Understanding Taylor Expansions of Gradients in Vector Calculus

[brydustin]

What does it mean to have a taylor expansion of a gradient (vector) about the position x?
I.e. taylor expansion of g(x + d) where g is the gradient and d is the small neighborhood.

 

[Some Pig]

Gradient is about a scalar function of a point, say $$\phi(x,y,z)$$
$$\triangledown\phi=\frac{\partial\phi}{\partial x}\vec i+\frac{\partial\phi}{\partial y}\vec j+\frac{\partial\phi}{\partial z}\vec k$$
So you should find the taylor expansions of the functions
$$\frac{\partial\phi}{\partial x},\ \frac{\partial\phi}{\partial y},\ \frac{\partial\phi}{\partial z}$$