Gradient

The gradient is a vector valued function \( \nabla f\colon \mathbb{R}^n \to \mathbb{R}^n \) defined as the unique vector field whose dot product with any unit vector \( \vec{v} \) at each \( x, y \) is the directional derivative of \( f \) along \( \vec{v} \). In \( \mathbb{R}^2 \) it can be shown the gradient of a scalar function of two variables is given by:

\[ \nabla f(x, y) = \langle \frac{\partial f}{\partial x}, \ \frac{\partial f}{\partial y} \rangle \]

With the obvious extension to multiple dimensions.

As you can see above, The nabla symbol: \( \nabla \) is used in the gradient function notation. The nabla or del operator is a vector whose components are operators

\[ \nabla = \langle \partial_x, \partial_y \rangle \]

It is not a vector in the typical sense in \( R^2 \) but is a very convenient abuse of notation. For instance the gradient can be denoted thusly:

\[ \nabla f = \langle \partial_x f, \partial_y f \rangle \]

Similar succinct expressions can be found for the divergence, curl and Laplace Operator of a vector field \( F \).

See: https://math.stackexchange.com/questions/2710328/what-does-the-symbol-nabla-indicate

The directional derivative at a point \( P = (x, y, f(x, y)) \) in the direction of the unit vector \( \vec{u} = ai + bj \) is defined as:

\[ \frac{\partial f}{\partial x} = {f_x}\left( {x,y} \right) = \lim_{h \to 0} \frac{f(x + ah, y +bh) - f(x, y)}{h}\]

Intuitively it is the slope of the surface at \( P \) in the direction \( \vec{u} \).