List of top Multivariable Calculus Questions on Functions of Two or Three Real Variables

Let \( f : \mathbb{R}^2 \to \mathbb{R} \) be given by

\[ f(x, y) = \begin{cases} \dfrac{x^2 y (x - y)}{x^2 + y^2}, & (x, y) \neq (0, 0) \\ 0, & (x, y) = (0, 0) \end{cases} \]

Then

\[ \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y}  \right) - \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x}  \right) \text{ at the point } (0, 0) \text{ is ..........} \]

Let \( f(x, y) = \sqrt{x^3 y} \sin \left( \dfrac{\pi}{2} e^{(\frac{y}{x} - 1)} \right) + xy \cos \left( \frac{\pi}{3} e^{(\frac{x}{y} - 1)} \right) \) for \( (x, y) \in \mathbb{R}^2, x > 0, y > 0 \). 
Then \( f_x(1, 1) + f_y(1, 1) = \) ..........