n dimensional vector space.
� 34. Forms on Rn Definition 34.1. A basic 1-form on Rn is a formal symbol dx1,dx2,...,dxn. A general 1-form on Rn is any expression of the form ω = f1 dx1 + f2 dx2 ++ fn dxn,··· where f1,f2,...,fn are smooth functions. Note that there are n basic 1-forms. If f is a smooth function, we get a 1-form using the formal rule, n� ∂f df =dxi. ∂xii=1 Definition 34.2. A basic 2-form on Rn is any formal symbol dxi ∧ dxj , where 1 ≤ i
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Vector space of n dimension is defined. The basic 1-form and the general 1-form for n smooth functions on the n dimensional vector space are defined and explained. In the similar way basic 2-form and hereby basic k-form and general 2- form and in turn general k- form are derived and analysed for a vector field of n dimension. The Hodge star operator is defined. These notation can be used to express Maxwell’s equations quite efficiently.
Prof. James McKernan, Maths, 18.022. Calculus of Several Variables, Fall 2010: 34. Forms on n to the power R: Massachusetts Institute of Technology: MIT OpenCourseWare),http://ocw.mit.edu (Accessed October19, 2011). License: Creative Commons BY-NC-SA:http://ocw.mit.edu/terms/#cc
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