MTH3008 Lecture 4

Recall previous definitions…

• Kronecker Delta,
• Alternating Tensor,
• Suffix Notation,
• Gradient,
• Divergence,
• Curl.

Then, we can start to consider combinations of these operators of the position vector, $\mathbf{r}=(x_1,x_2,x_3)$ with magnitude $\underline{r}=|\mathbf{r}|$.

Derivatives of the Position Vector

First, notice that the $\frac{\partial x_1}{\partial x_1}=1$ but $\frac{\partial x_1}{\partial x_2}=0$, so in general we have $\frac{\partial x_i}{\partial x_j}=\{\begin{array}{ll}1& \text{if }i=j\\ 0& \text{if }i\ne j\end{array}={\delta }_{ij}$.

Using this, and writing everything in index notation, we can derive each thing following their definitions to find…

• Gradient of vector position, $\nabla \underline{r}={\left[\frac{\mathbf{r}}{\underline{r}}\right]}_i$.
• Divergence of vector position, $\nabla \cdot \mathbf{r}=\frac{\partial r_i}{\partial x_i}=\frac{\partial x_i}{\partial x_i}={\delta }_{ii}=3$, by our convention.
• Curl of vector position, $(0{)}_i$ based on the definition of the Alternating Tensor.

Combinations of Grad, Div, and Curl

check this

There are only five combinations that we can have, due to domain/range mismatches:

1. Div grad, ${\nabla }^{2}f$ (the Laplacian operator).
2. Curl grad, $0$.
3. Grad div, $\frac{{\partial }^2{u}_j}{\partial x_i\partial x_j}$.
4. Div curl, $0$.
5. Curl curl, $\nabla (\nabla \cdot \mathbf{u})-\nabla \times (\nabla \times \mathbf{u})$.

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Pre-Lecture Notes from University Notes

• Recall past definitions:
  • Kronecker Delta.
  • Alternating Tensor.
  • Suffix Notation.
  • Gradient.
  • Divergence.
  • Curl.
• Then, just further definitions, like the position vector and its magnitude (same as normal), then exploring its relationships with the aforementioned definitions.
• Combining differential operators to get things like div grad, curl grade, grad div, div curl, and curl, finding them all in suffix notation.
  • The latter then allowing us to have a new description of the Laplacian operator.