In my multivariable calculus class, we briefly went over Taylor polynomial approximations for functions of two variables. My professor said that the second degree terms include any of the following: $$x^2, y^2, xy$$ What surprised me was the fact that [itex]xy[/itex] was listed as a nonlinear term. In my Ordinary Differential Equations class, an example of a linear differential equation was $$y'=xy$$ I was stuck wondering how one case could be linear and the other polynomial. I suppose the differences between the examples comes from the fact that the former example dealt with a function of two variables whereas the latter is only in terms of one. If anyone can provide an explanation as to why these two examples don't contradict each other, or on a broader scale, how the term "linear" is defined with more than one variable, I would greatly appreciate it.