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After seeing the following equation in a lecture about tensor analysis, I became confused.

$$ \frac{d\phi}{ds}=\frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds} $$

What exactly is the difference between $d$ and $\partial$?

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    [Total derivative](http://en.wikipedia.org/wiki/Total_derivative) vs [partial derivative](http://en.wikipedia.org/wiki/Partial_derivative).2012-08-16
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    @Jen, maybe elaborate a bit more in an answer? :)2012-08-16
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    I know, one is the partial and the other one is a total derivative. But isn't $\frac{\partial f}{\partial x}$ the same as $\frac{df}{dx}$?2012-08-16
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    @iblue, the first one treats all the other independent variables as if they were constants. The second one doesn't.2012-08-16
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    @iblue Read what Jennifer linked to, it will help.2012-08-16
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    She edited her comment, and added the link to the wikipedia article on total deriviates. Now I see...2012-08-16

2 Answers 2

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As mentioned $d$ means total and $\partial$ partial derivative and are not the same. Total derivative also counts $x$ dependencies in other variables. For instance: $$ f(x,v) = x^2 + v(x) \\ \frac{\partial f}{\partial x} = 2x \\ \frac{\partial f}{\partial v} = 1 \\ \frac{d f}{d x} = 2x + \frac{\partial v(x)}{\partial x} $$

Your formula most probably uses Einstein notation and is only a shorter way to write $$ \frac{d\phi}{ds}= \sum_m \frac{\partial \phi}{\partial x^m}\frac{dx^m}{ds} $$

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The difference is whether the rest of the variables of $f$ are considered constants or variables in $x.$ Former is partial, latter is total.

  • $\frac{\partial f}{\partial x}$ is the partial derivative: $f$ is differentiated w.r.t. to $x$ while all other variables are considered constants in $x.$

  • $\frac{d f}{d x}$ the is total derivative: $f$ is differentiated w.r.t. to $x$ while nothing is assumed about the other variables; they are considered variables in $x.$ (some variables might be, in fact, constants in $x.$)