# 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:

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

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.

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

Product rule

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

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

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

This function is invariant wrt the parametrization

**Proposition** If \(\gamma\) is parametrized by arc length, then

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?)

### 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

**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

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

**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

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

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

**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

## 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

#### 3.2.2. Surface Integral over a Vector Field

**Definition (surface integral over a vector field, flux)**

**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

**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

## 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 \((u_1, u_2, u_3)\)

The reverse transformation can also be mae

### 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

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 \(h_1 = |\frac{\partial \mathbf{r}}{\partial u_1}|\)

then

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

**volumene element** is

## 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.