Math S 21 a : Multivariable calculus

1 For f(x, y) = x − 6xy + y, we have fx(x, y) = 4x − 12xy, fxx = 12x − 12y, fy(x, y) = −12xy+4y, fyy = −12x +12y and see that fxx+ fyy = 0. A function which satisfies this equation is also called harmonic. The equation fxx + fyy = 0 is an example of a partial differential equation: it is an equation for an unknown function f(x, y) which involves partial derivatives with respect to more than one variables.