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0x031 Curve and Surface

This note covers the classical differential geometry over regular curves and surfaces

1. Curve

1.1. Parameterized Curve

Definition (parametrized curve) A parametrized curve in \(R^n\) is a smooth function: \(\gamma: I \to R^n\), where \(I \subset R\) is an open interval.

Note that curve is a map, which is different from its trace (defined to be curve's image set)

Definition (tangent vector, speed vector) speed at time \(t\) is defined to be \(\gamma'(t)\)

arc length between \(t_1, t_2\) is \(\int_{t_1}^{t_2} |\gamma'(t)|dt\)

A curve is called regular if its speed is always nonzero. It is called unit-speed or parametrized by arc length if its speed is always equal to 1

The image of a curve is called trace, which only represents the path of the moving object and it contains no time information. In many cases, we only care about the trace (e.g: curvature). It means we want those properties to be unchanged by a reparametrization

1.2. Curvature

Definition (curvature function) Let \(\gamma: I \to R^n\) be a regular curve. Its curvature function, \(\kappa: I \to [0, \infty)\), is defined as

\[\kappa(t) = \frac{|a^{\perp}(t)|}{|v(t)|^2}\]

This function is invariant wrt the parametrization, if \(\gamma\) is parametrized by arc length, then

\[\kappa(t) = |a(t)|\]

because \(v(t)=1\) unit speed and \(a(t) = a^{\perp}(t)\) because \(a \perp v\) for constant speed.

Definition (osculating plane) The plane which contains the tangent and principal normal

Definition (normal plane) The plane contains principal normal and binormal, it is the plane perpendicular to the tangent vector

Definition (rectifying plane) The plane contains the tangent and binormal, perp to the principal normal.

Theorem (Frenet-Serret) \(\(\frac{d}{ds}\left(\begin{array}{c} \mathbf{T} \\ \mathbf{N} \\ \mathbf{B} \end{array}\right) = \left( \begin{array}{ccc} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{array} \right) \left(\begin{array}{c} \mathbf{T} \\ \mathbf{N} \\ \mathbf{B} \end{array}\right)\)\)

where \(\mathbf{T}\) is the unit tangent vector, \(\mathbf{N}\) is the principal normal vector and \(\mathbf{B}\) is the binormal vector, \(\kappa\) is the curvature and \(\tau\) is the torsion, torsion measures how sharply it is twisting out of the plane of curvature. (curve up?)

Frenet frame

1.3. Line Integral

1.3.1. Line Integral over Scalar Field

Definition (arc length) Let \(\mathbf{x} \in C^1: [a,b] \to R^3\) where \(\mathbf{x}(t) = (x(t), y(t), z(t))\)with \(\mathbf{x}'(t) \neq 0\). The \(\mathbf{x}\) is called a smooth parametrization of \(C\). The length of the curve is

\[\int_C ds = \int_a^b ||\mathbf{x}'(t)|| dt\]

Definition (line integral of a scalar field) Let \(f\) be a continuous function on a smooth curve \(C\) parameterized by \(\mathbf{x}(t)\), the integral of \(f\) over \(C\) with respect to arc length is

\[\int_C f ds = \int_a^b f(\mathbf{x}(t))||\mathbf{x}'(t)|| dt\]

The illustration on Wikipedia is pretty good


The line integral over

1.3.2. Line Integral over Vector Field

The vector field line integral has two versions:

  • measuring the sum of tangent component (work)
  • meausring the sum of perpendicular component (flux)

Definition (line integral of a vector field, work) For a vector field \(\mathbf{F}: R^n \to R^n\), the line integral along a piecewise smooth curve \(C\), in the direction of \(r\), is defined as

\[\int_C \mathbf{F}(r) \cdot d\mathbf{r} = \int_a^b \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{r}'(t) dt\]

Theorem (fundamental theorem of calculus for line integrals, gradient theorem) Let \(\varphi: U \subseteq R^n \to R^n\) be a continuously differentiable function and \(\gamma\) any curve in \(U\) which starts at \(\mathbf{p}\) and ends at \(\mathbf{q}\). then

\[\int_\gamma \nabla \varphi(\mathbf{r}) \cdot d\mathbf{r} = \varphi(\mathbf{q}) - \varphi(\mathbf{p})\]

It implies the line integrals through a gradient field are path independent. In Physics, \(\varphi\) is a potential and \(\nabla \varphi\) is a conservative field: Work done by the conservative forces does not depend on the path, but only on the end points.

