Gradient and curl operators I have some troubles with vector identities for the gradient and curl operators, for example something like the gradient of the vector or the cross product :
Vector calculus identities
since i have not took any course in multivariate calculus (in fact i need such formulas in electromagnetic) , i am looking for some "trick" for remembering such identities for differential operators . 
 A: The most useful "trick" for this kind of thing is to think of the nabla (del) operator as a vector of differential operators.  In usual x,y,z Cartesian coordinates, the following is just such a vector:
$$
\nabla = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)
$$
With this in mind, the operations of the gradient, divergence, and curl are actually encoded by the notation we use.  For example, suppose you have a scalar function $\varphi(x,y,z)$.  The gradient of a scalar is written $\nabla\varphi$, which looks like you are multiplying a vector by a scalar.  So the "trick" is to write out what happens when you "multiply" a vector times a scalar.  Instead of multiplying of course, you apply the differential operators.
$$
\nabla\varphi = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\varphi = 
\left(\frac{\partial\varphi}{\partial x},\frac{\partial\varphi}{\partial y},\frac{\partial\varphi}{\partial z}\right)
$$
The same goes for the divergence, which we write out like a "dot product" between del and a vector.  Suppose you have a vector ${\bf A}=(A_x,A_y,A_z)$, with divergence $\nabla\cdot {\bf A}$, then you can expand this out using the usual dot product idea, where you multiply component by component and sum up, except applying differential operators replaces multiplication.
$$
\nabla\cdot {\bf A} = \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\cdot (A_x,A_y,A_z) = \frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}
$$
Finally, the curl is written like a cross product between del and $\bf A$, and the rules you learned about cross products as the determinant of a special matrix also still apply:
$$
\nabla\times{\bf A} = 
\left|
\begin{array}{ccc}
\hat x & \hat y & \hat z \\
\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\
A_x & A_y & A_z
\end{array}
\right|
$$
I'll leave expanding that determinant out to you (use the cofactor method on the row with unit vectors), and you'll see that you'll get the usual formula for the curl.  Similar logic holds for the laplacian ($\nabla^2$), which can be thought of as the dot product of two dels, which incidentally matches its definition/notation as the divergence of the gradient.
NB: The above is only for Cartesian coordinate systems.  For others, you are better off looking up the definitions of the operators in that coordinate system.
