The value of $\underset{(\mathrm{x},\mathrm{y})\to (-1,2)}{\lim }{\mathrm{x}}^3+2{\mathrm{x}}^2\mathrm{y}-3{\mathrm{y}}^2$ is _____.

The points on the curve ${\mathrm{y}}^2\mathrm{x}=2$ which is nearest to the origin are _____. 

The gradient of $\mathrm{f}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)={\mathrm{x}}^3\mathrm{yz}$ at $\left(-1,1,2\right)$ is _____.

A function $\mathrm{f}(\mathrm{x},\mathrm{y})$ is said to be continuous at point $(\mathrm{a},\mathrm{b})$ if _____.

Find ${\left(\nabla \mathrm{f}\right)}_{\mathrm{P}}·\frac{\overline{\mathrm{a}}}{\left|\overline{\mathrm{a}}\right|}$ , if $\mathrm{f}\left(\mathrm{x},\mathrm{y},\mathrm{z}\right)=\mathrm{xyz}$ , $\mathrm{P}=\left(-1,1,3\right) and$   $\overline{\mathrm{a}}=\left(1,-2,2\right).$

Number of third order partial derivatives of function of two independent variables is _____.

What is the unit vector in the direction of $\overline{\mathrm{a}}=\left(1,2,2\right)$ ?

If $\mathrm{u}=\mathrm{f}\left(\mathrm{r}\right)$ and ${\mathrm{r}}^2={\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2$ , then $\frac{\partial \mathrm{r}}{\partial \mathrm{y}}=$ _____.

If $\mathrm{f}(\mathrm{x},\mathrm{y},\mathrm{z})={\mathrm{x}}^3{\mathrm{y}}^2\mathrm{z}-2\mathrm{xz}+3{\mathrm{xy}}^2$ then ${\mathrm{f}}_{\mathrm{y}}=$ _____.

If $\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)=\mathrm{c}$ be an implicit function, then $\frac{d\mathrm{y}}{d\mathrm{x}}=$ _____.