At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
A power series is done pretty much the same as single variable, but you have to take all the variables into account. So in the linear term, you replace df/dx with di f/ di x + di f/di y. (I'm saying "di" in place of the curly d for partial derivatives because the equation editor doesn't seem to support partials.) To get the second derivative term you have to take the partial with respect to both variables of the original partials. So replace d^2 f/dx^2 with (di^2 f/di x^2 + di^2 f/di y^2 + 2 di^2 f/ di x di y). Third derivative is harder, and with more than two variables it gets really bad really quick, especially with trig functions or exponentials. So they often lump the second or third derivative terms together behind a capital script "O", meaning "on the order of", and then give the most significant power of the most significant variable, as in "O x^4", then prove that that is insignificant compared to the terms already listed, and ignore it. I first ran across this in an MIT paper on wave propagation in a taut string. That led directly to my review of calculus on OCW...