5 Definite Integral

5.1 The Concept of the Definite Integrals

Darboux Integral

Consider a bounded function $f$ on a closed interval $[a,b]$, and a partition of $[a,b]$:

$$\begin{equation*} P: \quad a = x_0 < x_1 < \cdots < x_N = b. \end{equation*}$$

For each subinterval $[x_{k-1}, x_k]$, define

$$\begin{equation*} M_k = \sup_{x \in [x_{k-1},x_k]} f(x), \qquad m_k = \inf_{x \in [x_{k-1},x_k]} f(x). \end{equation*}$$

The upper Darboux sum of $f$ with respect to the partition $P$ is

$$\begin{equation*} \overline{S}(f,P) = \sum_{k=1}^{N} M_k (x_k - x_{k-1}), \end{equation*}$$

and the lower Darboux sum is

$$\begin{equation*} \underline{S}(f,P) = \sum_{k=1}^{N} m_k (x_k - x_{k-1}). \end{equation*}$$

Define

$$\begin{equation*} \underline{S}(f) = \sup_P\, \underline{S}(f,P), \qquad \overline{S}(f) = \inf_P\, \overline{S}(f,P), \end{equation*}$$

where the supremum and infimum are taken over all partitions $P$ of $[a,b]$.

• $\underline{S}(f)$ is called the Darboux lower integral of $f$ on $[a,b]$;
• $\overline{S}(f)$ is called the Darboux upper integral of $f$ on $[a,b]$.

If

$$\begin{equation*} \underline{S}(f) = \overline{S}(f), \end{equation*}$$

then $f$ is said to be Darboux integrable on $[a,b]$. The common value is denoted by

$$\begin{equation*} \int_a^b f(x)\,dx, \end{equation*}$$

and is called the Darboux integral of $f$ on $[a,b]$.

Theorem 5.1. For any two partitions $P, Q$ of the interval $[a,b]$, let $P \lor Q$ denote the partition consisting of all points in $P \cup Q$. We call $P \lor Q$ the common refinement of $P$ and $Q$. Then

$$\begin{equation*} \underline{S}(f, Q) \;\le\; \underline{S}(f, P \lor Q) \;\le\; \overline{S}(f, P \lor Q) \;\le\; \overline{S}(f, P). \end{equation*}$$

Theorem 5.2. A bounded function $f$ on the closed interval $[a,b]$ is Darboux integrable if and only if for every $\varepsilon > 0$, there exists a partition $P$ of $[a,b]$ such that

$$\begin{equation*} \overline{S}(f,P) - \underline{S}(f,P) < \varepsilon. \end{equation*}$$

Theorem 5.3. A bounded function $f$ on the closed interval $[a,b]$ is Darboux integrable, and $I \in \mathbb{R}$ is its Darboux integral, if and only if for every $\varepsilon > 0$, there exists a partition $P$ of $[a,b]$ such that

$$\begin{equation*} I - \varepsilon < \underline{S}(f,P) \le I \le \overline{S}(f,P) < I + \varepsilon. \end{equation*}$$

Theorem 5.4. Every monotonic function on a bounded closed interval $[a,b]$ is Darboux integrable. More precisely, for every $\varepsilon > 0$, there exists $\delta > 0$ such that for any partition

$$\begin{equation*} a = x_0 < x_1 < \cdots < x_N = b, \end{equation*}$$

if

$$\begin{equation*} \|P\| := \max_{1 \le k \le N} (x_k - x_{k-1}) < \delta, \end{equation*}$$

then

$$\begin{equation*} \overline{S}(f,P) - \underline{S}(f,P) < \varepsilon . \end{equation*}$$

Theorem 5.5. If $f$ is a bounded function on the closed interval $[a,b]$, and $f$ is continuous on the open interval $(a,b)$, then $f$ is Darboux integrable on $[a,b]$. More precisely, for every $\varepsilon > 0$, there exists $\delta > 0$ such that for any partition $P: a = x_0 < x_1 < \cdots < x_N = b$,

if

$$\begin{equation*} \|P\| := \max_{1 \le k \le N} (x_k - x_{k-1}) < \delta, \end{equation*}$$

then

$$\begin{equation*} \overline{S}(f, P) - \underline{S}(f, P) < \varepsilon. \end{equation*}$$

Corollary 1. If a bounded function $f$ on the closed interval $[a,b]$ has only finitely many discontinuities, then $f$ is Darboux integrable on $[a,b]$.

