Thread: Change of coordinates (& notation)

1. Change of coordinates (& notation)

Hi,

I have a question regarding, well, notation basically. In my book of practice problems, this is the first "revision" problem in the partial differential equations section.

It involves the transformation from Cartesian to Polar coordinates ie $(x,y)=r\cdot(\cos\theta ,\sin\theta)$.

I am asked to find "the partial derivatives of r and theta with respect to x and y, expressing them in terns of r and theta. Deduce that in general, $\left(\frac{\partial r}{\partial x}\right)_y \left(\frac{\partial r}{\partial x}\right)_\theta\neq 1$.

What does the subscript y signify? Is this simply the variable in the function which is treated as constant?

Also:

$\theta = \arccos\left(\frac{x}{r}\right) \Rightarrow \frac{\partial\theta}{\partial x} = \frac{1}{r} \cdot \frac{-1}{\sqrt{1-\frac{x^2}{r^2}}} = -\frac{1}{\sqrt{r^2-x^2}} = -\frac{1}{r\sin\theta}$, yes?

but the answer says it is $-\frac{\sin\theta}{r}$. . .

2. well I think I have solved my problem...

the domain for arccos isn't really very big, so I would use the arctangent, differentiation of which gives the correct answer.

And it looks like the extra letter is indeed simply identifying the sleeping variable.