1. ## Checking Linear Operators

Given $\displaystyle u_x(x,y)-u_y(x+1,y)=xy$

$\displaystyle \displaystyle L(u)+L(v)=u_x(x,y)-u_y(x+1,y)+v_x(x,y)-v_y(x+1,y)=(u_x+v_x)(x,y)-(u_y+v_y)(x+1,y)=L(u+v)$

Is that how it should be written, the portion in red?

Or like this:

$\displaystyle \displaystyle L(u)+L(v)=u_x(x,y)-u_y(x+1,y)+v_x(x,y)-v_y(x+1,y)=u_x(x,y)+v_x(x,y)-(u_y(x+1,y)+v_y(x+1,y))=L(u+v)$

Thanks.

The color option isn't working. Which line is how it should be shown, thanks.

2. Everything you've written looks fine.

If you are worried about the expressions like:
$\displaystyle (u_x+v_x)(x,y)$

This is well understood to be addition of functions:
$\displaystyle (f + g)(z) = f(z) + g(z)$

So in this case:
$\displaystyle f = u_x, g = v_x, z = (x,y)$

which is exactly what you want.

However, you are missing the step rewriting:

$\displaystyle u_x+v_x = (u + v)_x$
$\displaystyle u_y+v_y = (u + v)_y$

This will allow the expression to match the linear operator format exactly without missing an intermediate step.