The ideas involved in Taylor series for functions of one variable can be generalized. For example, the Taylor series for
about x = a, y = b can be written
![\displaystyle f(x, y) = f(a, b) + f_x(a, b)(x -a) + f_y(a, b)(y -b)\\\vspace{0.2 in} \ \ \ + \frac{1}{2!}\big[f_{xx}(a, b)(x - a)^2 + 2f_{xy}(a, b)(x - a)(y - b) + f_{yy}(a, b)(y - b)^2\big] + \cdots](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+f%28x%2C+y%29+%3D+f%28a%2C+b%29+%2B+f_x%28a%2C+b%29%28x+-a%29+%2B+f_y%28a%2C+b%29%28y+-b%29%5C%5C%5Cvspace%7B0.2+in%7D++%5C+%5C+%5C+%2B+%5Cfrac%7B1%7D%7B2%21%7D%5Cbig%5Bf_%7Bxx%7D%28a%2C+b%29%28x+-+a%29%5E2+%2B+2f_%7Bxy%7D%28a%2C+b%29%28x+-+a%29%28y+-+b%29+%2B+f_%7Byy%7D%28a%2C+b%29%28y+-+b%29%5E2%5Cbig%5D+%2B+%5Ccdots&bg=T&fg=000000&s=0 "\displaystyle f(x, y) = f(a, b) + f_x(a, b)(x -a) + f_y(a, b)(y -b)\\\vspace{0.2 in} \ \ \ + \frac{1}{2!}\big[f_{xx}(a, b)(x - a)^2 + 2f_{xy}(a, b)(x - a)(y - b) + f_{yy}(a, b)(y - b)^2\big] + \cdots")
Taylor series for functions of two or more variables
with Database, Statistics and Nutrition