Riemann Integral

For a function $f:[a,b]\to \mathbb{R}$, a partition of the interval $[a,b]$ is

$$\begin{equation*} P:\ a = x_0 < x_1 < \cdots < x_N = b. \end{equation*}$$

Along with a set of sample points

$$\begin{equation*} \xi = \{\xi_i\}_{i=1}^{N}, \qquad (\xi_i \in [x_{i-1}, x_i]), \end{equation*}$$

the Riemann sum is defined as

$$\begin{equation*} S(f, P, \xi) = \sum_{i=1}^{N} f(\xi_i)(x_i - x_{i-1}). \end{equation*}$$

A function $f : [a,b] \to \mathbb{R}$ is said to be Riemann integrable on $[a,b]$ if there exists a real number $I \in \mathbb{R}$ such that for every $\varepsilon > 0$, there exists $\delta > 0$ with the property that for any tagged partition $(P,\xi)$ of $[a,b]$, if the norm of the partition satisfies $\|P\| < \delta$, then

$$\begin{equation*} \left| S(f, P, \xi) - I \right| < \varepsilon. \end{equation*}$$

In this case, we call $I$ the Riemann integral of $f$ on $[a,b]$, and write

$$\begin{equation*} I = \int_a^b f(x)\, dx. \end{equation*}$$

Theorem 5.6. For a function $f:[a,b]\to\mathbb{R}$, the following statements are equivalent:

$\text{(1)}\quad f$ is Riemann integrable on $[a,b]$;

$\text{(2)}\quad f$ is bounded on $[a,b]$, and Darboux integrable;

$\text{(3)}\quad f$ is bounded on $[a,b]$, and the set of all discontinuities of $f$ in $[a,b]$ forms a set of measure zero.

5.2 Properties and Computation of Definite Integrals

Theorem 5.7 (Newton-Leibniz Formula). If $f \in \mathcal{R}[a,b]$ and $F$ is an antiderivative of $f$, then

$$\begin{equation*} \int_a^b f(u)\,du = F(b) - F(a). \end{equation*}$$

Theorem 5.8. If $f$ is Riemann integrable on $[a,b]$ and on $[b,c]$, then $f$ is integrable on $[a,c]$, and

$$\begin{equation*} \int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx. \end{equation*}$$

Theorem 5.9. Assume $f \in \mathcal{C}[a,b]$, $g \in \mathcal{R}[a,b]$, and $g(x) \ge 0$ for all $x \in [a,b]$. Then there exists $\xi \in [a,b]$ such that

$$\begin{equation*} \int_a^b f(x)g(x)\,dx = f(\xi)\int_a^b g(x)\,dx. \end{equation*}$$

Theorem 5.10. Assume that $f \in \mathcal{R}[a,b]$. Define

$$\begin{equation*} F(x) = \int_a^x f(u)\,du. \end{equation*}$$

Then $F \in \mathcal{C}[a,b]$. Then

$\text{(1)}\quad$Then the integral with variable upper limit $F(x) = \int_a^x f(t) \mathrm{d}t$ is continuous on $[a, b]$.

$\text{(2)}\quad$Let function $f$ be continuous at a point $x_0 \in [a, b]$. Then $F$ is differentiable at $x_0$, and $F'(x_0) = f(x_0).$

Theorem 5.11. Let function $f$ be continuous on $[a, b]$. Then

$$\begin{equation*} \frac{\mathrm{d}}{\mathrm{d}x} \int_a^x f(t) \mathrm{d}t = f(x) \quad (a \leqslant x \leqslant b). \end{equation*}$$

Theorem 5.12. Let $f, g \in \mathcal{R}[a,b]$. Then

$$\begin{equation*} \left| \int_a^b f(x)g(x)\,dx \right| \,\le\, \sqrt{\int_a^b |f(x)|^2\,dx}\; \sqrt{\int_a^b |g(x)|^2\,dx}. \end{equation*}$$

Theorem 5.13. If $f^{(n+1)} \in \mathcal{R}[a,b]$, then for every $x \in [a,b]$,

$$\begin{equation*} f(x) = f(a) + f'(a)(x-a) + \cdots + \frac{f^{(n)}(a)}{n!} (x-a)^n + \int_{a}^{x} \frac{f^{(n+1)}(t)}{n!} (x - t)^n \, dt. \end{equation*}$$

