Could someone spell out the remainder in integral form for a first-order bivariate Taylor series?
Univariate Taylor: $ f(x) = f(a) + f'(a)(x-a) + \int_a^x (x - t)f''(t) dt $
Bivariate Taylor: $ f(x,y) = f(a, b) + f_x(a,b) (x -a) + f_y(a,b) (y -b) + \int_? \cdots ? $
Thanks!
$$f(x,b) = f(a,b) + f_x(a,b)(x-a) + \int_a^x f_{xx}(s,b)(x-s)\; ds$$ $$f(x,y) - f(x,b) = f_y(x,b)(y-b) + \int_b^y f_{yy}(x, t)(y-t)\; dt$$ $$f_y(x,b) = f_y(a,b) + \int_a^x f_{xy}(s,b)\; ds$$ So: $$f(x,y) - f(a,b) = f_x(a,b)(x-a) + f_y(a,b)(y-b) + \int_a^x f_{xx}(s,b)(x-s)\; ds + \int_a^x f_{xy}(s,b)(y-b)\; ds + \int_b^y f_{yy}(x,t)(y-t)\; dt $$