I have an exam later and I need to do Taylor expansions of functions. I have questions like:
Consider the map $F:\mathbb{R}^2_x \rightarrow \mathbb{R}^2_y$, given by the equations
$$y_1 = x_2 + x_1^2,$$ $$y_2 = x_1 - x_2^3.$$
Find the Taylor series of the local inverse map up to the terms of degree $4$.
and
Let $f(x,y,z) = x + y + z + x^2 + y^2 + z^2$. Consider the surface
$$S = \{f(x,y,z) = 0 \} \subset \mathbb{R}^3$$
near the origin. Compute the Taylor series of the function
$$g = (x + y + 3yz)|_S$$
at the origin in the coordinates $(y,z)$ up to degree $2$.
Now, I know how to do these questions, the only mistakes I make are algebraic ones, so I expand something wrong or mess up on my signs or whatever and I was just wandering if there was a way to check if my expansions are correct?