Search results for "Exterior derivative"
showing 4 items of 4 documents
The exterior derivative as a Killing vector field
1996
Among all the homogeneous Riemannian graded metrics on the algebra of differential forms, those for which the exterior derivative is a Killing graded vector field are characterized. It is shown that all of them are odd, and are naturally associated to an underlying smooth Riemannian metric. It is also shown that all of them are Ricci-flat in the graded sense, and have a graded Laplacian operator that annihilates the whole algebra of differential forms.
Graded Poisson structures on the algebra of differential forms
1995
We study the graded Poisson structures defined on Ω(M), the graded algebra of differential forms on a smooth manifoldM, such that the exterior derivative is a Poisson derivation. We show that they are the odd Poisson structures previously studied by Koszul, that arise from Poisson structures onM. Analogously, we characterize all the graded symplectic forms on ΩM) for which the exterior derivative is a Hamiltomian graded vector field. Finally, we determine the topological obstructions to the possibility of obtaining all odd symplectic forms with this property as the image by the pullback of an automorphism of Ω(M) of a graded symplectic form of degree 1 with respect to which the exterior der…
The General Stokes’s Theorem
2012
Let ω be a differential form of degree k - 1 and class C 1 in a neighborhood of a compact regular k-surface with boundary M of class C 2. The general Stokes’s theorem gives a relationship between the integral of ω over the boundary of M and the integral of the exterior differential dω over M. It can be viewed as a generalization of Green’s theorem to higher dimensions, and it plays a role not unlike that of the fundamental theorem of calculus in an elementary course of analysis. Particular cases of the general Stokes’s theorem that are of great importance are the divergence theorem, which relates a triple integral with a surface integral and what we know as the classical Stokes’s theorem, w…
Geometric Aspects of Thermodynamics
2016
This chapter deals with mathematical aspects of thermodynamics most of which will be seen to be primarily of geometrical nature. Starting with a short excursion to differentiable manifolds we summarize the properties of functions, of vector fields and of one-forms on thermodynamic manifolds. This summary centers on exterior forms over Euclidean spaces and the corresponding differential calculus. In particular, one-forms provide useful tools for the analysis of thermodynamics. A theorem by Caratheodory is developed which is closely related to the second law of thermodynamics. The chapter closes with a discussion of systems which depend on two variables and for which there is an interesting a…