https://haifei.pro/en/tags/exterior-algebra/Recent content in Exterior Algebra on Haifei's HomeHugoenhfwang132@gmail.com (hfwang132)hfwang132@gmail.com (hfwang132)This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.Fri, 26 May 2023 18:11:03 +0800https://haifei.pro/en/post_20230526_%E8%B5%9D%E7%9F%A2%E9%87%8F%E5%88%B0%E5%BA%95%E6%98%AF%E4%BB%80%E4%B9%88/Fri, 26 May 2023 18:11:03 +0800hfwang132@gmail.com (hfwang132)https://haifei.pro/en/post_20230526_%E8%B5%9D%E7%9F%A2%E9%87%8F%E5%88%B0%E5%BA%95%E6%98%AF%E4%BB%80%E4%B9%88/<p>In physics books, we often encounter the terms “pseudovector” (pseudo-vector) and “pseudoscalar” (pseudo-scalar).</p> <p>In fact, on a 3-dimensional manifold, a “pseudovector” is the exterior product of two tangent vectors \(v\in T_pM\wedge T_pM=\bigwedge^2(T_pM)\), while a “pseudoscalar” is the exterior product of three tangent vectors \(s\in T_pM\wedge T_pM\wedge T_pM=\bigwedge^3(T_pM)\).</p> <p>After equipping the space with an inner product (or a nondegenerate bilinear form), there is a Hodge duality relation between \(\bigwedge^2(T_pM)\) and \(\bigwedge^1(T_pM)\), so we “mistakenly regard” pseudovectors as vectors. Similarly, because there is a Hodge duality relation between \(\bigwedge^3(T_pM)\) and \(\bigwedge^0(T_pM)\) (a scalar field), we “mistakenly regard” pseudoscalars as scalars.</p>