3.5 KiB
Chapter 1 - Mathematics
1.5 - Dyads and Tensors
Definition. A dyadic is a representation of two-ish vectors.
\stackrel{\leftrightarrow}{\mathbf{D}} = \begin{matrix}
D_{xx} \hat{\mathbf{x}}\hat{\mathbf{x}} &+ D_{xy} \hat{\mathbf{x}}\hat{\mathbf{y}} &+ D{xz} \hat{\mathbf{x}}\hat{\mathbf{z}} \\
+ D_{yx} \hat{\mathbf{y}}\hat{\mathbf{x}} &+ D_{yy} \hat{\mathbf{y}}\hat{\mathbf{y}} &+ D{yz} \hat{\mathbf{y}}\hat{\mathbf{z}} \\
+ D_{zx} \hat{\mathbf{z}}\hat{\mathbf{x}} &+ D_{zy} \hat{\mathbf{z}}\hat{\mathbf{y}} &+ D{zz} \hat{\mathbf{z}}\hat{\mathbf{z}}
\end{matrix}
Definition. If a dyadic can be written as a composition of two vectors \mathbf{A} and \mathbf{B}, it is called a dyad.
\mathbf{AB} = \begin{matrix}
A_x B_x \hat{\mathbf{x}}\hat{\mathbf{x}} &+ A_x B_y \hat{\mathbf{x}}\hat{\mathbf{y}} &+ A_x B_z \hat{\mathbf{x}}\hat{\mathbf{z}} \\
+ A_y B_x \hat{\mathbf{y}}\hat{\mathbf{x}} &+ A_y B_y \hat{\mathbf{y}}\hat{\mathbf{y}} &+ A_y B_z \hat{\mathbf{y}}\hat{\mathbf{z}} \\
+ A_z B_x \hat{\mathbf{z}}\hat{\mathbf{x}} &+ A_z B_y \hat{\mathbf{z}}\hat{\mathbf{y}} &+ A_z B_z \hat{\mathbf{z}}\hat{\mathbf{z}}
\end{matrix}
The dot product of a dyad \stackrel{\leftrightarrow}{\mathbf{D}} = \mathbf{AB} and vector \mathbf{v} can be written as follows:
(\mathbf{AB}) \cdot \mathbf{v} = \mathbf{A} (\mathbf{B} \cdot \mathbf{v})
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:
\mathbf{F}(\mathbf{r}) = - \nabla{\mathbf{\Phi}(\mathbf{r})} + \nabla \times{\mathbf{A}(\mathbf{r})}
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
\mathbf{\Phi}(\mathbf{r}) = \frac{1}{4 \pi} \int_V \frac{D(\mathbf{r}')}{|{\mathbf{r}-\mathbf{r}'}|} d{V'} - \frac{1}{4 \pi} \int_{\partial V} \frac{\mathbf{F}(\mathbf{r}') \cdot \mathbf{n}'}{|{\mathbf{r}-\mathbf{r}'}|} d{S'}
\mathbf{A}(\mathbf{r}) = \frac{1}{4 \pi} \int_V \frac{C(\mathbf{r}')}{|{\mathbf{r}-\mathbf{r}'}|} d{V'} - \frac{1}{4 \pi} \int_{\partial V} \mathbf{n}' \times \frac{\mathbf{F}(\mathbf{r}')}{|{\mathbf{r}-\mathbf{r}'}|} d{S'}
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.
\mathbf{\Phi}(\mathbf{r}) = \frac{1}{4 \pi} \int_V \frac{D(\mathbf{r}')}{|{\mathbf{r}-\mathbf{r}'}|} d{V'}
\mathbf{A}(\mathbf{r}) = \frac{1}{4 \pi} \int_V \frac{C(\mathbf{r}')}{|{\mathbf{r}-\mathbf{r}'}|} d{V'}