Taylor series with partial derivatives

[Claire84]

We were gievn a question in tutorial last week asking us to calculate the Taylor series of the function f(x,y) = e^(x^(2) + y^(2)) to second order in h and k about the point x=0, y=0

I've got the formula here with all the h's and k's in it and have it written down, but it's actually how to work it out that's confusing me.

f(a,b) + 1/1! (hd/dx + kd/dy)f(a'b) etc...

My confusion is do you multiply out the brackets so you'd have-

f(a,b) + 1/1! (hdf(a,b)/dx + kdf(a,b)/dy)

So you do the derivatives and then sub in the values of x and y

Or, do you leave it as it is the first tiem I wrote it and end up with-

1+ (hd/dx + kd/dx) +0.5(hd/dx + kd/dy)^2 etc

I know there are more terms but I've so much trouble typing out mathematical terms on this computer! I know this is v.obvious etc but I just want to get this clear in my head cos I have a test this Wednesday at uni and I want to go in with a fighting chance! Thanks!

 

[matt grime]

Your first statement is correct. I'm not sure I see what you've done in the second.

(hd/dx +kd/dy) is an operator that acts on f, you then evaluate it at (0,0) in this case.the generic term of degree n in the expansion about (0,0) is

$$\frac{1}{n!}\sum_{r=0}^n \binom{n}{r}h^rk^{n-r}\frac{\partial^nf}{\partial x^r \partial y^{n-r}}$$ with the function understood to be evaluated at (0,0)

 

[Claire84]

Thanks very much!