# Multivariable Derivatives/Elliptic PDE

From http://en.wikipedia.org/wiki/Elliptic_operator:
"A nonlinear operator $$L(u) = F(x, u, (\partial^\alpha u)_{|\alpha| \le 2k})$$ is elliptic if its first-order Taylor expansion with respect to u and its derivatives about any point is a linear elliptic operator."

I'm a bit confused by what we mean by taylor expansion w.r.t. u and derivatives... So say we have an (made up) example where ##u:\mathbb{R}^3→\mathbb{R}##, and we have a second order operator given by ##L(u) = F(x, y, z, u, (\partial^\alpha u)_{|\alpha| \le 2}) = z^2uΔu##, what are the derivatives and taylor expansion of interest?

Are there simpler ways of classifying quasilinear/non-linear PDEs as being elliptic?

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