Let $$ n $$ be a non-zero natural number in $$ \mathbb{N}^{\ast} $$ and let $$ \left(\mathbb{R}^n, +, \cdot, \langle\cdot, \cdot\rangle\right) $$ be the standard Euclidean $$ n $$-space. Assuming the elements of the exterior algebra $$ \bigwedge(\mathbb{R}^n) $$ exist and are well-defined, define all of the following operations on $$ \bigwedge\left(\mathbb{R}^{n}\right) $$:
The wedge product, the regressive product, the interior product, the inner product, the Hodge dual, the cross product and the geometric product.
The elements of $$ \bigwedge(\mathbb{R}^n) $$ are interpreted to be multivectors (not multilinear forms). Give precise definitions.






The exterior algebra $ \bigwedge(\mathbb{R}^n) $ is a powerful algebraic structure built from the vector space $ \mathbb{R}^n $. It provides a framework for defining various operations on multivectors, which are elements of this algebra. Here, we will define the operations you mentioned: the wedge product, the regressive product, the interior product, the inner product, the Hodge dual, the cross product, and the geometric product.

### 1. Wedge Product (Exterior Product)

The **wedge product** (also known as the exterior product) is a bilinear, antisymmetric operation on multivectors. For vectors $ u, v \in \mathbb{R}^n $, the wedge product is defined as:

\[
u \wedge v = u \otimes v - v \otimes u
\]

For higher

Explore operations on exterior algebra and multivectors in Euclidean space, including the wedge product, regressive product, interior product, inner product, Hodge dual, cross product, and geometric product. Understand the definitions and applications of these operations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation