The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral

Throughout, $m \in \mathbb{N}$ with $m \ge 1$, and vector-valued functions, their components and their limits are as in Vector-valued functions $f : A \to \mathbb{R}^m$, their limits and continuity, with the dictionary to the metric notions.

The derivative

Let $A \subseteq \mathbb{R}$, let $f : A \to \mathbb{R}^{m}$ and let $c \in A$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$). The difference quotient of $f$ at $c$ is the vector-valued function

$q_{f,c} : A \setminus \{c\} \to \mathbb{R}^{m}, \qquad q_{f,c}(x) \;:=\; \frac{1}{x - c}\,\bigl(f(x) - f(c)\bigr),$

the scalar multiple being that of the vector space $\mathbb{R}^{m}$ (The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = \{0, 1, \dots, n-1\}$); the division is legitimate because $x \ne c$ gives $x - c \ne 0$. As in The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set, $c$ is a limit point of $A \setminus \{c\}$ as well, since a punctured neighbourhood of $c$ omits $c$.

$f$ is differentiable at $c$ when $\lim_{x \to c} q_{f,c}(x)$ exists in $\mathbb{R}^{m}$, and then the derivative is

$f'(c) \;:=\; \lim_{x\to c} q_{f,c}(x) \;\in\; \mathbb{R}^{m}.$

The notation denotes a single vector. At most one $L \in \mathbb{R}^{m}$ satisfies the limit condition, as proved in Vector-valued functions $f : A \to \mathbb{R}^m$, their limits and continuity, with the dictionary to the metric notions; this is the vector-valued form of the obligation At a limit point of the domain a function has at most one limit discharges for real-valued functions and A sequence in a metric space has at most one limit for sequences.

The intrinsic form is the definition; the componentwise form is a theorem. For $i < m$ the $i$-th component of $q_{f,c}(x)$ is $\bigl(f_i(x)-f_i(c)\bigr)/(x-c)$, which is the real difference quotient of $f_i$ at $c$ (The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set). So by A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions clause 2:

$f$ is differentiable at $c$ if and only if every $f_i$ is differentiable at $c$, and then $f'(c)_i = f_i'(c)$ for every $i<m$.

Nothing below reverses this order of presentation: the intrinsic limit is what is defined, and the coordinates are read off it.

Algebra of derivatives. If $f, g : A \to \mathbb{R}^{m}$ are differentiable at $c$ and $\lambda \in \mathbb{R}$, then $f + g$ and $\lambda f$ are differentiable at $c$ with $(f+g)'(c) = f'(c)+g'(c)$ and $(\lambda f)'(c) = \lambda f'(c)$: read componentwise through the displayed equivalence, these are clauses 1 and 2 of the published Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$, $(\alpha f)'(c) = \alpha f'(c)$, $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$, and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$.

The integral

Let $a, b \in \mathbb{R}$ with $a < b$ and let $f : [a,b] \to \mathbb{R}^{m}$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). $f$ is integrable on $[a,b]$ when every component $f_i : [a,b] \to \mathbb{R}$ is bounded (Lower bound, bounded below, bounded set) and Darboux integrable in the sense of The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$, and then

$\int_a^b f \;:=\; \text{the function } m \to \mathbb{R} \text{ sending } i \mapsto \int_a^b f_i .$

That really is an element of $\mathbb{R}^{m}$. In this library $\mathbb{R}^{m}$ is the set of functions $m \to \mathbb{R}$ (The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = \{0, 1, \dots, n-1\}$), not a set of tuples, so the displayed assignment is literally an element of it; each value $\int_a^b f_i$ is a single real by The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$, Darboux integrability as their equality, and the notation $\int_a^b f$. In the standard basis (The standard list $e : n \to F^{n}$ with $e_i(i) = 1_F$ and $e_i(j) = 0_F$ for $j \ne i$ is an ordered basis of $F^{n}$; hence $\dim_F F^{n} = n$, and $F^{0}$ is the zero space with basis $\varnothing$ and dimension $0$) the same object is $\int_a^b f = \sum_{i<m}\bigl(\int_a^b f_i\bigr)e_i$.

Oriented limits. Following The integral with oriented limits: $\int_a^a f := 0$ and $\int_b^a f := -\int_a^b f$ componentwise, set

$\int_a^a f \;:=\; 0 \in \mathbb{R}^{m}, \qquad \int_b^a f \;:=\; -\int_a^b f \quad (a < b),$

so that $\int_u^v f = -\int_v^u f$ for all $u,v$ in an interval on which $f$ is integrable. The clauses do not overlap with the case $a<b$, so nothing has to be checked for consistency, exactly as in The integral with oriented limits: $\int_a^a f := 0$ and $\int_b^a f := -\int_a^b f$.

Linearity. If $f, g : [a,b] \to \mathbb{R}^{m}$ are integrable and $\lambda,\mu \in \mathbb{R}$ then $\lambda f + \mu g$ is integrable with

$\int_a^b (\lambda f + \mu g) \;=\; \lambda\int_a^b f + \mu\int_a^b g ,$

since each side has $i$-th coordinate $\int_a^b(\lambda f_i + \mu g_i)$ and $\lambda\int_a^b f_i + \mu\int_a^b g_i$ respectively, and those agree by Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and $\int_a^b(\lambda f+\mu g) = \lambda\int_a^b f + \mu\int_a^b g$.

Restriction and splitting. If $f$ is integrable on $[a,b]$ then it is integrable on every nondegenerate closed subinterval $[c,d]$ with $a\le c<d\le b$, and for $a<c<b$, $\int_a^b f = \int_a^c f + \int_c^b f$; both are the componentwise readings of A function integrable on $[a,b]$ is integrable on every closed subinterval and For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then $\int_a^b f = \int_a^c f + \int_c^b f$; with the oriented form for arbitrary $a,b,c$, applied to each $f_i$ and reassembled coordinate by coordinate.

• The two halves are independent. The derivative clause needs no integral and the integral clause needs no derivative; they are collected in one item because they are the two constructions of the one-dimensional theory that transfer to $\mathbb{R}^{m}$ by the same move, and because If $f : [a,b] \to \mathbb{R}^m$ is differentiable with integrable $f'$ then $\int_a^b f' = f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz is what joins them.

• Why the intrinsic derivative is stated first. The componentwise formula is the one used in computations, but it is tied to the standard basis, whereas the limit of the difference quotient is not. The intrinsic form is the one that survives when the domain is enlarged from an interval to a subset of $\mathbb{R}^{n}$, which is a later page of this track.

• No norm appears in either definition. The derivative is a limit in $(\mathbb{R}^{m}, d_2)$, and by For $n \ge 1$ all norms on $\mathbb{R}^n$ are equivalent the same limits are obtained from any other norm, so the notion does not depend on the choice (The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$, The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$). The integral is defined coordinatewise and mentions no metric at all.

• Integrability of $\lVert f\rVert_2$ is a separate matter and is not part of this definition: it is proved, together with the inequality it belongs to, in For $a \le b$ and $f : [a,b] \to \mathbb{R}^m$ integrable when $a<b$, $\bigl\lVert\int_a^b f\bigr\rVert_2 \le \int_a^b \lVert f\rVert_2$; for $a<b$, $\lVert f\rVert_2$ is integrable.