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I'm having trouble with total differentials in relation to the attached picture (fluid deformation). I don't understand how the expressions for du and dv come about.

It looks like u = f(x) and v = f(y), so i'm not sure why the incremental increases du and dv involve partial differentials, rather than differentials of only one variable. picture

It's probably pretty obvious to someone in the know, all help gratefully received

Thanks

David

1 Answers1

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In general you have $u = u(x,y)$ so that

$$ {\rm d}u = \frac{\partial u}{\partial x}{\rm d}x + \frac{\partial u}{\partial y}{\rm d}y $$

However, if you move vertically in your diagram ${\rm d}y = 0$, so that this becomes

$$ u + {\rm d}u = u + \frac{\partial u}{\partial x}{\rm d}x $$

with a similar expression for $v = v(x,y)$

caverac
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