Pfaff form

In the mathematical sub-area of differential geometry , Pfaff's form (after Johann Friedrich Pfaff ) or differential form of degree 1 or 1-form for short denotes an object that is in a certain way dual to a vector field . Pfaff's forms are the natural integrands for path integrals .

context

Be it ${\displaystyle U}$

• an open subset of the ${\displaystyle \mathbb {R} ^{n}}$
• or more generally an open part of a differentiable submanifold of the ${\displaystyle \mathbb {R} ^{n}}$
• or in general an open part of an (abstract) differentiable manifold .

In each of these cases there is

• the concept of the differentiable function ; the space of infinitely differentiable functions on going with designated; ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle C^{\infty }(U)}$
• the concept of tangent space an at a point ; ${\displaystyle \mathrm {T} _{p}U}$ ${\displaystyle U}$ ${\displaystyle p\in U}$
• the concept of the directional derivative for a tangential vector and a differentiable function ; ${\displaystyle Xf}$ ${\displaystyle X\in \mathrm {T} _{p}U}$ ${\displaystyle f}$
• the concept of differentiable vector field on . The space of the vector fields is denoted by. ${\displaystyle U}$ ${\displaystyle U}$ ${\displaystyle \Gamma \mathrm {(} TU)}$

Elementary definition

A Pfaffian shape on assigns a linear shape to each point . Such linear forms are called cotangential vectors ; they are elements of the dual space of the tangential space . The space is called the cotangent space . ${\displaystyle \omega }$ ${\displaystyle U}$ ${\displaystyle p\in U}$ ${\displaystyle \omega _{p}\colon \mathrm {T} _{p}U\to \mathbb {R} }$ ${\displaystyle \mathrm {T} _{p}^{*}U}$ ${\displaystyle \mathrm {T} _{p}U}$ ${\displaystyle \mathrm {T} _{p}^{*}U}$

A Pfaffian form is therefore an image ${\displaystyle \omega }$

${\displaystyle \omega \colon U\to \bigsqcup _{p\in U}\mathrm {T} _{p}^{*}U,\quad p\mapsto \omega _{p}\in \mathrm {T} _{p}^{*}U.}$

Other definitions

• A differentiable Pfaffian form is a linear mapping. Continuous or measurable Pfaffian forms are defined analogously. ${\displaystyle C^{\infty }(U)}$ ${\displaystyle \Gamma \mathrm {(} TU)\to C^{\infty }(U).}$
• The amount given above is called the cotangent bundle . This is nothing other than the dual vector bundle of the tangent bundle . A Pfaffian shape can thus be defined as the intersection of the cotangent bundle. ${\displaystyle \textstyle \bigsqcup _{p\in U}\mathrm {T} _{p}^{*}U}$
• Pfaff's forms are precisely the covariant first order tensor fields .

Total differential of a function

See also : Total differential

The total differential or the external derivative of a differentiable function is Pfaff's form, which is defined as follows: If a tangential vector, then: is therefore equal to the directional derivative of in direction . ${\displaystyle \mathrm {d} f}$ ${\displaystyle f\colon U\rightarrow \mathbb {R} }$ ${\displaystyle X\in \mathrm {T} _{p}U}$ ${\displaystyle (\mathrm {d} f)_{p}(X)=Xf,}$ ${\displaystyle f}$ ${\displaystyle X}$

So is a way with and , so is ${\displaystyle \gamma \colon (-\varepsilon ,\varepsilon )\to U}$ ${\displaystyle \gamma (0)=p}$ ${\displaystyle {\dot {\gamma }}(0)=X}$

${\displaystyle (\mathrm {d} f)_{p}(X)=\left.{\frac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}f(\gamma (t)).}$

The following applies:

• ${\displaystyle \mathrm {d} \lambda =0,}$if is a constant function; ${\displaystyle \lambda \in \mathbb {R} }$
• ${\displaystyle \mathrm {d} (fg)=f\cdot \mathrm {d} g+g\cdot \mathrm {d} f}$for differentiable functions . ${\displaystyle f,g\in C^{\infty }(U)}$

If a scalar product is given, the total differential of can be represented with the help of the gradient : ${\displaystyle U}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle f}$

${\displaystyle (\mathrm {d} f)_{p}(X)=\langle \mathrm {grad} \,f,X\rangle .}$

Coordinate representation

Let it be a coordinate system . The coordinates can be used as functions ${\displaystyle x_{1},\ldots ,x_{n}}$ ${\displaystyle U}$

${\displaystyle x_{i}\colon U\to \mathbb {R} ,\quad p\mapsto x_{i}(p)}$

which assign its -th coordinate to a point . The total differentials of these functions form a local basis . That is, for each point it is ${\displaystyle i}$ ${\displaystyle \mathrm {d} x_{1},\ldots ,\mathrm {d} x_{n}}$ ${\displaystyle p\in U}$

