Exterior algebra Definition and 15 Threads

Do we really need concept of cross product at all? I always believed cross product to be sort of simplification of exterior product concept tailored for the 3D case. However, recently I encountered the following sentence «...but, unlike the cross product, the exterior product is associative»...

Maxwell's equations in differential form notation appeared as a motivating example in a mathematical physics book I'm reading. However, being a mathematical physics book it doesn't delve much into the physical aspects of the problem. It deduces the equations by setting dF equal to zero and d(*F)...

(A1−A2,B1−B2,C1−C2)∧(A1,B1,C1)(A1−A2,B1−B2,C1−C2)∧(A1,B1,C1) ##=((A1−A2)∗B1−(B1−B2)∗A1)∗(\hat x \wedge \hat y)+((C1−C2)∗A1−(A1−A2)∗C1)∗(\hat z \wedge \hat x)+((B1−B2)∗C1−(C1−C2)∗B1)∗(\hat y \wedge \hat z)## Is this the correct exterior product?

If we seek a bijection $$\wedge^p V \to \wedge^{n-p} V$$ for some inner product space ##V##, we might think of starting with the unit ##n##-vector and removing dimensions associated with the original vector in ##\wedge^p V ##. Might this be expressed as a sequence of steps by some binary...

The determinant of some rank 2 tensor can be expressed via the exterior product. $$T = \sum \mathbf{v}_i \otimes \mathbf{e}_i \;\;\; \text{or}\sum \mathbf{v}_i \otimes \mathbf{e}^T_i $$ $$ \mathbf{v}_1\wedge \dots \wedge \mathbf{v}_N = det(T) \;\mathbf{e}_1\wedge \dots \wedge\mathbf{e}_N$$ The...

No question this time. Just a simple THANK YOU For almost two years years now, I have been struggling to learn: differential forms, exterior algebra, calculus on manifolds, Lie Algebra, Lie Groups. My math background was very deficient: I am a 55 year old retired (a good life) professor of...

Hello I am a mechanical engineer who is teaching himself the math of exterior algebra and differential forms. It is not easy for me and I have had many SIMPLE stumbling blocks due to my not respecting algebra. May I ask for help on some simple aspects? (Please be patient with me.) My...

Hi, I am currently reading about differential forms in "Introduction to Smooth Manifolds" by J. M. Lee, and I was wondering exactly how you define the wedge product on the exterior algebra \Lambda^*(V) = \oplus_{k=0}^n\Lambda^k(V) of a vector space V. I understand how the wedge product is...

Hello, In R^3, the surface of the parallelogram determined by two vectors u and v is given by the norm of the cross product of u and v. For my research, I have to know if this can be generalized in the following manner: Let e_1,..,e_n be the canonical basis of R^n, and Ext_k be the exterior...

I'm reading Marsden's vector calculus. In the chapter of differential forms, it mentions the wedge product satisfies the laws: dy^dx=-dxdy. and for a 0-form f, f^w=fw. Does it have formal derivation? hope someone can give me a hint or even a link.

I understand that there is a way to find a basis \{e_1,...,e_n\} of a vector space V such that a 2-vector A can be expressed as A = e_1\wedge e_2 + e_3\wedge e_4 + ...+e_{2r-1}\wedge e_{2r} where 2r is denoted as the rank of A. However the way that I know to prove this seems sort of...

1.how to prove div(A × B) = (rot A)· B - A ·(rot B) 2.d(ω1(A) × ω1(B))=?

Can someone please thoroughly explain how the determinant comes from the wedge product? I'm only in Cal 3 and Linear at the moment. I'm somewhat trying to learn more about the Wedge Product in Exterior Algebra to understand the determinant on a more fundamental basis. A thorough website or...

Unfortunately there seems to be a misprint in the paper I'm reading which is an introduction to clifford algebra, it says:(I highlighted in red possible misprint, either one of them has to be true misprint if you know what I mean) The Clifford algebra C(V) is isomorphic to the tensor algebra...

Does anyone have any good refreshers/tutorials for exterior algebra? I need to reacquaint myself with differentials and wedge products specifically. Thanks.