If I have $\mathbf{x}=(x_1,...,x_n)\in\mathbb{R}^n$ and a multi-index $\mathbf{k}=(k_1,...,k_n)$ (where each entry is a non-negative integer), then $\mathbf{x^k}$ is defined (on Wiki) as $$x_1^{k_1}x_2^{k_1}...x_n^{k_n}.$$ I'm just checking I understand this correctly - so $\mathbf{x^k}$ is actually a one-dimensional element?
If so, is there a notation for the corresponding point in $\mathbb{R}^n$ $$(x_1^{k_1},...,x_n^{k_n})$$?
Also, does it make sense to write $\sin(\mathbf{x})$, where $\mathbf{x}$ is in $\mathbb{R}^n$?
The reals are closed under multiplication, and are closed under exponentiation with non-negative integer exponents (as this is repeated multiplication). So yes, $\mathbf{x}^\mathbf{k}$ is in $\mathbb{R}$.
You can write $\sin(\mathbf{x})$, and I would understand it as element-wise application of the sine function to $\mathbf{x}$.