Theorem 5.14. Suppose that $f$ is continuous on an interval $I$, $a, b \in I$, and $\varphi : [\alpha, \beta] \to I$ satisfies $\varphi(\alpha) = a$, $\varphi(\beta) = b$, and $\varphi' \in \mathcal{R}[\alpha, \beta]$. Then

$$\begin{equation*} \int_a^b f(x)\,dx = \int_\alpha^\beta f(\varphi(t))\, \varphi'(t)\, dt. \end{equation*}$$

5.3 Improper Integrals

Improper Integral

Let $f:[a,+\infty)\to \mathbb{R}$ be Riemann integrable on every bounded closed interval $[a,A]$.

If

$$\begin{equation*} \lim_{A\to+\infty} \int_a^A f(x)\,dx \end{equation*}$$

exists, then the improper integral $\displaystyle \int_a^{+\infty} f(x)\,dx$ is said to converge, and we define

$$\begin{equation*} \int_a^{+\infty} f(x)\,dx = \lim_{A\to+\infty} \int_a^A f(x)\,dx. \end{equation*}$$

If the limit does not exist, the improper integral is said to diverge.

Similarly, we define

$$\begin{equation*} \int_{-\infty}^a g(x)\,dx = \lim_{A\to-\infty} \int_A^a g(x)\,dx. \end{equation*}$$

Let $f:\mathbb{R}\to\mathbb{R}$ be Riemann integrable on every bounded closed interval $[a,b]$.

If both improper integrals

$$\begin{equation*} \int_a^{+\infty} f(x)\,dx, \qquad \int_{-\infty}^{a} f(x)\,dx \end{equation*}$$

converge, then the improper integral

$$\begin{equation*} \int_{-\infty}^{+\infty} f(x)\,dx \end{equation*}$$

is said to converge, and we define

$$\begin{equation*} \int_{-\infty}^{+\infty} f(x)\,dx = \int_{-\infty}^{a} f(x)\,dx + \int_{a}^{+\infty} f(x)\,dx. \end{equation*}$$

Let $f:[a,b)\to\mathbb{R}$ be unbounded, but Riemann integrable on every bounded closed interval $[a,b-\delta]$ for $\delta>0$. In this case, the integral $\displaystyle \int_a^b f(x)\,dx$ is called an improper integral, and $b$ is called an improper point.

If

$$\begin{equation*} \lim_{\delta\to 0^+} \int_a^{\,b-\delta} f(x)\,dx \end{equation*}$$

exists, then the improper integral $\displaystyle \int_a^b f(x)\,dx$ is said to converge, and we define

$$\begin{equation*} \int_a^b f(x)\,dx = \lim_{\delta\to 0^+} \int_a^{\,b-\delta} f(x)\,dx. \end{equation*}$$

Similarly, one may define improper integrals where the left endpoint is an improper point.

Examples

1. $$\begin{equation*} \int_{1}^{+\infty} \frac{1}{x^p} \mathrm{d}x \end{equation*}$$

   Solution:

   $$\begin{equation*} \int_{1}^{A} \frac{1}{x^p} \mathrm{d}x = \begin{cases} \frac{1 - A^{1-p}}{p-1}, & p \neq 1, \\ \ln A, & p = 1. \end{cases} \end{equation*}$$

   Therefore, the integral converges if and only if $p > 1$. When it converges, the limit is:

   $$\begin{equation*} \int_{1}^{+\infty} \frac{1}{x^p} \mathrm{d}x = \frac{1}{p-1} \end{equation*}$$
2. $$\begin{equation*} \int_{0}^{1} \frac{1}{x^p} \mathrm{d}x \end{equation*}$$

   Solution:

   $$\begin{equation*} \int_{\delta}^{1} \frac{1}{x^p} \mathrm{d}x = \begin{cases} \frac{1 - \delta^{1-p}}{1-p}, & p \neq 1, \\ -\ln \delta, & p = 1. \end{cases} \end{equation*}$$

   Therefore, the integral converges if and only if $p < 1$. When it converges, the limit is:

   $$\begin{equation*} \int_{0}^{1} \frac{1}{x^p} \mathrm{d}x = \frac{1}{1-p} \end{equation*}$$