${\displaystyle \{(\mathrm {d} x_{1})_{p},\ldots ,(\mathrm {d} x_{n})_{p}\}}$

a base of . ${\displaystyle \mathrm {T} _{p}^{*}U}$

This means that every Pfaff form can be clearly identified as a

${\displaystyle \omega =f_{1}\,\mathrm {d} x_{1}+\ldots +f_{n}\,\mathrm {d} x_{n}}$

write with functions . ${\displaystyle f_{i}\colon U\to \mathbb {R} }$

The outer derivative of any differentiable function has the representation ${\displaystyle f\colon U\to \mathbb {R} }$

${\displaystyle \mathrm {d} f={\frac {\partial f}{\partial x_{1}}}\cdot \mathrm {d} x_{1}+\ldots +{\frac {\partial f}{\partial x_{n}}}\cdot \mathrm {d} x_{n}.}$

Definition of the curve integral

Let there be a continuously differentiable path in and a 1-form . Then the integral of along is defined as: ${\displaystyle \varphi \colon [a,b]\rightarrow U}$ ${\displaystyle U}$ ${\displaystyle \omega }$ ${\displaystyle U}$ ${\displaystyle \omega }$ ${\displaystyle \varphi }$

${\displaystyle \int _{\varphi }\omega =\int _{a}^{b}\omega _{\varphi (t)}({\dot {\varphi }}(t))\,\mathrm {d} t}$

The derivation of after denotes the parameter . ${\displaystyle {\dot {\varphi }}}$ ${\displaystyle \varphi }$ ${\displaystyle t}$

Geometric interpretation of the curve integral

The parameterization of a space curve represents a continuously differentiable function . The parameter can be understood as a time parameter. At the time you are at the place . Then it is driven to the place along a certain path or curve . So at the point in time the end point of the curve has been reached. If the location of the crossing is noted at each point in time, the result is the illustration . ${\displaystyle \varphi \colon [a,b]\to \mathbb {R} ^{3}}$ ${\displaystyle t\in [a,b]}$ ${\displaystyle t=a}$ ${\displaystyle \varphi (a)}$ ${\displaystyle \varphi (b)}$ ${\displaystyle t=b}$ ${\displaystyle \varphi (b)}$ ${\displaystyle t}$ ${\displaystyle \varphi \colon [a,b]\rightarrow \mathbb {R} ^{3}}$

It is vividly clear that the same curve can be traversed in different ways. So constant speed is a possibility. Another results from a slow start and subsequent acceleration. There are different parameterizations for the same curve. The designation "curve integral" is justified because it can be shown that the value of the integral is independent of the selected parameterization of the curve. With one exception: If the start and end points of the curve are swapped, i.e. if the movement is from the end point back to the start point of the curve, the sign of the integral changes.

Curve parameterized according to the arc length. ${\displaystyle \varphi (t)}$

In the visualization space, tangential and cotangential vectors can be identified with one another using the scalar product: The vector for which corresponds to a cotangential vector ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \omega }$ ${\displaystyle {\vec {w}}}$

${\displaystyle \omega ({\vec {x}})=\langle {\vec {w}},{\vec {x}}\rangle }$ for all ${\displaystyle {\vec {x}}\in \mathbb {R} ^{3}}$

applies. So 1-forms can be identified with vector fields.

The integral of a 1-form corresponds to the (ordinary) integral over the scalar product with the tangent vector:

${\displaystyle \int _{\varphi }\omega =\int _{a}^{b}\langle {\vec {w}},{\dot {\varphi }}(t)\rangle \,\mathrm {d} t.}$

If the curve is parameterized according to the arc length , the integrand is the (directed) length of the projection of the vector onto the tangent to the curve: ${\displaystyle {\vec {w}}}$ ${\displaystyle \tau }$

${\displaystyle \int _{\varphi }\omega =\int \|{\vec {w}}\|\cdot \cos \angle ({\vec {w}},{\vec {\tau }})\,\mathrm {d} s.}$

Exact and closed forms

A continuously differentiable function is called an antiderivative of the 1-form if: ${\displaystyle F\colon U\rightarrow \mathbb {R} }$ ${\displaystyle \omega }$

${\displaystyle \mathrm {d} F=\omega .}$

A 1-form is called exact if it has an antiderivative.

