PX153 - G4 - taylor expansion of a function of two variables

consider a power series approximation to a function of two variables, $f (x, y)$ , about $x = a, y = b$

\begin{array}{r} f (x, y) ≊ f (a, b) + A_{1} (x - a) + B_{1} (y - b) + A_{2} (x - a)^{2} \\ + B_{2} (y - b)^{2} + C_{2} (x - a) (y - b) . . . \end{array}

to find $A, B, C, . . .$ , consecutive partial derivatives of the left and the right-hand sides need to be taken:

\frac{\partial f}{\partial x} = A_{1} + 2 A_{2} (x - a) + C_{2} (y - b) + . . .

evaluate at $x = a$ and $y = b$ :

⟹ A_{1} = \frac{\partial f}{\partial x} |_{_{y = b}^{x = a}}

similarly,

\frac{\partial f}{\partial y} = B_{1} + 2 B_{2} (y - b) + C_{2} (x - a) + . . .

⟹ B_{1} = \frac{\partial f}{\partial y} |_{_{y = b}^{x = a}}

$f_{x x} ⟹ A_{2} = \frac{1}{2!} \frac{\partial^{2} f}{\partial x^{2}} |_{_{y = b}^{x = a}}$
$f_{y y} ⟹ B_{2} = \frac{1}{2!} \frac{\partial^{2} f}{\partial y^{2}} |_{_{y = b}^{x = a}}$
$f_{x y} = f_{y x} ⟹ C_{2} = \frac{\partial^{2} f}{\partial x \partial y} |_{_{y = b}^{x = a}}$
this gives the expansion to second order, writing $(x - a) = Δ x$ , and $(y - b) = Δ y$ :

f (x, y) ≊ f (a, b) + f_{x} Δ x + f_{y} Δ y + \frac{1}{2!} f_{x x} Δ x^{2} + \frac{1}{2!} f_{y y} Δ y^{2} + f_{x y} Δ x Δ y + . . .

following a similar process, you can derive the full expression:

f (x, y) = \sum_{n = 0}^{\infty} \frac{1}{n!} [{(Δ x \frac{\partial}{\partial x} + Δ y \frac{\partial}{\partial y})}^{n} f (x, y)]

eg: expand the function $f (x, y) = e^{x y}$ tp 2nd order about $x = 1, y = 2$
$f (1, 2) = e^{2}$
$f_{x} (1, 2) = y e^{x y} |_{1, 2} = 2 e^{2}$
$f_{x x} (1, 2) = y^{2} e^{x y} |_{1, 2} = 4 e^{2}$
$f_{y} (1, 2) = x e^{x y} |_{1, 2} = e^{2}$
$f_{y y} (1, 2) = x^{2} e^{x y} |_{1, 2} = e^{2}$
$f_{x y} (1, 2) = (x y e^{x y} + e^{x y}) |_{1, 2} = 3 e^{2}$
- the taylor series is:

\begin{array}{r} f (x, y) ≊ e^{2} (1 + 2 (x - 1) + (y - 2) + \frac{1}{2!} 4 (x - 1)^{2} \\ + \frac{1}{2!} (y - 2)^{2} + 3 (x - 1) (y - 2) + . . .) \end{array}