Checking Linear Operators

• December 27th 2010, 07:02 PM
dwsmith
Checking Linear Operators
Given $u_x(x,y)-u_y(x+1,y)=xy$

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

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

Or like this:

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

Thanks.

The color option isn't working. Which line is how it should be shown, thanks.
• December 27th 2010, 08:18 PM
snowtea
Everything you've written looks fine.

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

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

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

which is exactly what you want.

However, you are missing the step rewriting:

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

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