List of practice Questions

Let \(f\) be a function of real variables \(x\) and \(y\), defined as:
\[ f(x,y) = x^2y + 3y^2x \]
The value of \( \frac{\partial^2 f}{\partial x^2} \) at \(x=1, y=1\) is

Choose one correct option.
Given \(w = f(ax+by)\), where \(a\) and \(b\) are constants. The value of
\[ \left(b\frac{\partial w}{\partial x} - a\frac{\partial w}{\partial y}\right) \] is:

If \[ \vec{v}=(x+y+1)\hat{i}+\hat{j}+(-x-y)\hat{k}, \] then the value of \[ \vec{v}\cdot(\nabla\times\vec{v}) \] is

If \(u=\sin^{-1}\left(\dfrac{x^{1/3}+y^{1/3}}{x^{1/2}-y^{1/2}}\right)^{1/2}\), then \(x\dfrac{\partial u}{\partial x}+y\dfrac{\partial u}{\partial y}=\)