DDG Exterior Algebra

Exterior Algebra: A Framework for Geometric Computations

Exterior algebra offers a robust mathematical framework for understanding and manipulating geometric objects such as volumes, areas, and orientations in higher-dimensional spaces. This post delves into three key concepts: the wedge product, the Hodge star, and coordinate representation, exploring how they allow us to express and compute signed volumes and their orthogonal complements.

1. The Wedge Product: Building Oriented k-Vectors

The wedge product is the cornerstone of exterior algebra, enabling the construction of higher-dimensional geometric objects by combining vectors.

Span

The concept of span is fundamental to understanding the wedge product. The span of a set of vectors is the collection of all possible linear combinations of those vectors:

\[\text{span}\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\} = \{a_1\mathbf{v}_1 + a_2\mathbf{v}_2 + \dots + a_k\mathbf{v}_k \mid a_i \in \mathbb{R}\}.\]

Geometrically, the span defines the subspace generated by the vectors. For instance:

The span of one vector is a line.
The span of two linearly independent vectors is a plane.

Definition

The wedge product \(\mathbf{u} \wedge \mathbf{v}\) represents the signed and oriented area of the parallelogram spanned by \(\mathbf{u}\) and \(\mathbf{v}\). More generally, the wedge product of \(k\) vectors spans a \(k\)-dimensional volume.

Antisymmetry

Swapping the order of vectors reverses the orientation: \(\mathbf{u} \wedge \mathbf{v} = -\mathbf{v} \wedge \mathbf{u}\).
If two vectors are parallel, their wedge product is zero because they cannot span a higher-dimensional volume.

k-Vectors

A k-vector is the wedge product of \(k\) vectors, representing an oriented geometric quantity. Examples include:

0-vectors, which are scalars.
1-vectors, which are ordinary vectors.
2-vectors, which represent oriented areas.
3-vectors, which represent oriented volumes.

Geometric Interpretation

The wedge product \(\mathbf{u} \wedge \mathbf{v}\) encodes both the magnitude and orientation of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{v}\), with the orientation determined by the order of the vectors.

2. The Hodge Star: Orthogonal Complements and Duality

The Hodge star operation complements the wedge product by mapping a k-vector to its orthogonal complement in the vector space.

Orthogonal Complement

The orthogonal complement of a subspace \(W\) in a vector space \(V\) consists of all vectors in \(V\) that are orthogonal to every vector in \(W\):

\[W^\perp = \{\mathbf{v} \in V \mid \langle \mathbf{v}, \mathbf{w} \rangle = 0 \ \text{for all } \mathbf{w} \in W\},\]

where \(\langle \cdot, \cdot \rangle\) is the inner product.

Geometrically, the orthogonal complement captures what a subspace excludes. For example:

In 3D, the orthogonal complement of a plane is a line perpendicular to it.
In 2D, the orthogonal complement of a line is another line perpendicular to it.

The Hodge Star

The Hodge star maps a k-vector to its \((n-k)\)-vector orthogonal complement, where \(n\) is the dimension of the space. This mapping depends on the inner product structure of the space.

Key Properties

Signed Complement: The Hodge star transforms a k-vector into a complementary volume with its orientation preserved.
Double Star: Applying the Hodge star twice yields a scalar multiple of the original vector:
\[\star (\star \mathbf{v}) = (-1)^{k(n-k)} \mathbf{v}.\]

Geometric Example

In 3D space:

The Hodge star of a 1-vector (line) is a 2-vector (plane orthogonal to the line).
The Hodge star of a 2-vector (plane) is a 1-vector (line orthogonal to the plane).

Applications

The Hodge star connects the wedge product and duality:

Wedge \(k\)-vectors to compute a signed volume.
Apply the Hodge star to find its complementary \((n-k)\)-vector.

For example, in 3D, the wedge product \(\mathbf{u} \wedge \mathbf{v}\) spans a plane (2-vector), and the Hodge star transforms it into a vector orthogonal to that plane.

3. Coordinate Representation

Exterior algebra gains its computational power by expressing vectors and their combinations in terms of basis elements. This section examines basis k-vectors and how the Hodge star maps between complementary dimensions, concluding with an example involving the wedge product and area.

