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spaghetti3451
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Is ##\text{d}^{2}=\text{d}\wedge\text{d}## a definition of the exterior algebra, or can it be derived from more fundamental mathematical statements?
That is neither a definition nor is it is true in general. For ##n##-dimensional space and for a ##p##-form, ##** = -(-1)^{p(n-p)}## in Minkowski space and ##** = (-1)^{p(n-p)}## in Euclidean space.failexam said:I know that ##**=-1##, but is this a definition, or can it be proved in two to three lines?
Matterwave said:I'm sorry if I'm mistaken as it has been a while since I've done differential geometry, but isn't ##\text{d}^2=0## one of the defining properties of the exterior derivative?
... and geometry, topology, and (homological) algebra.lavinia said:BTW: Differential forms and exterior derivatives do not require the idea of a metric so they are not specifically restricted to Differential Geometry but rather to Calculus on Manifolds.
fresh_42 said:... and geometry, topology, and (homological) algebra.
Geometry to me means measurement of angles at least and usually also distance. These ideas are not needed to do calculus. Differential forms are just calculus. For instance one can integrate a differential form on a smooth manifold that has no shape and is just a bunch of smoothly overlapping coordinate charts..Matterwave said:I was not aware that differential geometry required a metric? Wouldn't that fall under Riemannian geometry, or Semi-Riemannian geometry?
Fightfish said:That is neither a definition nor is it is true in general. For ##n##-dimensional space and for a ##p##-form, ##** = -(-1)^{p(n-p)}## in Minkowski space and ##** = (-1)^{p(n-p)}## in Euclidean space.
Matterwave said:I was not aware that differential geometry required a metric? Wouldn't that fall under Riemannian geometry, or Semi-Riemannian geometry?
The square of the exterior derivative is a mathematical operation that involves taking the exterior derivative of a differential form, and then taking the exterior derivative of the result. It is denoted as d².
The square of the exterior derivative is used to study the curvature and topology of spaces. It helps to identify closed and exact forms, and can also be used to define cohomology groups.
To calculate the square of the exterior derivative, you first take the exterior derivative of a differential form, and then take the exterior derivative of the resulting form. This can be represented mathematically as d²ω = d(dω).
The square of the exterior derivative is closely related to the Laplacian operator, which is a differential operator used to measure the curvature of a space. In fact, in certain cases, the square of the exterior derivative is equal to the negative of the Laplacian operator.
The square of the exterior derivative has various applications in fields such as differential geometry, topology, and physics. It is used to study the curvature of spaces, define cohomology groups, and also plays a role in the study of electromagnetic fields and gauge theories.