Taylor's Formula in Higher Dimension/Higher order Total differentials
1. The problem statement, all variables and given/known data
First write f(x,y) = x^2 + xy + y^2 in terms of powers of (x+1) and (y-1)
Then write the taylor's formula for f(x,y) a = (1,4) and p=3
2. Relevant equations
We write taylor's formula as:
f(x) = f(a) + sum[(1/k!)*D^(k)f(a;h)] + (1/p!)D^(p)f(c;h) where sum is from k=1 to p-1 and D^(k)f(a;h) is the kth total differential of f at a evaluated for change h=x-a.
3. The attempt at a solution
I'm not sure how to do the first part--I thought perhaps I could divide f(x,y) by x+1 or use the binomial theorem. Is there a calculational method to do this?
For the second part, By just plugging in to Taylor formula I have f(x,y) = 3 + D^(1)f(a;h) +1/2D^(2)f(a;h) + 1/6D^(3)f(c;h)
The problem is I'm not sure how to evaluate the n-order total derivative D^(n)f(a;h). For n=1, I thought Df(a;h) = df(a)/dx*h + df(a)/dy*k = h/2 + k/4.
Is this calculation right? Can someone help me with the evaluating of the total derivative?