Help with math terminology in tutorial

[hotvette]

I write math related tutorials and would appreciate comments/advice about terminology that may be potentially ambiguous or confusing. As an example, I use $f(x;\beta)$ to define a generic continuous function with independent variable $x$ and function parameters $\beta = (\beta_1, \beta_2, \dots, \beta_n)$ that are also considered variables (to be determined by an iterative optimization process). The partial derivative of $f(x;\beta)$ with respect to the $k$th component of $\beta$, or $\beta_k$ would be:
$$\frac{\partial f(x;\beta)}{\partial \beta_k} = .....$$
However, later on, $\beta_k$ is referred to as the $k$th iterate of the values of the vector $\beta$, for example: $\beta_{k+1} = \beta_k + \Delta \beta$. Thus, $\beta_k$ has a different interpretation based on context. I've seen use of the following to denote iterates, which I don't like (seems too complicated):
$$\beta^{(k+1)} = \beta^{(k)} + \Delta \beta \quad\text{or}\quad \beta_{(k+1)} = \beta_{(k)} + \Delta \beta$$
I've also thought about using a different symbol for the vector, so that:
$$\phi = (\beta_1, \beta_2, \dots, \beta_n)
\qquad
\frac{\partial f(x;\phi)}{\partial \beta_k} = .....
\qquad
\phi_{k+1} = \phi_k + \Delta \phi$$
which seems confusing to me because there appears to be no connection between $f$ and $\beta_k$. Lastly, I've thought about using a capital greek letter to refer to the vector, something like:
$$\Phi = (\phi_1, \phi_2, \dots, \phi_n)
\qquad
\frac{\partial f(x;\Phi)}{\partial \phi_k} = .....
\qquad
\Phi_{k+1} = \Phi_k + \Delta \Phi$$
I think the last choice is probably the most clear, but I don't like the bold symbols sprinkled around in the tutorial. What I really want to do is have two different meanings of $\beta_k$, one as vector component and the other as a vector iterate. Is this bad form?

 

[Hill]

the vector β

How about the vector notation, $\vec \beta$?
Accordingly, $\beta_k$ for components and $\vec \beta_k$ for iterations.

 

[hotvette]

Thanks for the suggestion!

 

[anuttarasammyak]

A proposal for simple and clear notation.

Intorducing n+1 dimension vector
\mathbf{x}=(x_0,x_1,x_2,...,x_n)
components of which correspond to the original variables as
x=x_0, \beta_1=x_1,\beta_2=x_2,...,\beta_n=x_n
the function is expressed as
f(\mathbf{x})=f(x_0,x_1,x_2,...,x_n)

Values of variables are identified by upper suffix
\mathbf{x}^{(i)}=(x_0^{(i)},x_1^{(i)},x_2^{(i)},...,x_n^{(i)})
VAlues of partial derivatives are
\frac{\partial f}{\partial x_k}(x_0^{(i)},x_1^{(i)},x_2^{(i)},...,x_n^{(i)})=\frac{\partial f}{\partial x_k}(\mathbf{x}^{(i)})

 

[hotvette]

Thanks, I'll chew on it.