Originally Posted by
Jhevon for a scalar function $\displaystyle f(x,y)$ of two variables, we define $\displaystyle \nabla f$ as the following:
$\displaystyle \nabla f = f_x(x,y) ~\bold{i} + f_y(x,y)~ \bold{j} $
similarly, for a function $\displaystyle f(x,y,z)$ of three variables,
$\displaystyle \nabla f = f_x(x,y,z) ~\bold{i} + f_y(x,y,z)~\bold{j} + f_z(x,y,z)~\bold{k}$
we call $\displaystyle \nabla f$ the gradient of $\displaystyle f$ and it is a vector field on $\displaystyle \mathbb{R}^2$ or $\displaystyle \mathbb{R}^3$ for two or three variables respectively. specifically, it is called a gradient vector field, as it assigns each point $\displaystyle (x,y)$ in domain $\displaystyle f(x,y)$ (respectively each point $\displaystyle (x,y,z)$ in the domain of $\displaystyle f(x,y,z)$) with a 2-dimensional vector in $\displaystyle \mathbb{R}^2$ (respectively, a 3-Dimensional vector in $\displaystyle \mathbb{R}^3$)