Aloha :)
zu a) Die Jacobi-Matrix einer Abbildung enhält die Gradienten der Komponenten-Funktionen als Zeilenvektoren.
Hier lauet die Funktion$$\vec f(r;\vartheta;\varphi)=\begin{pmatrix}f_1\\f_2\\f_3\end{pmatrix}=\begin{pmatrix}r\sin\vartheta\cos\varphi\\r\sin\vartheta\sin\varphi\\\cos\vartheta\end{pmatrix}$$
Die Jacobi-Matrix lauetet daher:$$J(\vec f)=\left(\begin{array}{c}\partial_r f_1 & \partial_\vartheta f_1 & \partial_\varphi f_1\\\partial_r f_2 & \partial_\vartheta f_2 & \partial_\varphi f_2\\\partial_r f_3 & \partial_\vartheta f_3 & \partial_\varphi f_3\end{array}\right)=\left(\begin{array}{rrr}\sin\vartheta\cos\varphi & r\cos\vartheta\cos\varphi & -r\sin\vartheta\sin\varphi\\\sin\vartheta\sin\varphi & r\cos\vartheta\sin\varphi & r\sin\vartheta\cos\varphi\\0 & -\sin\vartheta & 0\end{array}\right)$$
zu b) Hier rechnest du am besten von rechts nach links und jeden Term einzeln:
$$\frac{\partial^2G}{\partial r^2}=\frac{\partial}{\partial r}\,\frac{\partial G}{\partial r}=\frac{\partial}{\partial r}\left(\frac{\partial g}{\partial x}\,\frac{\partial x}{\partial r}+\frac{\partial g}{\partial y}\,\frac{\partial y}{\partial r}\right)=\frac{\partial}{\partial r}\left(\frac{\partial g}{\partial x}\,\cos\varphi+\frac{\partial g}{\partial y}\,\sin\varphi\right)$$$$\phantom{\frac{\partial^2G}{\partial r^2}}=\frac{\partial}{\partial r}\left(\red{\frac{\partial g}{\partial x}}\right)\cos\varphi+\frac{\partial}{\partial r}\left(\blue{\frac{\partial g}{\partial y}}\right)\sin\varphi$$$$\phantom{\frac{\partial^2G}{\partial r^2}}=\left(\frac{\partial}{\partial x}\,\red{\frac{\partial g}{\partial x}}\,\frac{\partial x}{\partial r}+\frac{\partial}{\partial y}\,\red{\frac{\partial g}{\partial x}}\,\frac{\partial y}{\partial r}\right)\cos\varphi+\left(\frac{\partial}{\partial x}\,\blue{\frac{\partial g}{\partial y}}\,\frac{\partial x}{\partial r}+\frac{\partial}{\partial y}\,\blue{\frac{\partial g}{\partial y}}\,\frac{\partial y}{\partial r}\right)\sin\varphi$$$$\phantom{\frac{\partial^2G}{\partial r^2}}=\left(\frac{\partial^2g}{\partial x^2}\,\cos\varphi+\frac{\partial^2g}{\partial y\,\partial x}\,\sin\varphi\right)\cos\varphi+\left(\frac{\partial^2g}{\partial x\,\partial y}\,\cos\varphi+\frac{\partial^2g}{\partial y^2}\,\sin\varphi\right)\sin\varphi$$$$\phantom{\frac{\partial^2G}{\partial r^2}}=\frac{\partial^2g}{\partial x^2}\,\cos^2\varphi+2\,\frac{\partial^2g}{\partial x\,\partial y}\,\sin\varphi\cos\varphi+\frac{\partial^2g}{\partial y^2}\,\sin^2\varphi$$
$$\frac{1}{r^2}\frac{\partial^2G}{\partial\varphi^2}=\frac{1}{r^2}\frac{\partial}{\partial\varphi}\,\frac{\partial G}{\partial\varphi}=\frac{1}{r^2}\frac{\partial}{\partial\varphi}\left(\frac{\partial g}{\partial x}\,\frac{\partial x}{\partial \varphi}+\frac{\partial g}{\partial y}\,\frac{\partial y}{\partial \varphi}\right)$$$$\phantom{\frac{1}{r^2}\frac{\partial^2G}{\partial\varphi^2}}=\frac{1}{r^2}\frac{\partial}{\partial \varphi}\left(\frac{\partial g}{\partial x}(-r\sin\varphi)+\frac{\partial