I have a function that looks like this $$f(x,y,z_1,z_2,z_3,\dots ,z_k) \tag{1}$$ and I figured I could write it compactly as $$f'(x,y,Z) \tag{2}$$ where $Z = [z_1,z_2,z_3,\dots,z_k]$, and $f'$ denotes that the function may need to be changed to accommodate taking a vector as an input instead of a series of variables (I'm not sure it needs to be changed, but I put the $'$ incase a change is necessary).
Are these equivalent? At first I thought they were but now I'm thinking that they may not be. That is, can i rewrite $(1)$ as $(2)$?
If the re-writing is indeed okay, how would I denote the partial derivative of $(2)$ w.r.t $z_i, 1\leq i \leq k$? Because if I use $(1)$ is is just $f_{i+2}(\cdot)$
I'm thinking now that it may be best to leave my function in the form of $(1)$, regardless of whether a transformation is possible...
Thanks.
A function $f:A^n\to\mathbb{R}$, where $A^n\subset \mathbb{R}^n$ can be expressed in any convenient way, either $f(x_1,...,x_n)$ or $f(\vec{x})$ where $\vec{x}=(x_1,...,x_n)$, that is $$ f(\vec{x})\equiv f(x_1,...,x_n)$$ both notation are equivalent and commonly used. The gradient (a vector of partial derivatives) of both is $$\vec{\nabla} f =\left(\frac{\partial f}{\partial x_1},...,\frac{\partial f}{\partial x_n}\right)$$