Let's say we have a function [itex]F(\vec{r})=F(s, \phi, z)[/itex]. Then (correct me if I'm wrong):(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\frac{dF}{dx}=\frac{\partial F}{\partial s}\frac{ds}{dx}+...[/itex]

So then what is [itex]\frac{\partial F}{\partial x}[/itex]? Is it zero because [itex]F[/itex] doesn't depend explicitly on x? Is it the same as [itex]\frac{dF}{dx}=\frac{\partial F}{\partial s}\frac{ds}{dx}+...[/itex]? Is it [itex]\frac{\partial F}{\partial x}=\frac{\partial F}{\partial s}\frac{\partial s}{\partial x}+...[/itex]? Is there even any difference between these last two things?

More generally:

If we have [itex]F(\vec{r})=F(e^1(\vec{r}), e^2(\vec{r}), e^3(\vec{r}))[/itex], then is this correct?:

[itex]\frac{dF}{dx}=\sum_i\frac{\partial F}{\partial e^i}\frac{de^i}{dx}[/itex]

and:

[itex]\frac{\partial F}{\partial x}=\sum_i\frac{\partial F}{\partial e^i}\frac{\partial e^i}{\partial x}[/itex]

I guess what my question is is what exactly is the difference between a partial and a full derivative? My previous understanding is that you should only take partial derivatives with respect to variables that are explicitly included in the expression, whereas you consider all implicit and explicit dependencies on a variable when you take a full derivative. However, I don't think this understanding of a partial is sufficient anymore. For example, the case above, where we are taking a partial derivative of something which doesn't depend explicitly on the variable we are considering. My past understanding would tell me that the partial derivative would be zero, but I'm fairly certain this is not the case.

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# Partial vs Full Derivative

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