Lecture 3
-
Vector calculus
- a scalars;
- b vector;
- c tensor;
- Index notation
- Examples - applictoins - identities
- Intorduction to fluid mechanics
- Clarification on HWK1
Tensor
$n^{th}$ order tensor. Scalar is the $0^{th}$ order tensor Vector is the $1^{th}$ order tensor
Tensors $n^{th}$
$\vec{u}=u_i\cdot\hat{e}_i$
$\delta_{ij}\cdot u_j=\delta_{i1}\cdot u_1 + \delta_{i2}\cdot u_2 + \delta_{i3}\cdot u_3 = u_i=(u_1, u_2, u_3)$
$\hat{e}i \cdot \hat{e}_j = \delta{ij}$ Scalar product of two vectors $\vec u, \vec v$ are the vectors.
$\epsilon_{ijk}=\epsilon_{jki}=\epsilon_{kij}$
$\vec{u}\times \vec{v} = \epsilon_{ijk} \cdot \hat{e}_i u_j v_k$
$\frac{\partial\phi}{\partial n}=\nabla\phi\cdot\hat n$
$\nabla\cdot\vec u = \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} + \frac{\partial w}{\partial y}=\frac{\partial u_j}{\partial x_j}$
$B_{ij}=B_{ji}$ symmetrie $B_{ij}=-B_{ji}$ anti-symmetrie $B_{ii}=-B-{ii} => B_{ii} = 0$ AsM: Zero Diorgonal
$A_{ij} = \frac{1}{2}A_{ij} + \frac{1}{2}A_{ij}=\frac{1}{2}A_{ij} + \frac{1}{2}A_{ji} + \frac{1}{2}A_{ij} - \frac{1}{2}A_{ji}=\frac{1}{2}(A_{ij} + A_{ji}) + \frac{1}{2}(A_{ij} - A_{ji})=symmetric + {anti-symmetric}$