g}{\partial y}\,r\cos\varphi\right)=\frac{1}{r}\frac{\partial}{\partial \varphi}\left(\red{\frac{\partial g}{\partial x}}(-\sin\varphi)+\blue{\frac{\partial g}{\partial y}}\,\cos\varphi\right)$$$$\phantom{\frac{1}{r^2}\frac{\partial^2G}{\partial\varphi^2}}=\frac{1}{r}\left(\left(\frac{\partial}{\partial x}\red{\frac{\partial g}{\partial x}}\frac{\partial x}{\partial\varphi}+\frac{\partial}{\partial y}\red{\frac{\partial g}{\partial x}}\frac{\partial y}{\partial\varphi}\right)(-\sin\varphi)+\red{\frac{\partial g}{\partial x}}(-\cos\varphi)\right)$$$$\phantom{\frac{1}{r^2}\frac{\partial^2G}{\partial\varphi^2}}+\frac{1}{r}\left(\left(\frac{\partial}{\partial x}\blue{\frac{\partial g}{\partial y}}\frac{\partial x}{\partial\varphi}+\frac{\partial}{\partial y}\blue{\frac{\partial g}{\partial y}}\frac{\partial y}{\partial\varphi}\right)\cos\varphi+\blue{\frac{\partial g}{\partial y}}(-\sin\varphi)\right)$$$$\phantom{\frac{1}{r^2}\frac{\partial^2G}{\partial\varphi^2}}=\frac{1}{r}\left(\left(\frac{\partial^2g}{\partial x^2}(-r\sin\varphi)+\frac{\partial^2g}{\partial y\partial x}\,r\cos\varphi\right)(-\sin\varphi)+\frac{\partial g}{\partial x}(-\cos\varphi)\right)$$$$\phantom{\frac{1}{r^2}\frac{\partial^2G}{\partial\varphi^2}}+\frac{1}{r}\left(\left(\frac{\partial^2g}{\partial x\partial y}(-r\sin\varphi)+\frac{\partial^2g}{\partial y^2}r\cos\varphi\right)\cos\varphi+\frac{\partial g}{\partial y}(-\sin\varphi)\right)$$$$\phantom{\frac{1}{r^2}\frac{\partial^2G}{\partial\varphi^2}}=\frac{1}{r}\left(\left(\frac{\partial^2g}{\partial x^2}(-r\sin\varphi)+\frac{\partial^2g}{\partial y\partial x}\,r\cos\varphi\right)(-\sin\varphi)+\frac{\partial g}{\partial x}(-\cos\varphi)\right)$$$$\phantom{\frac{1}{r^2}\frac{\partial^2G}{\partial\varphi^2}}+\frac{1}{r}\left(\left(\frac{\partial^2g}{\partial x\partial y}(-r\sin\varphi)+\frac{\partial^2g}{\partial y^2}r\cos\varphi\right)\cos\varphi+\frac{\partial g}{\partial y}(-\sin\varphi)\right)$$$$\phantom{\frac{1}{r^2}\frac{\partial^2G}{\partial\varphi^2}}=\frac{\partial^2g}{\partial x^2}\sin^2\varphi-2\frac{\partial^2g}{\partial x\partial y}\sin\varphi\cos\varphi+\frac{\partial^2g}{\partial y^2}\cos^2\varphi-\frac{\cos\varphi}{r}\frac{\partial g}{\partial x}-\frac{\sin\varphi}{r}\frac{\partial g}{\partial y}$$
$$\frac1r\frac{\partial G}{\partial r}=\frac1r\left(\frac{\partial g}{\partial x}\,\frac{\partial x}{\partial r}+\frac{\partial g}{\partial y}\,\frac{\partial y}{\partial r}\right)=\frac1r\left(\frac{\partial g}{\partial x}\,\cos\varphi+\frac{\partial g}{\partial y}\,\sin\varphi\right)$$$$\phantom{\frac1r\frac{\partial G}{\partial r}}=\frac{\cos\varphi}{r}\frac{\partial g}{\partial x}+\frac{\sin\varphi}{r}\frac{\partial g}{\partial y}$$
Wenn du nun die 3 Ergebnisse addierst, folgt die zu zeigende Gleichung:$$\frac{\partial^2G}{\partial r^2}+\frac{1}{r^2}\frac{\partial^2G}{\partial\varphi^2}+\frac1r\frac{\partial G}{\partial r}=\frac{\partial^2g}{\partial^2x}+\frac{\partial^2g}{\partial^2y}=\Delta g$$
