Question 3

a. If $y_1 = r\cos\theta_1$, $y_2 = r\sin\theta_1\cos\theta_2$, $y_3 = r\sin\theta_1\sin\theta_2\cos\theta_3$, …, $y_{n-1} = r\sin\theta_1\sin\theta_2\cdots\sin\theta_{n-2}\cos\theta_{n-1}$ and $y_n = r\sin\theta_1\sin\theta_2\cdots\sin\theta_{n-1}$, then find the value of$$\frac{\partial(y_1, y_2, \ldots, y_n)}{\partial(r, \theta_1, \theta_2, \ldots, \theta_{n-1})}.$$ (5 marks)

b. Discuss the maximum or minimum values of $u(x,y) = ax^3y^2 - x^4y^2 - x^3y^3$. (4 marks)