Definition. A symmetric/antisymmetric dyadic is defined the same way that a matrix is.
Definition. The identity dyadic is \(\stackrel{\leftrightarrow}{\mathbf{I}} = \hat{\mathbf{x}}\hat{\mathbf{x}} + \hat{\mathbf{y}}\hat{\mathbf{y}} + \hat{\mathbf{z}}\hat{\mathbf{z}}\).
Definition. For a tensor, with coordinates \(u^i\), we have two sets of basis vectors:
\[
\mathbf{e}_i = \pdv{\mathbf{r}}{u^i}
\]
\[
\mathbf{e}^i = \nabla{u^i}
\]
1.9 - Helmholtz Theorem
Given an arbitrary vector field \(\mathbf{F}(\mathbf(r))\), we can write said field as a composition of a curl-free component \(\mathbf{\Phi}(\mathbf{r})\) and a divergence-free component \(\mathbf{A}(\mathbf{r})\) as follows:
Definition. Here, the gradient of the scalar potential is \(\nabla{\mathbf{\Phi}(\mathbf{r})}\) and the curl of the vector potential is \(\nabla \times{\mathbf{A}(\mathbf{r})}\). Thus, the scalar potential is \(\mathbf{\Phi}(\mathbf{r})\) and the vector potential is \(\mathbf{A}(\mathbf{r})\).
Letting said field be over bounded volume \(V\) with closed surface \(\partial V\), and the functions \(\mathbf{C}(\mathbf{r}) = \nabla \times{\mathbf{F}(\mathbf{r})}\) and \(\mathbf{D}(\mathbf{r}) = \nabla \cdot \mathbf{F}(\mathbf{r})\) are known, we can say that
Now, assume that \(\lim(\frac{\mathbf{F}(\mathbf{r})}{\mathbf{r}}) = 0\) as \(\mathbf{r} \rightarrow \infty\), with a large enough volume, we see that the second terms vanish.