Criterion (conservative field is irrotational) A necessary and sufficient condition that a field \(\mathbf{F}\) to be conservative is that

\[\nabla \times \mathbf{F} = 0\]

Theorem (Green) Let \(\mathbf{F}\) be a continuously differentiable vector field, and \(D\) a domain in \(R^2\), then

\[\int_{\partial D} \mathbf{F} \cdot d{\mathbf{r}} = \iint_{D} (\frac{\partial Q}{\partial x}-\frac{\partial{P}}{\partial y}) dA\]

Definition (line integral of a vector field, flux) For a vector field \(\mathbf{F}(x,y) = (P(x,y), Q(x,y))\), the line integral across a curve \(C\) is defined in terms of a piecewise smooth parametrization \(\mathbf{r}(t) = (x(t), y(t))\) as

\[\int_C \mathbf{F}(r) \cdot d\mathbf{r}^{\perp} = \int_a^b (P(x,y), Q(x,y))(y', -x')dt = \int_a^b -Q dx + P dy\]

2. Surface

2.1. Parameterized Surface

Definition (smooth surface) Let \(\mathbf{x}\) be a smoothly bounded set \(D: \mathbb{R}^2 \to \mathbb{R}^3\), denoted \(\mathbf{x}(u, v) = (x(u,v), y(u,v), z(u,v))\), Suppose \(\mathbf{x}\) satisfies the following conditions on the interior of \(D\)

  • \(\mathbf{X}\) is 1-to-1
  • the partial deriavtives are bounded
  • partial derivatives are linearly independent, \(\mathbf{x}_u(u,v) \times \mathbf{x}_v(u,v) \neq 0\).

Then the range of \(\mathbf{x}\) is called a smooth surface, parametrized by \(\mathbf{x}\)

2.2. Surface Integral

2.2.1. Surface Integral over a Scalar field

Definition (surface integral over a scalar field) If a surface \(S\) is parametrized by the continuously differentiable function \(x(u,v)\) over the domain \(D\) in the uv-plane, then the scalar surface integral of the function \(f(x)\) over \(S\) is

\[\int_S f(x) dS = \int_D f(x(u,v)) || x_u \times x_v || du dv\]

2.2.2. Surface Integral over a Vector Field

Definition (surface integral over a vector field, flux)

\[\iint_S \mathbf{F}(x) \cdot d\mathbf{S} = \iint_S (\mathbf{F}(x) \cdot \mathbf{n}) dS\]

Theorem (divergence, Gauss) Let \(F\) be a \(C^1\) vector field on regular set \(D\) and let \(N\) be the unit normals to \(\partial D\) that point out of \(D\), then

\[\int_{\partial D} (F \cdot N) d\sigma = \int_D (\nabla \cdot F) dV\]

Theorem (curl, Stokes) Let \(G\) be a vector field that is \(C^1\) on a piece-wise smooth oriented surface \(S\) whose boundary \(\partial S\) is a piecewise smooth curve

\[\int_{\partial S} (G \cdot T) ds = \int_{S} (\nabla \times G) dS \]

3. n-dimensional Manifold

4. Reference

[1] Tapp, Kristopher. Differential geometry of curves and surfaces. Berlin: Springer, 2016.

[2] Lax, Peter D., and Maria Shea Terrell. Multivariable Calculus with Applications. Springer, 2017.

[3] Differential geometry of curves and surfaces