Blundell Thermal Physics · Problem 11.4

Energy

Physics — Statistical Mechanics · partial-derivatives-polar-coordinates

Problem

In polar coordinates,

x = r\cos\theta

and

y = r\sin\theta

. The definition of

x

implies that

\frac{\partial x}{\partial r} = \cos\theta = \frac{x}{r}. \tag{11.40}

But we also have

x^2 + y^2 = r^2

, so differentiating with respect to

r

gives

2x\frac{\partial x}{\partial r} = 2r \implies \frac{\partial x}{\partial r} = \frac{r}{x}. \tag{11.41}

But eqns 11.40 and 11.41 imply that

\frac{\partial x}{\partial r} = \frac{\partial r}{\partial x}. \tag{11.42}

What's gone wrong?