Meaning of the following, partial derivatives..

Question

What is the meaning of $${\partial^kG \over \partial t^k} \in C$$ how is this function explained $G(t,s)$, does it mean that the k-th derivative of $G$ is continuous. I've done some studying on this subject a while back, just this notation is confusing me..$\partial t^k$ and the meaning of $$\left.{\partial^{n-1}G \over \partial t^{n-1}}\right|_{t=s-0}^{t=s+0}=1?$$ aswell...

Answer 1

The definition of the partial derivative is the following limit, if it exists:

$$\frac{\partial G}{\partial t}(t, s) = \lim_{h \to 0} \frac{G(t+h, s) - G(t, s)}{h}$$

What we're doing here is holding the variable $s$ constant, so that the resulting function is a function of $t$ only, and then taking its derivative. This defines a new function which depends on $t$ and $s$, so we can take derivatives of this new function as well (if they exist). The notation $$\frac{\partial^k G}{\partial t^k}$$ means the $k^{\text{th}}$ partial derivative of $G$ with respect to $t$. If the notation confuses you, it make make more sense if you remember that $\frac{\partial}{\partial t}$ is the "take the partial derivative with respect to $t$ operator," so if we want to take $k$ such partial derivatives of $G$, we get $$\frac{\partial}{\partial t} \ldots \frac{\partial}{\partial t} G(t, s) = \left( \frac{\partial}{\partial t}\right)^k G(t, s) = \frac{\partial^k G}{\partial t^k}(t, s)$$

If there is some domain $D \subset \mathbb{R}^2$ on which $\frac{\partial^k G}{\partial t^k}(t, s)$ is defined, we write $\frac{\partial^k G}{\partial t^k} \in C(D) = C^0(D)$ if $\frac{\partial^k G}{\partial t^k}$ is continuous on $D$.

I'm not sure about the last bit of notation. Perhaps they are comparing the following limits:

$$\lim_{t \to 0^+} \frac{\partial^{n-1} G}{\partial t^{n-1}}(t, t) \text{ versus } \lim_{t \to 0^-} \frac{\partial^{n-1} G}{\partial t^{n-1}}(t, t)$$

i.e. one-sided limits of $\frac{\partial^{n-1} G}{\partial t^{n-1}}(t, s)$ along the diagonal line $t=s$.

Answer 2

It means that the $(n-1)$th partial derivative is continuous about the point $s$.

${\partial^kG \over \partial t^k} \in C$ generally means that it is an element of a class of continuous functions.