Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates:
where the are smooth functions. From this perspective, a one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field.
The most basic non-trivial differential one-form is the "change in angle" form This is defined as the derivative of the angle "function" (which is only defined up to an additive constant), which can be explicitly defined in terms of the atan2 function. Taking the derivative yields the following formula for the total derivative:
While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative -axis – which reflects the fact that angle cannot be continuously defined, this derivative is continuously defined except at the origin, reflecting the fact that infinitesimal (and indeed local) changes in angle can be defined everywhere except the origin. Integrating this derivative along a path gives the total change in angle over the path, and integrating over a closed loop gives the winding number times
In the language of differential geometry, this derivative is a one-form, and it is closed (its derivative is zero) but not exact (it is not the derivative of a 0-form, that is, a function), and in fact it generates the first de Rham cohomology of the punctured plane. This is the most basic example of such a form, and it is fundamental in differential geometry.
Let be open (for example, an interval ), and consider a differentiable function with derivative The differential of at a point is defined as a certain linear map of the variable Specifically, (The meaning of the symbol is thus revealed: it is simply an argument, or independent variable, of the linear function ) Hence the map sends each point to a linear functional This is the simplest example of a differential (one-)form.
Differential form – Expression that may appear after an integral sign
Inner product – Generalization of the dot product; used to define Hilbert spacesPages displaying short descriptions of redirect targets
Reciprocal lattice – Fourier transform of real-space lattices, important in solid-state physics
Tensor – Algebraic object with geometric applications
References
^"2 Introducing Differential Geometry‣ General Relativity by David Tong". www.damtp.cam.ac.uk. Retrieved 2022-10-04.
^McInerney, Andrew (2013-07-09). First Steps in Differential Geometry: Riemannian, Contact, Symplectic. Springer Science & Business Media. pp. 136–155. ISBN978-1-4614-7732-7.
March 10, 2023
form, differential, geometry, form, redirects, here, confused, with, form, linear, algebra, differential, geometry, form, differentiable, manifold, smooth, section, cotangent, bundle, equivalently, form, manifold, displaystyle, smooth, mapping, total, space, t. One form redirects here Not to be confused with One form linear algebra In differential geometry a one form on a differentiable manifold is a smooth section of the cotangent bundle 1 Equivalently a one form on a manifold M displaystyle M is a smooth mapping of the total space of the tangent bundle of M displaystyle M to R displaystyle mathbb R whose restriction to each fibre is a linear functional on the tangent space 2 Symbolically a T M R a x a T x M T x M R displaystyle alpha TM rightarrow mathbb R quad alpha x alpha T x M T x M rightarrow mathbb R where a x displaystyle alpha x is linear Often one forms are described locally particularly in local coordinates In a local coordinate system a one form is a linear combination of the differentials of the coordinates a x f 1 x d x 1 f 2 x d x 2 f n x d x n displaystyle alpha x f 1 x dx 1 f 2 x dx 2 cdots f n x dx n where the f i displaystyle f i are smooth functions From this perspective a one form has a covariant transformation law on passing from one coordinate system to another Thus a one form is an order 1 covariant tensor field Contents 1 Examples 2 Differential of a function 3 See also 4 ReferencesExamples EditThe most basic non trivial differential one form is the change in angle form d 8 displaystyle d theta This is defined as the derivative of the angle function 8 x y displaystyle theta x y which is only defined up to an additive constant which can be explicitly defined in terms of the atan2 function Taking the derivative yields the following formula for the total derivative d 8 x atan2 y x d x y atan2 y x d y y x 2 y 2 d x x x 2 y 2 d y displaystyle begin aligned d theta amp partial x left operatorname atan2 y x right dx partial y left operatorname atan2 y x right dy amp frac y x 2 y 2 dx frac x x 2 y 2 dy end aligned While the angle function cannot be continuously defined the function atan2 is discontinuous along the negative y displaystyle y axis which reflects the fact that angle cannot be continuously defined this derivative is continuously defined except at the origin reflecting the fact that infinitesimal and indeed local changes in angle can be defined everywhere except the origin Integrating this derivative along a path gives the total change in angle over the path and integrating over a closed loop gives the winding number times 2 p displaystyle 2 pi In the language of differential geometry this derivative is a one form and it is closed its derivative is zero but not exact it is not the derivative of a 0 form that is a function and in fact it generates the first de Rham cohomology of the punctured plane This is the most basic example of such a form and it is fundamental in differential geometry Differential of a function EditMain article Differential of a function Let U R displaystyle U subseteq mathbb R be open for example an interval a b displaystyle a b and consider a differentiable function f U R displaystyle f U to mathbb R with derivative f displaystyle f The differential d f displaystyle df of f displaystyle f at a point x 0 U displaystyle x 0 in U is defined as a certain linear map of the variable d x displaystyle dx Specifically d f x 0 d x f x 0 d x displaystyle df x 0 cdot dx mapsto f x 0 dx The meaning of the symbol d x displaystyle dx is thus revealed it is simply an argument or independent variable of the linear function d f x 0 displaystyle df x 0 cdot Hence the map x d f x displaystyle x mapsto df x sends each point x displaystyle x to a linear functional d f x displaystyle df x cdot This is the simplest example of a differential one form In terms of the de Rham cochain complex one has an assignment from zero forms scalar functions to one forms that is f d f displaystyle f mapsto df See also EditDifferential form Expression that may appear after an integral sign Inner product Generalization of the dot product used to define Hilbert spacesPages displaying short descriptions of redirect targets Reciprocal lattice Fourier transform of real space lattices important in solid state physics Tensor Algebraic object with geometric applicationsReferences Edit 2 Introducing Differential Geometry General Relativity by David Tong www damtp cam ac uk Retrieved 2022 10 04 McInerney Andrew 2013 07 09 First Steps in Differential Geometry Riemannian Contact Symplectic Springer Science amp Business Media pp 136 155 ISBN 978 1 4614 7732 7 Retrieved from https en wikipedia org w index php title One form differential geometry amp oldid 1121864122, wikipedia, wiki, book, books, library,