How can I convert an action in terms of differential forms to tensors?
Answers
Answered by
0
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
For instance, the expression f(x) dx from one-variable calculus is an example of a 1-form, and can be integrated over an interval [a, b] in the domain of f:
{\displaystyle \int _{a}^{b}f(x)\,dx.}
Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dx ∧ dz + h(x, y, z) dy ∧ dzis a 2-form that has a surface integral over an oriented surface S:
{\displaystyle \int _{S}f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dx\wedge dz+h(x,y,z)\,dy\wedge dz.}
The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz represents a volume element that can be integrated over a region of space. In general, a k-form is an object that may be integrated over k-dimensional sets, and is homogeneous of degree k in the coordinate differentials.
The algebra of differential forms is organized in a way that naturally reflects the orientationof the domain of integration. There is an operation d on differential forms known as the exterior derivative that, when acting on a k-form, produces a (k + 1)-form. This operation extends the differential of a function, and is directly related to the divergence and the curl of a vector field in a manner that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the generalized Stokes' theorem.
For instance, the expression f(x) dx from one-variable calculus is an example of a 1-form, and can be integrated over an interval [a, b] in the domain of f:
{\displaystyle \int _{a}^{b}f(x)\,dx.}
Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dx ∧ dz + h(x, y, z) dy ∧ dzis a 2-form that has a surface integral over an oriented surface S:
{\displaystyle \int _{S}f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dx\wedge dz+h(x,y,z)\,dy\wedge dz.}
The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz represents a volume element that can be integrated over a region of space. In general, a k-form is an object that may be integrated over k-dimensional sets, and is homogeneous of degree k in the coordinate differentials.
The algebra of differential forms is organized in a way that naturally reflects the orientationof the domain of integration. There is an operation d on differential forms known as the exterior derivative that, when acting on a k-form, produces a (k + 1)-form. This operation extends the differential of a function, and is directly related to the divergence and the curl of a vector field in a manner that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the generalized Stokes' theorem.
Answered by
0
Explanation:
Wikipedia Search
Differential form
Read in another language
Watch this page
Edit
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
For instance, the expression f(x) dx from one-variable calculus is an example of a 1-form, and can be integrated over an interval [a, b] in the domain of f:
Similar questions