Help with understanding taylor approximation

[MaxManus]

Homework Statement

I'm reading about fluid mechanics and in one of the examples they have approximated the velocity field. The field is two dimensional u = (u,v)
I have never seen this before so cold someone tell me what it is called so I can look it up?
The notes I am reading are hand written so there may be small mistakes, but it seems likes some of the ts are primed.

u = u(x,y,t)

The Attempt at a Solution

u(0,0,t) = u(o,o,t) + $$\int_0^t$$u(0,0,t')dt'[$$\nabla$$u]_{x = 0}

 

[lippyka]

It looks like this is a two-dimensional velocity field. The notation u(x,y,t) indicates that the velocity is a function of three variables: x, y, and time t. The notation \nablau]x indicates the gradient of the velocity field in the x direction.