Flux Integral

A flux integral measures transport through space.

Description

Flux is a measure of transport through a surface or line. Especially when movement is described by a vector field, flux is measured using integrals.

Note that if a vector field is conservative, there is zero flux over a closed curve or surface.

Line Integrals

For a smooth curve C and a vector field given as F = <P,Q>, the flux integral is a line integral. More intuitively though, it is the integral of the directional derivative of F in the orthogonal direction.

where n̂ is the unit normal vector. Trivially though, given r = <x, y>, it must be that n = <-y, x>.

Parameterize F using r(t) for a ≤ t ≤ b to get:

where n(t) is derived from r'(t): if r'(t) = <r_x , r_y> then n(t) = <r_y , -r_x>.

Note alternatively that, given a line as <x, y>, the trivial normal vector is <-y, x>. If this is then normalized, the original formula may be simpler to solve.

Closed Line Integrals

For a smooth and closed curve C with counterclockwise orientation, the flux integral is a line integral as:

where n̂ is the unit normal vector.

When the vector field is given as F = <P(x,y), Q(x,y)>, Green's theorem gives a method for evaluating this. Specifically, the vector form of the theorem that uses divergence, sometimes known as the 'normal' or 'flux' form.

Recall that divergence in 2 dimensions is evaluated as:

Note the closely related circulation form of the theorem.

Surface Integrals

For a smooth surface S and a vector field given as F = <P,Q,R>, the flux integral is a surface integral. Note that the orientation of S should be given, because an orientable surface has two possible orientations and neither is more 'correct'. It is however important that the surface be orientable; complex surfaces like the Möbius strip are not orientable.

Recall that a generic surface integral looks like:

Also recall that this integral is evaluated by parameterizing and knowing that dS = ||r_u × r_v||.

The flux integral along a surface differs in that the integrand is the dot product of a vector field F and the unit normal vector n̂, not a simple function f.

However, the integrand immediately simplifies because n̂ is normalized by ||r_u × r_v||. Therefore the surface flux integral is:

Note that, in some texts, the expression n̂ dS is rewritten as dS or dS⃗. This term represents the differential surface area vector.

Normal Vectors

In the case that S is a plane, there may be several shortcuts to identifying the normal vector.

• If the plane is defined like z=0, i.e. the surface is the xy-plane, then the trivial normal vector is [0, 0, 1].

• If the plane is defined by a function (ax + by + cz = d), the normal vector is trivially [a, b, c].

• If the plane is defined by 3 points (A, B, and C), then tangent vectors can be calculated as B-A and C-A. The points can be specially crafted to give simple tangent vectors. The normal vector is then given by the cross product of these.

In other cases, the normal vector is given by a complex expression. For example, when working in spherical coordinates where θ is the azimuthal angle, the normal vector of a sphere is given by [ρ²cos(θ)sin²(φ), -ρ²sin(θ)sin²(φ), ρ²sin(φ)cos(φ)].

Parameterization

Consider a slightly more complex surface, e.g. the triangle formed by A = (1,0,0), B=(0,1,0), and C=(0,0,1).

The surface is given by z=1-x-y, leading to a clear parameterization as r(u,v) = <u,v,1-u-v>. The parameterization is key because it projects the 3 dimensional surface on a 2 dimensional plane. For this triangle, the projection is:

Note that this new surface is bounded by v=0 and v=1-u over the range 0 ≤ u ≤ 1. This sets up a clear region of integration.

The original surface is S, and must be integrated with respect to dS. The projected surface is D and is just a double integral with respect to dA = dudv. (Although if there was a change in coordinate systems during parameterization, a Jacobian determinant also needs to be attached to dA.)

Orientation

Importantly, these formula are valid for surfaces with outward orientation. A

Closed Surface Integrals

Consider a smooth closed surface S as completely encompassing a solid E. If S has outward orientation (relative to E) and E is also encompassed by the domain of vector field F, then divergence theorem gives a method for evaluating the integral:

Recall that divergence is evaluated as:

Note that, in some texts, the expression n̂ dS is rewritten as dS or dS⃗. This term represents the differential surface area vector.

Unclosed Surfaces

If the surface does not completely enclose the solid E, these formula are not valid. However, it remains true that the surface integral of a closed surface is equal to the sum of the surface integrals of all faces. Therefore, the surface integral of a solid excluding a face can be calculated like:

As an example, consider the vector field given by F = [a, b, 2y+3z] and consider the surface S around a unit cube E, except one face of the unit cube is excluded.

All 6 of a unit cube's faces are defined by:

1. x = 1

2. y = 1

3. z = 1

4. x = 0

5. y = 0

6. z = 0

Trivially the first three faces have unit normal vectors of [1 0 0], [0 1 0], and [0 0 1]. The latter three are oppositely oriented, so the unit normal vectors should be expressed as [-1 0 0], [0 -1 0], and [0 0 -1].

Suppose the excluded face is the one corresponding to z = 0. The integrand is the dot product of [a, b, 2y+3z] and [0 0 -1], which clearly is -(2y+3z). Integrating this with respect to x and y gives -(1+3z). Finally, substitute in the surface's equation (i.e., z = 0). Therefore, the surface integral of the unclosed surface is given by evaluating the divergence theorem then substracting -1 (a.k.a. adding 1).

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