Theory

Surfaces, fluxes and the theorems of Gauss and Stokes

Regular parametric surfaces, normal vector and area, surface integrals and flux of a field, divergence and curl, the divergence theorem and Stokes' theorem.

On this page

Parametric surfaces
Normal vector and tangent plane

Parametric surfaces

1. Parametric surface

A parametric surface is a map $r : D \subseteq R^{2} \to R^{3}$ of class $C^{1}$ ,

r (u, v) = (x (u, v), y (u, v), z (u, v)), (u, v) \in D,

with $D$ closed and bounded, the union of a connected open set and its boundary. The image $Σ = r (D)$ is called the trace of the surface. A curve depends on one parameter, therefore its trace is a one-dimensional object; a surface depends on two parameters and its trace is a two-dimensional object in space. Fixing $v = v_{0}$ , the map $u \mapsto r (u, v_{0})$ is a curve drawn on the surface, called a coordinate line; the same holds with $u = u_{0}$ fixed.

Sphere and cylinder

The sphere centered at the origin with radius $R$ is parametrized by spherical coordinates

r (φ, θ) = (R sin φ cos θ, R sin φ sin θ, R cos φ), (φ, θ) \in [0, π] \times [0, 2 π],

where $φ$ is the colatitude measured from the $z$ -axis and $θ$ the longitude. The lateral surface of the cylinder $x^{2} + y^{2} = R^{2}$ with $0 \leq z \leq h$ is parametrized by

r (θ, t) = (R cos θ, R sin θ, t), (θ, t) \in [0, 2 π] \times [0, h] .

These two parametrizations appear in most exercises on areas and fluxes.

The parameter rectangle

[0, π] \times [0, 2 π]

of the sphere: the segment

φ = φ_{0}

becomes a parallel, the segment

θ = θ_{0}

a meridian

2. Regular surface

The derivative vectors of the parametrization are

r_{u} = (\frac{\partial x}{\partial u}, \frac{\partial y}{\partial u}, \frac{\partial z}{\partial u}), r_{v} = (\frac{\partial x}{\partial v}, \frac{\partial y}{\partial v}, \frac{\partial z}{\partial v}) .

The surface is regular if $r$ is injective at the interior points of $D$ and if

r_{u} \land r_{v} \neq = 0 at the interior points of D .

The vector $r_{u}$ is tangent to the coordinate line with $v$ fixed, the vector $r_{v}$ to the one with $u$ fixed. The condition $r_{u} \land r_{v} \neq = 0$ says that the two tangent vectors are not parallel, so at every interior point the surface has a well-defined tangent plane, with no cusps or edges. All the formulas of this chapter, from area to flux, are built with the vector $r_{u} \land r_{v}$ , therefore regularity is the hypothesis present in every statement.

Regularity of the sphere

For the sphere of radius $R$ we compute $∥ r_{φ} \land r_{θ} ∥ = R^{2} sin φ$ , which vanishes only for $φ = 0$ and $φ = π$ , that is, at the two poles, which are images of points on the boundary of the parameter rectangle. At the interior points the regularity condition is satisfied.

Normal vector and tangent plane

1. Normal vector

Let $Σ$ be a regular surface and $P_{0} = r (u_{0}, v_{0})$ a point corresponding to $(u_{0}, v_{0})$ interior to $D$ . The normal vector to $Σ$ at $P_{0}$ is

N (u_{0}, v_{0}) = r_{u} (u_{0}, v_{0}) \land r_{v} (u_{0}, v_{0}),

and the unit normal vector is $n = \frac{N}{∥ N ∥}$ . The cross product is orthogonal to both factors, so $N$ is orthogonal to the two tangent vectors $r_{u}$ and $r_{v}$ , therefore it is orthogonal to the surface at $P_{0}$ . Computing $N$ is the first step of almost every exercise: its direction gives the tangent plane, its norm the area element, its orientation the orientation in flux computations.

2. Tangent plane to a surface

The tangent plane to $Σ$ at $P_{0} = (x_{0}, y_{0}, z_{0})$ is the plane through $P_{0}$ orthogonal to $N = (N_{1}, N_{2}, N_{3})$ , with equation

N_{1} (x - x_{0}) + N_{2} (y - y_{0}) + N_{3} (z - z_{0}) = 0.

The tangent plane contains the tangent vectors to all the curves drawn on the surface and passing through $P_{0}$ , so it approximates the surface near the point just as the tangent line approximates the graph of a function of one variable.

Example

For $r (u, v) = (u cos v, u sin v, 2 v)$ at the point $(u_{0}, v_{0}) = (1, \frac{π}{2})$ we have $P_{0} = (0, 1, π)$ , $r_{u} = (0, 1, 0)$ , $r_{v} = (- 1, 0, 2)$ , so

N = r_{u} \land r_{v} = (2, 0, 1),

and the tangent plane has equation $2 x + (z - π) = 0$ , that is, $2 x + z = π$ .

3. Cartesian surface

The graph of a function $f : D \subseteq R^{2} \to R$ of class $C^{1}$ is a surface, called Cartesian, with the parametrization

r (x, y) = (x, y, f (x, y)), (x, y) \in D .

From $r_{x} = (1, 0, f_{x})$ and $r_{y} = (0, 1, f_{y})$ we obtain

N = r_{x} \land r_{y} = (- f_{x}, - f_{y}, 1), ∥ N ∥ = 1 + f_{x}^{2} + f_{y}^{2} .

The third component of $N$ equals $1 > 0$ , so a Cartesian surface is always regular and its normal vector points upward. The resulting tangent plane, $z = f (x_{0}, y_{0}) + f_{x} (x - x_{0}) + f_{y} (y - y_{0})$ , coincides with the tangent plane introduced in the chapter on differential calculus for differentiable functions.

Cross-section with the plane

y = 0

of the graph of

f (x, y) = 2 - x^{2}

: the tangent vector

r_{x}

and the normal vector

N

, which has third component

1

and points upward

The theory continues with 6 more sections (area of a surface, surface integrals of scalar functions, oriented surfaces and flux…), plus all worked examples, the solutions and the PDF.

Unlock now