Question

what is the power series form for quadratic terms?(in two variable calculus)

Answer 1

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...