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November 4th, 2016, 06:44 AM   #1
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Simple chainrule exercise, help?

Equation: 2*Z'x + 3*Z'y = 0
Simplify through variable change: u = x, v = 3x - 2y

I'm very confused here.
First, I don't know the function at hand, how am I suppose to solve it?
Secondly, am I suppose to replace x with u and then solve y = (v-3x)/2 and replace y with that?!

Very confusing exercise, all help is appreciated.
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November 4th, 2016, 09:12 AM   #2
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I presume your "Z'x" indicates the partial derivative with respect to x.
What function, that you don't know, are you talking about? If you mean z, this is an equation involving the "unknown" z with variables x and y and you are to convert it to an equivalent equation involving the "unknown" z with variable u and v.

When you have two variable, here x and y, and each depends on two other variables, x(u,v) and y(u,v) then
$\displaystyle \frac{\partial z}{\partial x}= \frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+ \frac{\partial z}{\partial v}\frac{\partial v}{\partial x}$
and
$\displaystyle \frac{\partial z}{\partial y}= \frac{\partial z}{\partial u}\frac{\partial u}{\partial y}+ \frac{\partial z}{\partial v}\frac{\partial v}{\partial y}$

$\displaystyle 2\frac{\partial z}{\partial x}+ 3\frac{\partial z}{\partial y}= 2\left(\frac{\partial z}{\partial u}\frac{\partial u}{\partial x}+ \frac{\partial z}{\partial v}\frac{\partial v}{\partial x}\right)+ 3\left(\frac{\partial z}{\partial u}\frac{\partial u}{\partial y}+ \frac{\partial z}{\partial v}\frac{\partial v}{\partial y}\right)= 0$

Here, here u= x so $\displaystyle \frac{\partial u}{\partial x}= 1$ and $\displaystyle \frac{\partial v}{\partial y}= 0$.

v= 3x- 2y so $\displaystyle \frac{\partial v}{\partial x}= 3$ and $\displaystyle \frac{\partial v}{\partial y}= -2$

So $\displaystyle \frac{\partial z}{\partial x}= \frac{\partial z}{\partial u}+ 3\frac{\partial z}{\partial v}$ and $\displaystyle \frac{\partial z}{\partial y}= -2\frac{\partial z}{\partial v}$.

Then $\displaystyle 2\frac{\partial z}{\partial x}+ 3\frac{\partial z}{\partial y}= 2\left(\frac{\partial z}{\partial u}+ 3\frac{\partial z}{\partial v}\right)+ 3\left(-2\frac{\partial z}{\partial v}\right)= 0$.
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November 4th, 2016, 09:36 AM   #3
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This is what I don't get, lets say they meant z = z(x(u,v), y(u,v))
Shouldn't dz/dx = dz/dx?
I mean, where asking: How much does z change when x changes?
We don't ask anything about any u,v. They do change x but that's not what we're asking for.

Also: du/dx what is that? x is dependent on u and v, u isnt dependent on x so where you get this du/dx from?

I just fail to see the connection here.. :/
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November 4th, 2016, 12:49 PM   #4
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In your original post you wrote "u = x, v = 3x - 2y" so u and v are certainly dependent on x and y (u isn't actually "dependent" of on y so the partial derivative of y is $\displaystyle \frac{\partial u}{\partial y}= 0$).
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November 4th, 2016, 04:19 PM   #5
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Oh so the function z should be thought of as:
z = z(u(x,y), v(x,y)) ?
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November 5th, 2016, 03:35 AM   #6
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Yes, that is what the notation means.
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November 5th, 2016, 06:41 AM   #7
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Thanks a ton man!
Finally understood this abit better!
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