How do I solve for partial differentiation in multivariable calculus?

[UbikPkd]

Ok here goes...

$$z=x^{2}+y^{2}$$

$$x=rcos\vartheta$$

$$y=r sin\vartheta$$

Find:

$$\[ \frac{\partial z}{\partial x}_{y}, \[ \frac{\partial z}{\partial \vartheta}_{x}, \[ \frac{\partial z}{\partial r}_{y}, \[ \frac{\partial z}{\partial r}_{ \vartheta},$$

___________________________

$$\[ \frac{\partial z}{\partial x}_{y} = 2x$$

this seems right to me (though I'm not

sure if I'm supposed to use a chain rule),

it's the next ones I'm not sure about...
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$$\[ \frac{\partial z}{\partial \vartheta}_{x}=$$

I've been at this one for hours, i think i

can use the following, but I'm getting

nowhere.

$$dz= \[ \frac{\partial z}{\partial x}_{y}dx + \[ \frac{\partial z}{\partial y}_{x}dy$$


$$dx= \[ \frac{\partial x}{\partial r}_{\vartheta}dr + \[ \frac{\partial x}{\partial \vartheta}_{r}d\vartheta$$


$$dy= \[ \frac{\partial y}{\partial r}_{\vartheta}dr + \[ \frac{\partial y}{\partial \vartheta}_{r}d\vartheta$$

so...

$$dz= \[ \frac{\partial z}{\partial x}_{y} \left[\[ \frac{\partial x}{\partial r}_{\vartheta}dr + \[ \frac{\partial x}{\partial \vartheta}_{r}d\vartheta\right]+ \[ \frac{\partial z}{\partial y}_{x}\left[\[ \frac{\partial y}{\partial r}_{\vartheta}dr + \[ \frac{\partial y}{\partial \vartheta}_{r}d\vartheta\right]$$

can i divide through by $$\partial \vartheta_{x}$$ and then work it all

out to get
$$\[ \frac{\partial z}{\partial \vartheta}_{x}$$ ?

please help, i don't have a clue what I'm

doing!

 

[Dick]

The direct way to do this is to simply write z as a function only of x and theta. This is pretty easy, since y/x=tan(theta). Try it that way before you go back to struggling with the chain rule.

 

[UbikPkd]

okay thanks for the tip, i worked it out using chain rules in the end before i got to read it, the answers i got were:

$$\[ \frac{\partial z}{\partial x}_{y}=2x$$

$$\[ \frac{\partial z}{\partial \vartheta}_{x}=0$$

$$\[ \frac{\partial z}{\partial r}_{y}=2r$$

$$\[ \frac{\partial z}{\partial r}_{\vartheta}=2r$$

I think these are right, thanks for your help!

 

[Dick]

Well, e.g. for the second one, y/x=tan(theta), so z=x^2+x^2*tan^2(theta). It doesn't look to me like the derivative wrt theta at constant x is 0.

 

[UbikPkd]

Well, e.g. for the second one, y/x=tan(theta), so z=x^2+x^2*tan^2(theta). It doesn't look to me like the derivative wrt theta at constant x is 0.

sorry! arghh I've made such a mess of this, yer the last two are right, as $$z=r^2$$ so it doesn't matter what's being held constant


$$\[ \frac{\partial z}{\partial \vartheta}_{x}=2r^{2}tan\vartheta$$

not 0, sorry, here's my full working:

$$\[ \frac{\partial z}{\partial \vartheta}_{x}=2y\[ \frac{\partial y}{\partial \vartheta}_{x}$$

_________

$$y=rsin\vartheta$$

$$\[ \frac{\partial y}{\partial \vartheta}_{x}=rcos\vartheta+\[ \frac{\partial r}{\partial \vartheta}_{x}sin\vartheta$$

________

$$x=rcos\vartheta$$

$$0=-rsin\vartheta+\[ \frac{\partial r}{\partial \vartheta}_{x}cos\vartheta$$

$$\[ \frac{\partial r}{\partial \vartheta}_{x}=r\[ \frac{sinv}{cos\vartheta}=rtan\vartheta$$

________

$$\[ \frac{\partial z}{\partial \vartheta}_{x}=2y(rcos\vartheta+rsin\vartheta tan\vartheta)$$

$$\[ \frac{\partial z}{\partial \vartheta}_{x}=2r^{2}\frac{sin\vartheta}{cos\vartheta}=2r^{2}tan\vartheta$$

I did it that way, as I thought differentiating via the $$tan^{2}\vartheta$$ way would probably go wrong for me somehow! I think the above is OK, seems like that's the way they want me to do it as well. Thanks for your help!

 

[Dick]

Doing it the other way gives you 2x^2*sin(theta)/cos^3(theta), which if you put x=r*cos(theta) gives you the same thing you got. So I think your answer is right.

 

[UbikPkd]

Doing it the other way gives you 2x^2*sin(theta)/cos^3(theta), which if you put x=r*cos(theta) gives you the same thing you got. So I think your answer is right.

cool thanks!