MTH3008 Lecture 3

Tom Ward

nothing, he wasn’t there :(

We’ll now use that Suffix Notation throughout tensor analysis, along with the definitions:

• A scalar field is a map that assigns a real number to every point in space: $\varphi :{\mathbb{R}}^n\to \mathbb{R}$.
• A vector field is a map that assigns a vector to every point in space: $\varphi :{\mathbb{R}}^n\to {\mathbb{R}}^n$.
• And note that we’ll relabel the Cartesian coordinate system $(x,y,z)$ as $(x_1,x_2,x_3)$.

Differential Operators

We’ll consider three differential operators in these fields, each of which can be expressed in suffix notation:

1. The gradient (multiplied with $\nabla$),
2. The divergence (dot product with $\nabla$), and
3. The curl (cross product with $\nabla$).

First, the gradient of a scalar field is $\nabla f=\left(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\frac{\partial f}{\partial x_3}\right)$, where the $i$-th component of the gradient is the partial derivative with respect to $x_i$, hence in suffix notation is $[\nabla f{]}_i=\frac{\partial f}{\partial x_i}$. Practically, to find these gradients then you simply calculate each component, often then specifying a value at a specific point.

Example

Find the gradient $\nabla \varphi$ of $\varphi (x_1,x_2,x_3)=\frac{1}{2}{x}_2^2{x}_3^3-3x_1{x}_2+x_2^5{x}_3+1$ at the point $P=(3,1,0)$…

First, the general gradient is $\nabla \varphi =\left(-3x_2,x_2{x}_3^3-3x_1+6x_2^4{x}_3,\frac{3}{2}{x}_2^2{x}_3^2+x_2^5\right)$, by finding the partial derivatives for each component. Then, at the point $P$, we find that $\nabla \varphi {|}_P=(-3,-9,1)=\boxed{-3\mathbf{i}-9\mathbf{j}+\mathbf{k}}$.

Second, the divergence of a vector field $\mathbf{u}$ is $\nabla \cdot \mathbf{u}=\frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}+\frac{\partial u_3}{\partial x_3}=\frac{\partial u_j}{\partial x_j}={\nabla }_i{u}_i$. Note the dummy index $j$ for the summation $j=1,2,3$, and the complete lack of free indices (as the divergence of a vector is a scalar quantity). If positive, it indicates a source, if negative, it indicates a sink.

Example

Find $\nabla \cdot \mathbf{A}$ at the point $Q=(2,1,2)$ for $\mathbf{A}=\frac{1}{2}{x}_1^3{x}_2\mathbf{i}-(4x_1{x}_2^5+1)\mathbf{j}+x_2{x}_3^3\mathbf{k}$…

$\nabla \cdot \mathbf{A}=\frac{3}{2}{x}_1^2{x}_2+20x_1{x}_2^4+3x_2{x}_3^2:\nabla \cdot \mathbf{A}{|}_{(2,1,2)}=6+40+12=\boxed{58}$.

Third, the curl of a vector field $\mathbf{u}$ is…

$\nabla \times \mathbf{u}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\partial }{\partial x_1}& \frac{\partial }{\partial x_2}& \frac{\partial }{\partial x_3}\\ u_1& u_2& u_3\end{array}\right|=\left(\frac{\partial u_3}{\partial x_2}-\frac{\partial u_2}{\partial x_3},\frac{\partial u_1}{\partial x_3}-\frac{\partial u_3}{\partial x_1},\frac{\partial u_2}{\partial x_1}-\frac{\partial u_1}{\partial x_2}\right)$.

Or, alternatively using the alternating tensor to represent the curl, each component is then $[\nabla \times \mathbf{u}{]}_i={ϵ}_{ijk}{\nabla }_j{u}_k={ϵ}_{ijk}\frac{\partial u_k}{\partial x_j}$. Here, we can see the two dummy indices $j$ and $k$ indicating a double sum, and the free index $i$ indicating that the result is a vector quantity.

Example

…

${\nabla }_j$ is just the partial derivative of the $j$-th component (of the following function).

---

Pre-Lecture Notes from University Notes

• First, recap the last lecture:
  • Kronecker Delta (definition, properties).
  • Alternating Tensor (definition, properties, relations) - plus, some practice.
• The Levi-Civita Symbol - a more generalised version of the Alternating Tensor, defined as ${ϵ}_{i_1{i}_2{i}_3\dots i_n}$ where if any two indices are interchanges then the symbol is negated, and if any are equal then the symbol equals zero. Hence, the ${ϵ}_{ijk}$ is just the Levi-Civita symbol in 3D space.
• This $ϵ$ relates to $\delta$ with: ${ϵ}_{ijk}{ϵ}_{k\ell m}={\delta }_{i\ell }{\delta }_{jm}-{\delta }_{im}{\delta }_{j\ell }$, where there are four free indices ($i$, $j$, $\ell$, $m$) and the repeated dummy index $k$; simply comes from observation.
  • Can be used to show even more relationships and simply even more expressions!
• Vector differential operators…
  • Scalar/vector field definition
  • Three differential operators: gradient, divergence, and curl - definitions in suffix notation and brief explanations of intuition for understanding.