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0x030 Foundation

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

1. Foundation

The vector differential operator del, written in \(\nabla\), is defined as follows:

\[\nabla = \frac{\partial}{\partial x} \mathbf{i} + \frac{\partial}{\partial y} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k}\]

Definition (gradient) Gradient acting on a scalar field to produce vector field.

\[\nabla T = \frac{\partial{T}}{\partial x}\mathbf{x} + \frac{\partial{T}}{\partial y}\mathbf{y} + \frac{\partial{T}}{\partial z}\mathbf{z}\]

The gradient \(\nabla T\) points in the direction of maximum increase of the function \(T\), its slop along this direction is given by \(|\nabla T|\)

Definition (divergence) Divergence acting on a vector field to produce scalar field.

\[\nabla \cdot \mathbf{v} = \frac{\partial{v_x}}{\partial x} + \frac{\partial{v_y}}{\partial y} + \frac{\partial{v_z}}{\partial z}\]

Divergence is a measure of how much the vector \(v\) spreads out from each point (therefore a scalar field). In fluid dynamics, if there is no loss of fluid anywhere, then \(\nabla \cdot \mathbf{b} = 0\). This is called the continuity equation for an incompressible fluid. It is said to have no sources or sinks. A vector field whose divergence is zero is sometimes called solenoidal.

Lemma (properties of divergence) Divergence is a linear operator

\[\nabla \cdot (a\mathbf{f} + b\mathbf{g}) = a(\nabla \cdot \mathbf{f}) + b(\nabla \cdot \mathbf{g})\]

Product rule

\[\nabla \cdot (\phi \mathbf{A}) = (\nabla \phi) \cdot \mathbf{A} + \phi (\nabla \cdot \mathbf{A})\]

Definion (curl) curl acting on a vector field to produce another vector field.

\[\nabla \times \mathbf{v} = (\frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z}) \mathbf{x} + (\frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x}) \mathbf{y} + (\frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}) \mathbf{z}\]

curl is a measure of how much the vector \(v\) swirls around the point in question. A vector field wihtout rotation (i.e.: \(\nabla \times \mathbf{v}=0\)) is called irrotational, a field that is not irrotational is sometimes called a vortex field.

Furthermore, if the vector field is irrotational \(\nabla \times \mathbf{v} = 0\), then it is a conservative vector field, which means there exists a scalar field \(\phi\) such that

\[\mathbf{v} = \nabla \phi\]

In conservative vector field, the line integral is path independent.

Lemma (properties of curl)

  • \(\nabla \times (\phi \mathbf{A}) = (\nabla \phi) \times \mathbf{A} + \phi (\nabla \times \mathbf{A})\)
  • \(\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})\)
  • \(\nabla \times (A \times B) = (B \cdot \nabla) A - B(\nabla \cdot A) - (A \cdot \nabla)B + A(\nabla \cdot B)\)
  • \(\nabla(A \cdot B) = (B \cdot \nabla) A + (A \cdot \nabla) B + B \times (\nabla \times A) + A \times (\nabla \times B)\)

2. Curve

2.1. 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 interval

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

2.2. 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

Proposition 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

2.3. Line Integral

2.3.1. 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

2.3.2. 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

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

3. Surface

3.1. 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 \(\mathbf{x}\)

3.2. Surface Integral

3.2.1. 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

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

3.2.2. 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

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

4. Curvature of a Surface

4.1. Gauss Map

5. Curvilinear Coordinates

Suppose we change from the Cartesian coordinate \((x_1, x_2, x_3)\) to the curvilinear coordinate \((u_1, u_2, u_3)\)

\[u_1 = u_1(x_1, x_2, x_3) \\ u_2 = u_2(x_1, x_2, x_3) \\ u_3 = u_3(x_1, x_2, x_3)\]

The reverse transformation can also be mae

\[x_i = x_i(u_1, u_2, u_3)\]

5.1. Orthogonal Curvilinear Coordinates

Suppose the point \(p\) has position \(\mathbf{r} = \mathbf{r}(u_1, u_2, u_3)\), consider the tangent vector to \(u_1\) curve is

\[\frac{\partial \mathbf{r}}{\partial u_1} = h_1 e_1\]

where \(e_1\) is the unit tangent vector in this direction: \(e_1 = \frac{\partial \mathbf{r}}{\partial u_1}/|\frac{\partial \mathbf{r}}{\partial u_1}|\) and \(h_1 = |\frac{\partial \mathbf{r}}{\partial u_1}|\)


\[\begin{aligned}d\mathbf{r} &= \frac{\partial \mathbf{r}}{\partial u_1}{du_1} + \frac{\partial \mathbf{r}}{\partial u_3}{du_3} + \frac{\partial \mathbf{r}}{\partial u_3}{du_3} \\ &= h_1 du_1 \mathbf{e}_1 + h_2 du_2 \mathbf{e}_2 + h_3 du_3 \mathbf{e}_3\end{aligned}\]

The arc length is determined by \(ds^2 = d\mathbf{r} \cdot d\mathbf{r}\)

\[ds^2 = h_1^2 du_1^2 + h_2^2 du_2^2 + h_3^2 du_3^2\]

volumene element is

\[dV = h_1h_2h_3 du_1du_2du_3\]

6. 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.