Theorems 5.15. Suppose $\int_{a}^{\omega} f'(x)g(x) \mathrm{d}x$ converges and $\lim\limits_{x \to \omega^-} f(x)g(x)$ converges. Then $\int_{a}^{\omega} f(x)g'(x) \mathrm{d}x$ converges, and:

$$\begin{equation*} \int_{a}^{\omega} f(x)g'(x) \mathrm{d}x = f(x)g(x) \bigg|_{a}^{\omega} - \int_{a}^{\omega} f'(x)g(x) \mathrm{d}x. \end{equation*}$$

Theorem 5.16. The improper integral $\int_{a}^{+\infty} f(x) \mathrm{d}x$ converges if and only if for any $\varepsilon > 0$, there exists $N_{\varepsilon} > 0$ such that for any $A_2 > A_1 > N_{\varepsilon}$, the following holds:

$$\begin{equation*} \left| \int_{A_1}^{A_2} f(x) \mathrm{d}x \right| < \varepsilon. \end{equation*}$$

Absolutely Convergent

An improper integral $\int_{a}^{+\infty} f(x) \mathrm{d}x$ is said to be absolutely convergent, or $f$ is absolutely integrable on the interval $[a, +\infty)$, if the improper integral $\int_{a}^{+\infty} |f(x)| \mathrm{d}x$ converges.

From the inequality:

$$\begin{equation*} \left| \int_{A_1}^{A_2} f(x) \mathrm{d}x \right| \leq \int_{A_1}^{A_2} |f(x)| \mathrm{d}x \end{equation*}$$

It is known that any absolutely convergent improper integral is also convergent.

Definition. An improper integral is said to be conditionally convergent if it converges but is not absolutely convergent.

Theorem 5.17. Let $f$ be Riemann integrable on any bounded closed interval $[a, A]$, and let $g$ be monotonic. If:

• $\int_{a}^{A} f(x) \mathrm{d}x$ is bounded for all $A > a$,
• $\lim\limits_{x \to +\infty} g(x) = 0$,

then the improper integral $\int_{a}^{+\infty} f(x)g(x) \mathrm{d}x$ converges.

Theorem 5.18. Let $f$ be Riemann integrable on any bounded closed interval $[a, A]$, and let $g$ be monotonic. If:

• The improper integral $\int_{a}^{+\infty} f(x) \mathrm{d}x$ converges,
• $g(x)$ is bounded on the interval $[a, +\infty)$,

then the improper integral $\int_{a}^{+\infty} f(x)g(x) \mathrm{d}x$ converges.

5.4 Applications of Definite Integrals

The Gamma Function

For any $\alpha > 0$:

$$\begin{equation*} \Gamma(\alpha) = \int_{0}^{+\infty} x^{\alpha-1}e^{-x} \mathrm{d}x \end{equation*}$$

The function satisfies $\Gamma(\alpha+1) = \alpha\Gamma(\alpha)$ and $\Gamma(1) = 1$. Therefore, for any positive integer $n$:

$$\begin{equation*} \Gamma(n+1) = n! \end{equation*}$$

The Beta Function

For any $\alpha > 0, \beta > 0$:

$$\begin{equation*} B(\alpha, \beta) = \int_{0}^{1} x^{\alpha-1}(1-x)^{\beta-1} \mathrm{d}x \end{equation*}$$

It satisfies $B(\alpha+1, \beta) = \frac{\alpha}{\beta} B(\alpha, \beta+1)$.

• The relationship between the Beta function and the Gamma function is given by the following identity:

$$\begin{equation*} B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} \end{equation*}$$

The area of a Plane Region

Area formula for a simple closed parametric curve

For a curve given by $\gamma: x = x(t),\, y = y(t),\, \alpha \le t \le \beta$, the area enclosed by (\gamma) is

$$\begin{align*} \text{Area} &= - \int_{\alpha}^{\beta} y(t)\, x'(t)\, dt = \int_{\alpha}^{\beta} x(t)\, y'(t)\, dt. \end{align*}$$

Averaging these two expressions yields

$$\begin{align*} \text{Area} &= \frac{1}{2} \int_{\alpha}^{\beta} \bigl( x(t) y'(t) - x'(t) y(t) \bigr)\, dt. \end{align*}$$

