Higher Derivatives on Open Subsets of ℝⁿ: Notation

(i) Base. This setting extends Multivariable Calculus: Notation: its conventions and notation are in force.

(ii) First-order derivatives. For a function $f:U\to\mathbb{R}$, a point $a\in U$ and $i\in[n]$, the partial derivative $\partial_{i}f(a)$ and, when it exists at every point of $U$, the partial derivative function $\partial_{i}f:U\to\mathbb{R}$ are as in Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (ii) and Partial Derivative of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (iii); the gradient $\nabla f(a)$ and the gradient function $\nabla f$ are as in Gradient of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (i) and Gradient of a Real-Valued Function on an Open Subset of $\mathbb{R}^n$ (ii). For a function $f:U\to\mathbb{R}^{m}$, the Jacobian matrix $Df(a)$ is as in Jacobian Matrix of a Function Between Euclidean Spaces (i). Differentiability at a point, on a subset of $U$ and on $U$ is as in Differentiability at a Point for Functions Between Euclidean Spaces (iii). For a function differentiable at $a$, $Df(a)$ is its derivative matrix and $\nabla f(a)$ its derivative vector, by The Derivative Is the Jacobian Matrix, and Is Unique (ii) and The Derivative Is the Jacobian Matrix, and Is Unique (i). Background: Differentiable Functions Are Continuous, Rules for Derivatives of Functions on Open Subsets of ℝⁿ, The Derivative of a Vector-Valued Function as a Limit of Difference Quotients in Norm, Chain Rule for Differentiable Functions Between Euclidean Spaces and Continuous Partial Derivatives Imply Differentiability.

(iii) Other domains. For an open subset $V$ of $\mathbb{R}^{m}$, everything above is used likewise for functions on $V$, with $V$ in place of $U$ and $m$ in place of $n$ as the dimension of the domain; such functions take values in $\mathbb{R}$ or in $\mathbb{R}^{p}$, $p\in\mathbb{N}$.