Consider approximating solutions to a system of PDE of the form
$$\dfrac{d}{dt}f_i(x,t) = g_i(\dfrac{d}{dx} f_i(x,t), \dfrac{d}{dy} f_i(x,t), \dfrac{d}{dz} f_i(x,t), t), ~~~~~~~1\le i\le 3$$
with $g_i(x,t):\mathbb{R}^3\times [0,\infty)\rightarrow \mathbb{R}^3$ smooth and differentiable for all $(x,t)$.
Can we extend Euler's approach of approximating ODES with its derivative in one dimension to vectors. For example use a scheme such as
$$f_i^n(x,t) = f_i^{n-1}(x,t) + \dfrac{d}{dt}f_i^{n-1}(x,t)* h$$ with h being a step size and $n$ being index of iteration (which we should take considerably smaller than time itself). How can we validate this approximation when solutions exist.