Taylor's Expansion: Breaking Down a Monster Equation

[Somefantastik]

I thought I was familiar with Taylor's Expansion, and then this monster popped up:

$$v(y) = v(x) + \sum_{j=1}^{2} \frac{\partial v}{\partial dx_{j}}(x)(y_{j}-x_{j}) + R(x,y)$$

where $$R(x,y) = \frac{1}{2} \sum_{i,j=1}^{2}\frac{\partial^2 v}{\partial x_{i} y_{j}}(\xi)(y_{i}-x_{i})(y_{j}-x_{j})$$

Can someone break this down for me?

 

[mathman]

One way to get it is to hold x₂ fixed and get a the first couple of terms for the series in x₁. Then for each term of this expansion, get the Taylor series around x₂. The expression for R(x,y) is the 2-d analog of the remainder term for Taylor series.

 

[Somefantastik]

so when it's expanded, it's only expanded in one direction at a time...?

 

[mathman]

so when it's expanded, it's only expanded in one direction at a time...?

Not necessarily. What I suggested was a computational method.