The gradient of a scalar function $f(x,y,z)$ is given by:
$$ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) $$
The gradient shows how the values of the function change. For example, if the scalar funcion is height in two dimensions, $H(x,y)$ then $\nabla H$ is a two-dimensional vector field that points in the direction of greatest rate of increase of $H$, and whose magnitue gives the slope in that direction.
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
