\boldsymbol{x},t)\), we have:

\[ \frac{\rm d}{{\rm d}t}\iint_{\Omega} \boldsymbol{f} \, d\Omega = \iint_{\Omega} \frac{\partial \boldsymbol{f}}{\partial t} \, d\Omega + \int_{\Gamma} (\boldsymbol{n}\cdot\boldsymbol{v})\boldsymbol{f} \, d\Gamma \]

where \(\boldsymbol{v}\) is the velocity of the boundary \(\Gamma\) and the second term on the RHS is the convection term.

The integration by parts states that

\[ \iint_\Omega \nabla u\cdot\boldsymbol{v} \, d\Omega = \int_{\Gamma} u(\boldsymbol{v}\cdot\boldsymbol{n}) \, d\Gamma - \iint_\Omega u(\nabla\cdot\boldsymbol{v}) \, d\Omega \]

If we replace \(\boldsymbol{v}\) with \(\nabla v\), we have the following Green's identity:

\[ \iint_\Omega \nabla u\cdot\nabla v \, d\Omega = \int_{\Gamma} u(\nabla v\cdot\boldsymbol{n}) \, d\Gamma - \iint_\Omega u\nabla^2 v \, d\Omega \]