User:Tetonca test bird

Instantiate. (A public sandbox is not useful for private experimentation -- a per-user sandbox would be better. Haven't seen one, so this is mine.)

Samples

 * $$\mathbf{u}=c_1x_1+c_2x_2+c_3x_3 \cdots c_nx_n$$


 * $$\mathbf{u}=\sum_{i=1}^n c_ix_i$$


 * $$\mathbf{u}=c_ix_i$$


 * $$\mathbf{u} = u_x \mathbf{i} + u_y \mathbf{j} + u_z \mathbf{k}$$


 * $$\mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + u_3 \mathbf{e}_3

= \sum_{i = 1}^3 u_i \mathbf{e}_i$$


 * $$\mathbf{u} \cdot \mathbf{v} = \sum_{i = 1}^3 u_i \mathbf{e}_i \cdot

\sum_{j = 1}^3 v_j \mathbf{e}_j = u_i \mathbf{e}_i \cdot v_j \mathbf{e}_j $$


 * $$\mathbf{u} \cdot \mathbf{v}

= \sum_{i = 1}^3 \sum_{j = 1}^3 u_i v_j ( \mathbf{e}_i \cdot \mathbf{e}_j ) = u_i v_j ( \mathbf{e}_i \cdot \mathbf{e}_j ) $$


 * $$ \mathbf{e}_i \cdot

\mathbf{e}_j = \delta_{ij} $$


 * $$\mathbf{u} \cdot \mathbf{v} = u_i v_j\delta_{ij}= u_i v_i = u_j v_j $$


 * $$ \mathbf{u} \times \mathbf{v}= \sum_{j = 1}^3 u_j \mathbf{e}_j \times

\sum_{k = 1}^3 v_k \mathbf{e}_k = u_j \mathbf{e}_j \times v_k \mathbf{e}_k = u_j v_k (\mathbf{e}_j \times \mathbf{e}_k ) = \epsilon_{ijk} \mathbf{e}_i u_j v_k $$

where $$ \mathbf{e}_j \times \mathbf{e}_k = \epsilon_{ijk} \mathbf{e}_i$$ and $$\ \epsilon_{ijk}$$ is the Levi-Civita symbol defined by:

--- truncated --- Above is from Einstein_notation.

Other
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