I'm brushing up on differentiating multi-variable functions subject to a constraint and was curious about the notation. In particular, why the derivatives change from complete to partial derivates. I've illustrated the question with an example, below. My specific question w.r.t. the example is in bold.(adsbygoogle = window.adsbygoogle || []).push({});

For example, if [itex]w = w(x,y,z)[/itex] is subject to the constraint [itex]g(x,y,z) = c[/itex], where c is a constant. To find [itex] \left(\frac{\partial w}{\partial y}\right)_z[/itex] using differentials, we would first write:

[itex] dw = w_x dx + w_y dy + w_z dz[/itex]

Because we are interested in the case where [itex]z[/itex] is held constant [itex]dz = 0[/itex], which leaves us with:

- [itex] dw = w_x dx + w_y dy[/itex].

Now, in order to find [itex] \left(\frac{\partial w}{\partial y}\right)_z[/itex], we use the constraint to find an expression for [itex]dx[/itex] in terms of [itex]dy[/itex], skipping the steps, this comes out to be:

- [itex] dx = -\frac{g_y}{g_x}dy[/itex],

where [itex] g_{\alpha} = \frac{\partial g}{\partial \alpha}[/itex] is just convenient shorthand. Plugging this expression for [itex]dx[/itex] into #1, rearranging and factoring out the [itex]dy[/itex], we have:

- [itex] dw = \left[w_y-w_x\frac{g_y}{g_x}\right]dy[/itex]

My question arises here: as I understand, the differentials in #3 are "complete" (i.e. not partial); however, if we want to go from #3 to the desired expression,viz.[itex]\left(\frac{\partial w}{\partial y}\right)_z[/itex], they becomepartialdifferentials - why?

A related question that interests me is how one would interpret these two elements on the RHS of #3 - I understand where each comes from entirely, so I'm not looking for a literal translation from the math to English, I'm trying to understand the relevance of the terms. I see that [itex]w_y[/itex] is simply the partial derivative w.r.t. [itex]y[/itex] in the case that the variables were all independent, but how to understand the other term and why it has the form that it does?

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# Partial Vs. Complete differentials when dealing with non-independent variables

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