Equivalently, using the determinant notation,

$$\begin{align*} \text{Area} &= \frac{1}{2} \int_{\alpha}^{\beta} \begin{vmatrix} x(t) & x'(t) \\ y(t) & y'(t) \end{vmatrix} dt. \end{align*}$$

Length of a Curve

Let $\mathbf{x}(t) = (x_1(t), \dots, x_n(t)) \in \mathbb{R}^n$ be a parametrization of the curve $\gamma$, where $\alpha \le t \le \beta$. A partition of the interval $[\alpha, \beta]$ is given by

$$\begin{equation*} \mathcal{P} : \alpha = t_0 < t_1 < \cdots < t_n = \beta. \end{equation*}$$

For points on the curve defined by

$$\begin{equation*} P_k = \mathbf{x}(t_k), \end{equation*}$$

the length of the polygonal line $P_0 P_1 \cdots P_n$ is

$$\begin{equation*} L(\mathcal{P}) = \sum_{k=1}^{n} \| P_{k-1} P_k \| = \sum_{k=1}^{n} \| \mathbf{x}(t_k) - \mathbf{x}(t_{k-1}) \|. \end{equation*}$$

In coordinates, this becomes

$$\begin{equation*} L(\mathcal{P}) = \sum_{k=1}^{n} \sqrt{ \bigl( x_1(t_k) - x_1(t_{k-1}) \bigr)^2 + \cdots + \bigl( x_n(t_k) - x_n(t_{k-1}) \bigr)^2 }. \end{equation*}$$

It is clear that as the partition is refined, the quantity $L(\mathcal{P})$ increases.

If the supremum $\displaystyle \sup_{\mathcal{P}} L(\mathcal{P})$ exists, then the curve $\gamma$ is said to be rectifiable, and

$$\begin{equation*} L(\gamma) = \sup_{\mathcal{P}} L(\mathcal{P}) \end{equation*}$$

is called the length of the curve $\gamma$.

$$\begin{equation*} L(\gamma) = \int_{\alpha}^{\beta} \| \mathbf{x}'(t) \| \, dt. \end{equation*}$$

