I would like to know what does it mean to have if:

a) $\frac{dV}{dx}$ < 0 and $\frac{dU}{dy}$ > 0;

b) $\frac{dU}{dy}$ < 0 and $\frac{dV}{dx}$ > 0;

c) both $\frac{dV}{dy}$ and $\frac{dU}{dx}$ are less than 0;

d) both $\frac{dV}{dy}$ and $\frac{dV}{dx}$ are greater than 0.

Now proceed to implement a). For example, from your starting arrows, go to the next grid point eastwards, and implement $\frac{dV}{dx}<0$, i.e. the upwards pointing arrows become smaller as you go eastwards. Do the same for U. Repeat for enough grid points until you see a pattern. Now connect $U$ and $V$ components into the final, two-dimensional velocity vector $\vec v$.

In this way you will possibly self-answer your question. I don't know if I can help you more, because I don't know what you mean with \

I would like to know what does it mean to have

The end result will be the graphical representation of a vector field. The shear tensor simply measure the amount of shear in each vector field. You will discover that some field structures surprisingly possess shear, while others don't.