Definition of curl Curl(F)=$\nabla\times F$
I am finding it difficult to understand why cross product gives Curl. Is this like torque equation $\tau=R\times F$? What is the direction of $\vec{\nabla}$?
In case of a scalar field $F=F(x,y)$, $\nabla F$ gives a vector in the $x-y$ plane which points along the direction of maximum increase of $F$.
 A: This is just a symbolic notation. You can always think of $\nabla$ as the "vector"
$$\nabla = \left( \frac{\partial}{\partial x} ,  \frac{\partial}{\partial y},  \frac{\partial}{\partial z}\right).$$
Well this is not a vector, but this notation helps you remember the formula. For example, the gradient of a function $f$ is a vector 
$$\left( \frac{\partial f}{\partial x} ,  \frac{\partial f}{\partial y},  \frac{\partial f}{\partial z}\right),$$
which you get think of it as 
$$\nabla f =  \left( \frac{\partial}{\partial x} ,  \frac{\partial}{\partial y},  \frac{\partial}{\partial z}\right) f$$
(Like multiplying $f$ to the vector $\nabla$). The divergence of $F = (A, B, C)$ is given by 
$$ \text{div} F = \frac{\partial A}{\partial x} + \frac{\partial B}{\partial y} + \frac{\partial C}{\partial z},$$
which can also be written as 
$$ \nabla \cdot F = \left( \frac{\partial}{\partial x} ,  \frac{\partial}{\partial y},  \frac{\partial}{\partial z}\right) \cdot (A, B, C).$$
On the other hand, the Curl can also be given by 
$$\begin{split} \nabla \times F &= \left|\begin{matrix} i & j & k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A & B& C \end{matrix} \right|\\
& = \left|\begin{matrix} \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\  B& C  \end{matrix}\right| i - \left|\begin{matrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial z} \\ A& C  \end{matrix}\right| j + \left|\begin{matrix} \frac{\partial}{\partial x} & \frac{\partial}{\partial y} \\  A& B  \end{matrix}\right| k \\
&=\left( \frac{\partial C}{\partial y} - \frac{\partial B}{\partial z}\right) i -\left( \frac{\partial C}{\partial x} - \frac{\partial A}{\partial z}\right)  j + \left( \frac{\partial B}{\partial x} - \frac{\partial A}{\partial y}\right)  k
\end{split}$$
In my opinion, this just give you a way to remember the formulas. 
