What is my lecturer trying to say here?

Question

My lecturer has a bad habit of not explaining his steps in his notes, so even if I want to do further research, I wouldn't know where to start.

I'm reading his revision notes on The Physical Representation of Divergence (Vector Calculus). Here's what they say:

A fluid is flowing in space with velocity $\vec{v}$.

Figure 2.4 shows a small parallelepiped having centre at $P(x,y,z)$. Let the fluid velocity at $P$ be represented by the vector quantity: $ \vec{v} = v_1 \hat{\imath} + v_2 \hat{\jmath} + v_3 \hat{k} $.

Here is where he completely lost me.

Fluid velocity at centre of face $ABCD$ is, in the x-direction:

$$ v_1 \bigg(x + {1\over 2} dx, y, z \bigg) = v_1 + {1\over 2} dx {\partial v_1 \over \partial x } + O\big(dx^2 \big)$$

What does that even mean?

No explanation as to how he got to that. Just straight to it.

Can someone please explain?

Answer 1

This is just linear approximation (or the first-order Taylor polynomial) for $v_1$ (since we're assuming $v_1$ is differentiable). (Make $dx$ instead $\Delta x$.)