Problem 1 - Analysis

Concerning the transformation between the orthogonal coordinates and polar coordinates, answer the following questions.

1.

Represent $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ in terms of $\frac{\partial f}{\partial r}$ and $\frac{\partial f}{\partial \theta}$.

2.

Represent $\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}$ in terms of the partial derivatives of $f$ with respect to $r$ and $\theta$.

1.

$$f(x,y)=f(r\cos \theta,r\sin \theta)$$

$$\frac{\partial f}{\partial r} = \frac{\partial f}{\partial x}\fra... ...rac{\partial f}{\partial x}\cos\theta + \frac{\partial f}{\partial y}\sin\theta$$

$$\frac{\partial f}{\partial \theta} = \frac{\partial f}{\partial x... ...c{\partial f}{\partial x}r\sin\theta + \frac{\partial f}{\partial y}r\cos\theta$$

$$r\sin\theta\frac{\partial f}{\partial r} = \frac{\partial f}{\partial x}r\sin\theta\cos\theta + \frac{\partial f}{\partial y}r\sin^2\theta$$

$$\cos\theta\frac{\partial f}{\partial \theta} = -\frac{\partial f}{\partial x}r\sin\theta\cos\theta + \frac{\partial f}{\partial y}r\cos^2\theta$$

$$\boxed{\frac{\partial f}{\partial y} = \sin\theta\frac{\partial f}{\partial r} + \frac{\cos\theta}{r}\frac{\partial f}{\partial \theta}}$$

$$r\cos\theta\frac{\partial f}{\partial r} = \frac{\partial f}{\partial x}r\cos^2\theta + \frac{\partial f}{\partial y}r\cos\theta\sin\theta$$

$$-\sin\theta\frac{\partial f}{\partial \theta} = \frac{\partial f}{\partial x}r\sin^2\theta - \frac{\partial f}{\partial y}r\sin\theta\cos\theta$$

$$\boxed{\frac{\partial f}{\partial x}=\cos\theta\frac{\partial f}{\partial r}-\frac{\sin\theta}{r}\frac{\partial f}{\partial \theta}}$$

2.

$$\frac{\partial^2 f}{\partial r^2} = \cos\theta\left(\frac{\partia... ...ial x\partial y}\cos\theta + \frac{\partial^2 f}{\partial y^2}\sin\theta\right)$$

$$\frac{\partial^2 f}{\partial r^2} =\cos^2\theta\frac{\partial^2 f... ...{\partial y^2} + 2\sin\theta\cos\theta\frac{\partial^2 f}{\partial x\partial y}$$

$$r^2\frac{\partial^2 f}{\partial r^2} =r^2\left(\cos^2\theta\frac{... ...y^2}\right) + 2r^2\sin\theta\cos\theta\frac{\partial^2 f}{\partial x\partial y}$$

$$\frac{\partial^2 f}{\partial \theta^2} = -\left(-\frac{\partial^2... ...al y^2}r\cos\theta\right)r\cos\theta - r\sin\theta\frac{\partial f}{\partial y}$$

$$\frac{\partial^2 f}{\partial \theta^2} = r^2\left(\sin^2\theta\fr... ...s\theta\frac{\partial f}{\partial x} - r\sin\theta\frac{\partial f}{\partial y}$$

$$r^2\frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\parti... ...s\theta\frac{\partial f}{\partial x} - r\sin\theta\frac{\partial f}{\partial y}$$

$$r^2\frac{\partial^2 f}{\partial r^2} + \frac{\partial^2 f}{\parti... ...^2} + \frac{\partial^2 f}{\partial y^2}\right) - r\frac{\partial f}{\partial r}$$

$$\boxed{\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\p... ...\frac{\partial^2 f}{\partial \theta^2} + r\frac{\partial f}{\partial r}\right)}$$