# Linearity for a PDE

1. Feb 11, 2012

### wumple

1. The problem statement, all variables and given/known data
Is the operator

Lu = du/dx + u * du/dy

linear?

2. Relevant equations

Linearity occurs for L[u+cv] = L + cL[v]

3. The attempt at a solution

I know this isn't linear because of the second term, but I don't understand why I can't write the operator as

L = (d/dx + u * d/dy)

which then seems to almost work out, except that I don't know what to make 'u' in the operator when applying the linearity condition since the linearity condition uses two different functions instead of only 'u'.

2. Feb 11, 2012

### xaos

a function or operator f is linear if f(x+y)=f(x)+f(y) and f(cx)=cf(x), for all x+y and cx in the vector space. in this case it is a vector space of functions such as u(x,y) or u(x,y) + c*v(x,y) and the operator is defined as L(u(x,y))=d/dx(u(x,y) + u(x,y)*d/dy(u(x,y), for arbitrary u(x,y) in the vectorspace.