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I'm trying to figure out how to go from: $${1 \over {r}}{({{∂P} \over{∂r}} + r {{∂^2 P} \over{∂r^2} })}= {1 \over {n}} {{∂P} \over {∂t}}$$

to:

$${1 \over {r}}{ ∂ \over{∂r} }{(r \times {{∂P} \over{∂r} })}= {1 \over {n}} {{∂P} \over {∂t}}$$

I'm rusty on my calculus and I'm trying to understand the deduction of the diffusivity equation.

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    Just a tip: try not to use the \times symbol for multiplication. The symbol has different meanings in different contexts (i.e curl operator in linear algebra). Even though it was clear in your question what you were asking, you are best off just writing r \partial_{r} P instead of r \times \partial_{r} P. – 2017-02-10
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    I'll keep that in mind. I'm new here and I'm still getting used to everything but I'll most definitely do it next time. – 2017-02-10
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    All good mate, it just means you won't get asked questions about notation and will save you time. – 2017-02-10

1 Answers 1

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\begin{eqnarray} { ∂ \over{∂r} }{(r \times {{∂P} \over{∂r} })} &=& \frac{\partial r}{\partial r}.\frac{\partial P}{\partial r}+r\frac{\partial }{\partial r}\frac{\partial P}{\partial r}=\frac{\partial P}{\partial r}+r\frac{\partial^2P}{\partial r^2} \end{eqnarray} so \begin{eqnarray} \frac1r{ ∂ \over{∂r} }{(r \times {{∂P} \over{∂r} })} &=& \frac1r\left(\frac{\partial P}{\partial r}+r\frac{\partial^2P}{\partial r^2}\right)={1 \over {n}} {{∂P} \over {∂t}} \end{eqnarray}

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    You are welcome. – 2017-02-10