Basis Vectors and k-Vectors

Basis 1-Vectors:

In \(\mathbb{R}^n\), the basis 1-vectors \(\{\mathbf{e}\_1, \mathbf{e}\_2, \dots, \mathbf{e}\_n\}\) are standard unit vectors that span the entire space. Any vector \(\mathbf{v}\) can be expressed as:

\[\mathbf{v} = v_1 \mathbf{e}_1 + v_2 \mathbf{e}_2 + \dots + v_n \mathbf{e}_n,\]

where \(v_i\) are scalar coefficients.

Basis 2-Vectors:

A 2-vector is the wedge product of two basis 1-vectors, such as \(\mathbf{e}\_i \wedge \mathbf{e}\_j\) (with \(i < j\)). These represent oriented areas in the planes spanned by the two vectors. For \(\mathbb{R}^3\), there are \(\binom{3}{2} = 3\) basis 2-vectors:

\[\{\mathbf{e}_1 \wedge \mathbf{e}_2, \mathbf{e}_1 \wedge \mathbf{e}_3, \mathbf{e}_2 \wedge \mathbf{e}_3\}.\]

Basis 3-Vectors:

A 3-vector is the wedge product of three basis 1-vectors, representing oriented volumes. In \(\mathbb{R}^3\), there is only one basis 3-vector (up to scaling):

\[\mathbf{e}_1 \wedge \mathbf{e}_2 \wedge \mathbf{e}_3.\]

General Basis k-Vectors:

In \(\mathbb{R}^n\), a basis k-vector is the wedge product of \(k\) distinct basis 1-vectors:

\[\mathbf{e}_{i_1} \wedge \mathbf{e}_{i_2} \wedge \dots \wedge \mathbf{e}_{i_k}, \quad 1 \leq i_1 < i_2 < \dots < i_k \leq n.\]

The total number of basis k-vectors is \(\binom{n}{k}\), the number of ways to choose \(k\) vectors from \(n\).

Hodge Star and Basis k-Vectors

The Hodge star maps a basis k-vector in \(\mathbb{R}^n\) to its complementary \((n-k)\)-vector. For \(\mathbb{R}^3\):

The Hodge star of a basis 1-vector maps it to a basis 2-vector:
\[\star \mathbf{e}_1 = \mathbf{e}_2 \wedge \mathbf{e}_3, \quad \star \mathbf{e}_2 = \mathbf{e}_3 \wedge \mathbf{e}_1, \quad \star \mathbf{e}_3 = \mathbf{e}_1 \wedge \mathbf{e}_2.\]
The Hodge star of a basis 2-vector maps it to a basis 1-vector:
\[\star (\mathbf{e}_1 \wedge \mathbf{e}_2) = \mathbf{e}_3, \quad \star (\mathbf{e}_2 \wedge \mathbf{e}_3) = \mathbf{e}_1, \quad \star (\mathbf{e}_3 \wedge \mathbf{e}_1) = \mathbf{e}_2.\]
The Hodge star of a basis 3-vector maps it to a scalar (0-vector):
\[\star (\mathbf{e}_1 \wedge \mathbf{e}_2 \wedge \mathbf{e}_3) = 1.\]

Example: Wedge Product and Area

To see how the wedge product relates to area, consider two vectors \(\mathbf{u} = u_1 \mathbf{e}\_1 + u_2 \mathbf{e}\_2 + u_3 \mathbf{e}\_3\) and \(\mathbf{v} = v_1 \mathbf{e}\_1 + v_2 \mathbf{e}\_2 + v_3 \mathbf{e}\_3\) in \(\mathbb{R}^3\).

The wedge product \(\mathbf{u} \wedge \mathbf{v}\) expands to:

\[\mathbf{u} \wedge \mathbf{v} = (u_1 v_2 - u_2 v_1) \mathbf{e}_1 \wedge \mathbf{e}_2 + (u_2 v_3 - u_3 v_2) \mathbf{e}_2 \wedge \mathbf{e}_3 + (u_3 v_1 - u_1 v_3) \mathbf{e}_3 \wedge \mathbf{e}_1.\]

The coefficients \(u_i v_j - u_j v_i\) represent the signed areas of the parallelograms projected onto the coordinate planes. The magnitude of \(\mathbf{u} \wedge \mathbf{v}\) gives the parallelogram’s total area, while the orientation is encoded in the basis 2-vectors.

Exterior algebra provides a systematic and elegant way to describe and compute geometric objects, making it a cornerstone in fields like physics, geometry, and computer graphics.