Vector and Tensor Calculus
Vector and Tensor Calculus
This chapter summarizes the vector and tensor calculus notation used throughout the course.
Derivatives
Given a function \(f(x,y)\), we write the partial derivative with respect to \(x\) as \[\frac{\partial f}{\partial x} = \partial_x f = f_{,x}.\] All variables following a comma in a subscript are derivatives.
The second derivative with respect to \(x\) is \[\frac{\partial ^2 f}{\partial x^2} = \partial ^2_x f = f_{,xx}.\]
Mixed derivatives are written as \[\frac{\partial ^2 f}{\partial x \partial y} = \partial_x\partial_y f = f_{,xy}.\]
The total derivative is indicated with the letter \(\mathrm{d}\), e.g. \[\frac{\mathrm{d}f}{\mathrm{d}t} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial t} =f_{,x}x_{,t}+f_{,y}y_{,t}\] for \(f=f(x,y)\), \(x=x(t)\) and \(y=y(t)\).
Alternative notations
- The prime is used to indicate a derivative, e.g. \(f'(x)=\mathrm{d}f/\mathrm{d}x\) is the derivative of \(f\).
- It is common to indicate the derivative with respect to time by a dot, i.e. given \(f(t)\) the derivative \(\dot{f}(t)=\mathrm{d}f/\mathrm{d}t\).
- Higher order derivatives are sometimes indicated by the order in parentheses, e.g. \(f^{(n)}(x)=\mathrm{d}^n f/\mathrm{d}x^n\) – for \(n=2\) giving \(f^{\prime \prime }(x)=f^{(2)}(x)=\mathrm{d}^2 f/\mathrm{d}x^2\).
Writing the differential operator explicitly is less ambiguous. In particular, for functions of more than one variable it allows us to distinguish clearly between total and partial derivatives.
Einstein summation convention
Einstein summation is an implicit summation over repeated indices (i.e. a summation where the \(\sum\)-sign is omitted). As a simple example, consider the total derivative of a function \(f(\vec{r})\) with \(\vec{r}=(x,y)\). We can write this as \[\frac{\mathrm{d}f}{\mathrm{d}t} = f_{,x}r_{x,t}+f_{,y}r_{y,t} = \sum_{i=x,y} f_{,i} r_{i,t} = f_{,i} r_{i,t}\] where in the right hand side the sum is implicit because the index \(i\) is repeated.
Examples of Einstein summation
- Divergence of stress tensor: \(\nabla \cdot \underline{\sigma} = \partial_j \sigma_{ij} = \sum_j \partial_j \sigma_{ij}\)
- Hydrostatic stress: \(3\sigma_h = \sigma_{kk} = \sum_k \sigma_{kk} = \text{tr}\,\underline{\sigma}\)
- Traction vector: \(t_i = \sigma_{ij} n_j = \sum_j \sigma_{ij} n_j\)
Tensor notation
We use explicit arrows, \(\vec{v}\), to indicate first-order tensors (vectors) and underline second-order tensors (matrices), \(\underline{M}\). A fourth-order tensor is underlined twice, \(\underline{\underline{C}}\). Unit vectors (vectors of length one) are denoted by a hat, \(\hat{n}\).
| Order | Symbol | Example |
|---|---|---|
| Scalar | Regular letter | \(f\), \(E\), \(\nu\) |
| Vector (1st order tensor) | Arrow | \(\vec{u}\), \(\vec{F}\), \(\vec{r}\) |
| 2nd order tensor (matrix) | Single underline | \(\underline{\sigma}\), \(\underline{\varepsilon}\), \(\underline{R}\) |
| 4th order tensor | Double underline | \(\underline{\underline{C}}\) |
| Unit vector | Hat | \(\hat{n}\), \(\hat{e}_x\) |
Note that we chose this notation over using, e.g., bold font to indicate vectors because it is blackboard friendly. It can be used on blackboards and typeset notes alike.
Nabla operator
In an \(n\)-dimensional Euclidean space equipped with Cartesian coordinates, the nabla operator reads \[\nabla =\left (\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\dots ,\frac{\partial }{\partial x_n} \right )=\sum_{i=1}^{n}\hat{e}_i\frac{\partial }{\partial x_i}\] where \(\hat{e}_i\) is the unit vector pointing in the \(i\)-th Cartesian direction.
Common operations:
- Gradient of a scalar field: \(\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)\)
- Divergence of a vector field: \(\nabla \cdot \vec{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}\)
- Laplacian: \(\nabla^2 f = \nabla \cdot (\nabla f)\)
When applied to a second-rank tensor, our divergence operator “acts on the right”, i.e. \(\nabla \cdot \underline{\sigma} = \sigma_{ij,j}\) – the derivative is taken with respect to the second index. This is entirely a convention and you may encounter textbooks or publications where a different convention is applied.
Divergence theorem
The divergence theorem is an important result of vector analysis. It converts an integral over a volume \(V\) into an integral over the surface \(S\) of this volume. For a vector field \(\vec{f}(\vec{r})\), the divergence theorem states that \[\int_V \mathrm{d}^3 r\, \nabla \cdot \vec{f}(\vec{r}) = \int_{S} \mathrm{d}^2 r\, \vec{f}(\vec{r}) \cdot \hat{n}(\vec{r})\] where \(\hat{n}(\vec{r})\) is the normal vector pointing outward on the surface \(S\) of the volume \(V\).
Inner and outer products
An inner product (scalar product) is expressed by a point \(\cdot\), as in \(\vec{a}\cdot\vec{b}\). The expression \(\vec{a}\vec{b}\) is not a scalar product but the outer product. We recommend using a specific symbol for the outer product to avoid ambiguity, e.g. \(\vec{a}\otimes\vec{b}\).
Each point (in the scalar product or the double contraction \(\underline{A}:\underline{B}\)) represents a sum.