Home > LaTeX, Mathematics > Going with the Flow

Going with the Flow

Let
\vec{a}=(\vec{v}\cdot\nabla)\vec{v}

\vec{b}=\frac{\nabla(\vec{v}\cdot\vec{v})}{2}

\vec{c}=(\nabla\times\vec{v})\times\vec{v}

Now looking at components:
a_i =v_j\frac{\partial v_i}{\partial x_j}

b_i=v_j\frac{\partial v_j}{\partial x_i}

c_n=\epsilon_{kmn}\epsilon_{kij}\frac{\partial v_j}{\partial x_i}v_m=(\delta_{mi}\delta_{nj}-\delta_{mj}\delta_{ni})\frac{\partial v_j}{\partial x_i}v_m
or
c_n =v_m\frac{\partial v_n}{\partial x_m}-v_m\frac{\partial v_m}{\partial x_n}

Relabelling:

c_i =v_j\frac{\partial v_i}{\partial x_j}-v_j\frac{\partial v_j}{\partial x_i}

Therefore,
a_i=b_i+c_i
and
(\vec{v}\cdot\nabla)\vec{v}=\frac{\nabla(\vec{v}\cdot\vec{v})}{2}+(\nabla\times\vec{v})\times\vec{v}

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Categories: LaTeX, Mathematics
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