Curvature and Torsion of a Curve

$$\begin{equation*} l = l(t) = \int_{\alpha}^{t} \|\mathbf{x}'(s)\|\, ds \end{equation*}$$ $$\begin{equation*} l'(t) = \|\mathbf{x}'(t)\| > 0 \end{equation*}$$ $$\begin{equation*} \tilde{\mathbf{x}}(l) = \mathbf{x}(t(l)) \quad \text{(the reparametrization by arc length)} \end{equation*}$$ $$\begin{equation*} \tilde{\mathbf{x}}'(l) = \mathbf{x}'(t(l))\, t'(l) = \frac{\mathbf{x}'(t)}{l'(t)} = \frac{\mathbf{x}'(t)}{\|\mathbf{x}'(t)\|} \quad \text{is the unit tangent vector} \end{equation*}$$ $$\begin{equation*} \sin\frac{\Delta \theta}{2} = \frac{1}{2}\, \|\tilde{\mathbf{x}}'(l+\Delta l) - \tilde{\mathbf{x}}'(l)\| \end{equation*}$$ $$\begin{equation*} = \frac{1}{2}\, \|\tilde{\mathbf{x}}''(l)\,\Delta l + o(\Delta l)\| \end{equation*}$$ $$\begin{equation*} = \frac{1}{2}\, \|\tilde{\mathbf{x}}''(l)\|\,|\Delta l| + o(\Delta l) \end{equation*}$$ $$\begin{equation*} \Delta \theta = \|\tilde{\mathbf{x}}''(l)\|\,\Delta l + o(\Delta l) \end{equation*}$$ $$\begin{equation*} \lim_{\Delta l \to 0} \frac{\Delta \theta}{\Delta l} = \|\tilde{\mathbf{x}}''(l)\| \end{equation*}$$ $$\begin{equation*} \tilde{\mathbf{x}}''(l) = \frac{d^2}{dl^2}\mathbf{x}(t(l)) = \frac{d}{dt}\left( \frac{d}{dt}\mathbf{x}(t)\, \frac{dt}{dl} \right)\frac{dt}{dl} \end{equation*}$$ $$\begin{equation*} = \left( \frac{\mathbf{x}'(t)}{\|\mathbf{x}'(t)\|} \right)' \frac{1}{\|\mathbf{x}'(t)\|} \end{equation*}$$ $$\begin{equation*} = \left( \frac{\mathbf{x}''(t)}{\|\mathbf{x}'(t)\|} - \frac{1}{2}\, \frac{\mathbf{x}'(t)\, 2\langle \mathbf{x}'(t),\mathbf{x}''(t)\rangle}{\|\mathbf{x}'(t)\|^3} \right) \frac{1}{\|\mathbf{x}'(t)\|} \end{equation*}$$ $$\begin{equation*} = \left( \mathbf{x}''(t) - \frac{\mathbf{x}'(t)}{\|\mathbf{x}'(t)\|} \left\langle \frac{\mathbf{x}'(t)}{\|\mathbf{x}'(t)\|}, \mathbf{x}''(t) \right\rangle \right) \frac{1}{\|\mathbf{x}'(t)\|^2} \end{equation*}$$ $$\begin{equation*} \lim_{\Delta l \to 0} \frac{\Delta \theta}{\Delta l} = \|\tilde{\mathbf{x}}''(l)\| = \kappa \quad\text{(curvature)} \end{equation*}$$ $$\begin{equation*} \tilde{\mathbf{x}}''(l): \ \text{curvature vector} \qquad \kappa = \|\tilde{\mathbf{x}}''(l)\|: \ \text{curvature} \end{equation*}$$ $$\begin{equation*} \mathbf{T}(l) = \tilde{\mathbf{x}}'(l): \ \text{unit tangent vector of the curve} \end{equation*}$$ $$\begin{equation*} \mathbf{N}(l) = \frac{\tilde{\mathbf{x}}''(l)}{\|\tilde{\mathbf{x}}''(l)\|}: \ \text{(unit) principal normal vector of the curve} \end{equation*}$$ $$\begin{equation*} \mathbf{T}'(l) = \kappa \mathbf{N}(l) \end{equation*}$$ $$\begin{equation*} 0 = \langle \mathbf{N}(l), \mathbf{N}(l) \rangle' = 2\langle \mathbf{N}(l), \mathbf{\dot{N}}(l) \rangle \qquad \Longrightarrow \qquad \mathbf{\dot{N}}(l) \ \text{is orthogonal to } \mathbf{N}(l) \end{equation*}$$ $$\begin{equation*} 0 = \langle \mathbf{T}(l), \mathbf{N}(l) \rangle' = \kappa \langle \mathbf{N}(l), \mathbf{N}(l)\rangle + \langle \mathbf{T}(l), \mathbf{\dot{N}}(l) \rangle = \kappa + \langle \mathbf{T}(l), \mathbf{\dot{N}}(l) \rangle \end{equation*}$$

Fundamental Equations of Plane Curve in $\mathbb{R}^2$

$$\begin{equation*} \dot{\mathbf{N}}(l) = -\kappa \mathbf{T}(l) \end{equation*}$$

In arc-length parametrization, the moving frame $(\mathbf{T}, \mathbf{N})$ of a plane curve in $\mathbb{R}^2$ satisfies the fundamental equations:

$$\begin{equation*} (\mathbf{T} \ \mathbf{N})' = (\mathbf{T} \ \mathbf{N}) \begin{pmatrix} 0 & -\kappa \\ \kappa & 0 \end{pmatrix} \end{equation*}$$

Given a curvature function $\kappa(l)$ and any orthonormal initial vectors $\mathbf{T}_0, \mathbf{N}_0$, the above system has a unique solution.

The curve is then obtained from

$$\begin{equation*} \mathbf{x}(l) = \mathbf{x}_0 + \int_0^l \mathbf{T}(s)\, ds \end{equation*}$$

A plane curve is uniquely determined by its curvature.

