From mono-dimensional theory of function series, we know that the following term by term differentiation theorem holds:
Theorem 1. Suppose that $u_k : [a, b]\to \mathbb{R}$, for each $k = 1, 2,\dots$, has continuous derivative on $[a,b]$. Suppose further that
(i) the numerical series $\sum_{1}^{+\infty}u_k(x_0)$ converges at some point $x_0 \in [a, b]$;
(ii) the series of derivatives $\sum_{1}^{+\infty}{u_{k}^{'}}(x)$ converges uniformly on $[a,b]$, to $u(x) =\sum_{1}^{+\infty}{u_{k}^{'}}(x)$ say.
Then
(1) the series $\sum_{1}^{+\infty}u_{k}(x)$ converges at every $x \in [a,b]$ and the sum $U(x) = \sum_{1}^{+\infty}u_{k}(x)$ is differentiable with $U′(x) = u(x)$ for each $x \in [a,b]$;
(2) moreover, the convergence of $\sum_{1}^{+\infty}u_{k}(x)$ to $U(x)$ is uniform on $[a,b]$.
I would like to know if it is possible to generalize the previous theorem to the case of multivariable functions, with respect to partial differentiation.
For instance, let's suppose we have a sequence of scalar functions $\{f_k=f_k(x,t)\}$ defined of some domain $W\subset\mathbb{R}^2$. Let's define the generic partial differentiation operator $D:=\frac{\partial^{p+q}}{\partial x^p \partial t^q}$ where $p,q$ are nonnegative integers such that $p+q\geq1$. Further, suppose that $Df_k=\frac{\partial^{p+q}}{\partial x^p \partial t^q} f_k$ exist and is continuous on $W$ for every $k$. Moreover suppose that
(i) the numerical series $\sum_{1}^{+\infty}f_k(x_0,t_0)$ converges at some point $(x_0,t_0) \in W$;
(ii) the series of partial derivatives $\sum_{1}^{+\infty}{Df_k(x,t)}$ converges uniformly on $W$, to $f(x,t) =\sum_{1}^{+\infty}Df_k (x,t)$ say;
Can we conclude that the following sentences
(1) the series $\sum_{1}^{+\infty}f_{k}(x,t)$ converges at every point of $W$ and the sum $F(x,t) = \sum_{1}^{+\infty}f_{k}(x,t)$ is such that $DF$ exist on $W$ with $DF(x,t) = f(x,t)$ for each $(x,t) \in W$;
(2) moreover, the convergence of $\sum_{1}^{+\infty}f_{k}(x,t)$ to $F(x,t)$ is uniform on $W$,
are true?
Any hint would be really appreciated. Thanks a lot in advance.