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

[raxAdaam]

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.

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

$dw = w_x dx + w_y dy + w_z dz$

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

1. $dw = w_x dx + w_y dy$.
   
   Now, in order to find $\left(\frac{\partial w}{\partial y}\right)_z$, we use the constraint to find an expression for $dx$ in terms of $dy$, skipping the steps, this comes out to be:
   
2. $dx = -\frac{g_y}{g_x}dy$,
   
   
   where $g_{\alpha} = \frac{\partial g}{\partial \alpha}$ is just convenient shorthand. Plugging this expression for $dx$ into #1, rearranging and factoring out the $dy$, we have:
   
3. $dw = \left[w_y-w_x\frac{g_y}{g_x}\right]dy$

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. $\left(\frac{\partial w}{\partial y}\right)_z$, they become partial differentials - 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 $w_y$ is simply the partial derivative w.r.t. $y$ 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?

 

[raxAdaam]

I managed to answer the second question (I think / I hope!) & thought I'd post it in case anyone was curious.

The first term $w_y = \frac{\partial w}{\partial y}$ is just the usual partial, as mentioned above.

Now the significance of the second term becomes clearer if one writes out the equivalent form of $\frac{g_y}{g_x}$, viz. because $g(x,y,z) = 0$ we have (holding $z$ constant: $dg = 0 = g_x dx + g_y dy$ which gives us $\frac{g_y}{g_x} = -\left(\frac{\partial x}{\partial y}\right)_z$ and the answer above becomes:

$\left( \frac{\partial w}{\partial y} \right)_z = w_y + w_x\cdot \left(\frac{\partial x}{\partial y}\right)_z$

so in the second term we are adding the additional change in $w$ caused by $x$ varying in response to the change in $y$ (which is what we would expect, I realize now - but the negative sign was confusing).

If there is a more salient (or geometric) interpretation - I'd be very interested to hear!

 

[raxAdaam]

Any thoughts or clarifications on this? Am very keen to understand what's happening here ...

 

[algebrat]

#3 $dw = \left[w_y-w_x\frac{g_y}{g_x}\right]dy$

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. $\left(\frac{\partial w}{\partial y}\right)_z$, they become partial differentials - 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 $w_y$ is simply the partial derivative w.r.t. $y$ 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?

As far as the differentials not being partial, two things. First thing, we have the notation $dx$, $\frac{\partial}{\partial x}$, and $\frac{d}{dx}$. No idea why derivatives have a regular and partial version, and differentials don't. That is an interesting question. Second thing is, that partial also has a z on it, which the differential doesn't have that either. I'm just going to make a wild guess and say that it has something to with theoretical mathematician's realizing that differentials are more well behaved, and don't require a lot of policing, but I'm not sure. I think you were observant to notice that, and it's good that you are seeing all the mechanisims that are going on. And the notation in math is not perfect, you are welcome to make your own system, maybe notes in the margins about what is being held fixed when.

But here's another way I might help, it may even help with the differentials. I like to be careful about naming things. For instance, from the constraint, we are to solve for x as a function y and z, so enforcing g=c implies we can find for x(y,z). So we have w(y,z)=w(x(y,z),y,z). The left side doesn't look right, so I like to rename it, v(y,z)=w(x(y,z),y,z). So we really want v_y. Which is w_x*x_y+w_y. So renaming things got me to where I want to go a little quicker.

If there is a more salient (or geometric) interpretation - I'd be very interested to hear!

Umm, geometrically, you are constraining yourself to move along the g=c surface. Now you take a z-slice, and the intersection of the z-slice with the g=c surface is a curve. Now let y vary, and staying on that curve tells you how the other dependent variables change, which in turn detemrines how the f value changes.