Fundamental Equations of Plane Curve in $\mathbb{R}^3$

$$\begin{equation*} \mathbf{B}(l) = \mathbf{T}(l) \times \mathbf{N}(l): \ \text{the (unit) binormal vector of the curve} \end{equation*}$$

$\dot{\mathbf{N}}(l)$ is orthogonal to $\mathbf{N}(l)$

$$\begin{equation*} \langle \mathbf{T}(l), \dot{\mathbf{N}}(l) \rangle = -\kappa \end{equation*}$$ $$\begin{equation*} \dot{\mathbf{N}}(l) = -\kappa \mathbf{T}(l) + \tau \mathbf{B}(l) \end{equation*}$$ $$\begin{equation*} \tau = \langle \dot{\mathbf{N}}(l), \mathbf{B}(l) \rangle \quad \text{(torsion of the curve)} \end{equation*}$$ $$\begin{equation*} \dot{\mathbf{B}}(l) = \dot{\mathbf{T}}(l) \times \mathbf{N}(l) + \mathbf{T}(l) \times \dot{\mathbf{N}}(l) = \tau\, \mathbf{T}(l) \times \mathbf{B}(l) = -\tau\, \mathbf{N}(l) \end{equation*}$$

In arc-length parametrization, the moving frame $(\mathbf{T}, \mathbf{N}, \mathbf{B})$ of a space curve in $\mathbb{R}^3$ satisfies:

$$\begin{equation*} (\mathbf{T}\ \mathbf{N}\ \mathbf{B})' = (\mathbf{T}\ \mathbf{N}\ \mathbf{B}) \begin{pmatrix} 0 & -\kappa & 0 \\ \kappa & 0 & -\tau \\ 0 & \tau & 0 \end{pmatrix} \end{equation*}$$

A space curve in $\mathbb{R}^3$ is uniquely determined by its curvature and torsion.

Frenet–Serret Formulas

$$\begin{equation*} \mathbf{T} = \frac{\mathbf{x}'}{\|\mathbf{x}'\|} \end{equation*}$$ $$\begin{equation*} \frac{dt}{dl} = \frac{1}{\|\mathbf{x}'\|} \end{equation*}$$ $$\begin{equation*} \mathbf{T}' = \frac{\mathbf{x}''}{\|\mathbf{x}'\|} - \frac{\mathbf{x}'}{\|\mathbf{x}'\|^{3}} \langle \mathbf{x}', \mathbf{x}'' \rangle = \frac{1}{\|\mathbf{x}'\|} \left( \mathbf{x}'' - \mathbf{T}\langle \mathbf{T}, \mathbf{x}'' \rangle \right) \end{equation*}$$ $$\begin{equation*} \dot{\mathbf{T}} = \mathbf{T}'\frac{dt}{dl} \quad\text{(curvature vector)} \end{equation*}$$ $$\begin{equation*} \kappa = \frac{1}{\|\mathbf{x}'\|^{2}} \left\|\mathbf{x}'' - \mathbf{T}\langle\mathbf{T}, \mathbf{x}''\rangle\right\| \end{equation*}$$ $$\begin{equation*} \mathbf{N} = \frac{\dot{\mathbf{T}}}{\|\dot{\mathbf{T}}\|} = \frac{\mathbf{x}'' - \mathbf{T}\langle \mathbf{T},\mathbf{x}''\rangle} {\left\|\mathbf{x}'' - \mathbf{T}\langle \mathbf{T},\mathbf{x}''\rangle\right\|} \end{equation*}$$ $$\begin{equation*} \mathbf{B} = \frac{\mathbf{x}' \times \mathbf{x}''}{\|\mathbf{x}' \times \mathbf{x}''\|} \end{equation*}$$ $$\begin{equation*} \mathbf{B}' = \frac{\mathbf{x}' \times \mathbf{x}''' - \mathbf{B} \langle \mathbf{B}, \mathbf{x}' \times \mathbf{x}''' \rangle} {\|\mathbf{x}' \times \mathbf{x}''\|} \end{equation*}$$ $$\begin{equation*} \dot{\mathbf{B}} = \mathbf{B}'\frac{dt}{dl} = - \tau \mathbf{N} \end{equation*}$$ $$\begin{equation*} \tau = -\langle \dot{\mathbf{B}}, \mathbf{N} \rangle \end{equation*}$$ $$\begin{equation*} \tau = \frac{\langle \mathbf{x}''',\mathbf{T}\times\mathbf{N}\rangle} {\|\mathbf{x}'\times\mathbf{x}''\|} = \frac{\langle\mathbf{x}''',\mathbf{B}\rangle} {\|\mathbf{x}'\times\mathbf{x}''\|} = \frac{\langle \mathbf{x}''', \mathbf{x}' \times \mathbf{x}'' \rangle} {\|\mathbf{x}' \times \mathbf{x}''\|^{2}} \end{equation*}$$