A basic question on the Del symbol / gradient I have a very basic question regarding the $\nabla$ (Del) symbol / gradient on a field (which I have called f). I have seen two definitions of this symbol:


*

*$\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})$

*$\nabla f = \frac{\partial f}{\partial x} \hat{\imath} + \frac{\partial f}{\partial y} \hat{\jmath} + \frac{\partial f}{\partial z} \hat{k}$ (where $\hat{\imath}$, $\hat{\jmath}$ and $\hat{k}$ are unit vectors)


Are (1) and (2) basically the same thing? If so, how do you get from (2) to (1)?
Many thanks!
 A: Yes, they're the same.  In the first definition, you're writing the gradient in standard 3D coordinate notation $(a,b,c)$.  In vector algebra/calculus, you start to think about these three numbers as being components of a vector with respect to some basis (these are linear algebra terms).  The "standard basis" of 3D vector calculus consists of the three vectors 
$$
\hat{\boldsymbol{\imath}} = \left[\begin{array}{c} 1\\0\\0\end{array}\right],\quad \hat{\boldsymbol{\jmath}} = \left[\begin{array}{c} 0\\1\\0\end{array}\right],\quad 
\hat{\boldsymbol{k}} = \left[\begin{array}{c} 0\\0\\1\end{array}\right]
$$  Then, the coordinates $(a,b,c)$ correspond to a column vector as follows:
$$
\left[\begin{array}{c} a\\b\\c\end{array}\right] = a\left[\begin{array}{c} 1\\0\\0\end{array}\right] + b\left[\begin{array}{c} 0\\1\\0\end{array}\right]+c\left[\begin{array}{c} 0\\0\\1\end{array}\right] = a \hat{\boldsymbol{\imath}} +b\hat{\boldsymbol{\jmath}} + c\hat{\boldsymbol{k}} 
$$  Thinking of the gradient as a column vector, you'll see why they're the same!
A: You have already gotten good explanation by @icarays1 on the notation so I'll skip that part.

$\nabla$ is an operator. It contains spatial derivatives with respect to all basis vectors. If operating on a scalar valued function it creates a vector.
The divergence of a vector field is the inner product between the nabla operator and a vector valued function:
$$\nabla \cdot {\bf v} = \left(\begin{array}{ccc}\frac{d}{dx}&\frac{d}{dy}&\frac{d}{dz}\end{array}\right){\bf v} = \left(\begin{array}{ccc}\frac{d}{dx}&\frac{d}{dy}&\frac{d}{dz}\end{array}\right)  \left(\begin{array}{c}{v_x}\\v_y\\v_z\end{array}\right) = \sum_{k \in \{x,y,z\}}\frac{d(v_k)}{d_k}$$
The gradient of a scalar field is the outer product between the nabla operator and a scalar valued function:
$$\nabla f = \left(\begin{array}{ccc}\frac{d}{dx}\\\frac{d}{dy}\\\frac{d}{dz}\end{array}\right)f =  \left(\begin{array}{ccc}\frac{d(f)}{dx}\\\frac{d(f)}{dy}\\\frac{d(f)}{dz}\end{array}\right)$$