A 1-form is called closed if: ${\displaystyle \textstyle \omega =\sum _{i=1}^{n}f_{i}\,\mathrm {d} x_{i}}$

${\displaystyle {\frac {\partial f_{i}}{\partial x_{j}}}={\frac {\partial f_{j}}{\partial x_{i}}}}$for everyone . ${\displaystyle i,j}$

More generally, a total differential can be defined that assigns a 2-form to every 1-form . A form is called closed if and only if applies. It follows from Schwarz's theorem that every exact shape is closed. ${\displaystyle \mathrm {d} \omega }$ ${\displaystyle \mathrm {d} \omega =0}$

Curve integral of the total differential

The following applies to the curve integral of the total differential along a path : ${\displaystyle \mathrm {d} F}$ ${\displaystyle \varphi \colon [a,b]\to U}$

${\displaystyle \int _{\varphi }\mathrm {d} F=F(\varphi (b))-F(\varphi (a)).}$

The integral of the total differential does not depend on the curve shape, but only on the end points of the curve. The integral over a closed curve, i.e. , is therefore equal to zero: ${\displaystyle \varphi (a)=\varphi (b)}$

${\displaystyle \oint _{\varphi }\mathrm {d} F=0.}$

In the special case and the fundamental theorem of analysis results because the integral is on the left ${\displaystyle U\subset \mathbb {R} }$ ${\displaystyle \varphi (t)=t}$

${\displaystyle \int _{\varphi }\mathrm {d} F=\int _{a}^{b}F'(t)\,\mathrm {d} t}$

is. The above statements can be traced back directly to the fundamental theorem.

Existence of an antiderivative

• As mentioned earlier, cohesion is a necessary condition for accuracy. The Poincaré lemma says that the obstacles to the inversion are global in nature: In a simply connected , especially in every star-shaped region , every closed Pfaff form has an antiderivative. In particular, every closed Pfaff form is locally exact. ${\displaystyle U\subset \mathbb {R} ^{n}}$

• A continuous Pfaff form in a domain has an antiderivative if and only if the integral of vanishes along every closed curve in . ${\displaystyle \omega }$ ${\displaystyle U\subset \mathbb {R} ^{n}}$ ${\displaystyle \omega }$ ${\displaystyle \varphi }$ ${\displaystyle U}$

Physical examples of Pfaffian forms

First example "force field"

A force field describes the force that is exerted on an object at any location . For example, the earth moves in the force field of the sun. The force field assigns a force vector to each point . Each force vector can be assigned a linear mapping which, by means of the scalar product, maps any vector linearly onto the number field . Based on this interpretation, the force field can be understood as Pfaff's form or first-order differential form. ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {r}}\in \mathbb {R} ^{3}}$ ${\displaystyle {\vec {F}}({\vec {r}})}$ ${\displaystyle {\vec {F}}({\vec {r}})}$ ${\displaystyle \left\langle {\vec {F}},\cdot \right\rangle }$ ${\displaystyle \left\langle \cdot ,\cdot \right\rangle }$ ${\displaystyle {\vec {r}}}$ ${\displaystyle \mathbb {R} }$

If the force field is represented in Cartesian coordinates, where with or the unit vectors are in Cartesian coordinates, then the following applies to the coordinate representation of the Pfaff form: ${\displaystyle {\vec {e}}_{i}}$ ${\displaystyle i=1,2}$ ${\displaystyle 3}$

${\displaystyle \left\langle {\vec {F}},\cdot \right\rangle =\sum _{i=1}^{3}F_{i}dx_{i}}$.

The differentials are simply the corresponding basis vectors of the dual space, so: ${\displaystyle dx_{i}}$

${\displaystyle \left\langle {\vec {e}}_{i},\cdot \right\rangle =dx_{i}}$.

Work has to be done to move an object in a force field along a path from place to place . The amount of work done is given by the curve integral along the path: ${\displaystyle {\vec {\varphi }}:\left[a,b\right]\rightarrow \mathbb {R} ^{3}}$ ${\displaystyle {\vec {\varphi }}(a)}$ ${\displaystyle {\vec {\varphi }}(b)}$ ${\displaystyle W}$

${\displaystyle W=\int _{\vec {\varphi }}{\vec {F}}=\int _{a}^{b}\left\langle {\vec {F}}({\vec {\varphi }}(s)),{\dot {\vec {\varphi }}}(s)\right\rangle ds}$

In a conservative force field, the size of the work done is path-independent. A conservative force does no work in a closed way. ${\displaystyle W}$

The antiderivative of a conservative force field is called the potential or potential energy of the force . So the total differential of the potential again represents the force . The following applies: ${\displaystyle V}$ ${\displaystyle {\vec {F}}}$ ${\displaystyle V}$ ${\displaystyle {\vec {F}}}$

${\displaystyle {\vec {F}}=-\mathrm {grad} \,V=-\nabla \,V}$

The sign is just a convention.

Second example "entropy"

Another important application of the theory of differential forms is in the field of thermodynamics. According to Clausius' inequality:

${\displaystyle \oint _{c}{\frac {1}{T}}{\overrightarrow {\delta Q}}\leq 0}$

${\displaystyle T}$represents the temperature of the thermodynamic system and the heat exchange contact of the system with its environment. The thermodynamic system can, for example, represent a gas whose independent state variables are temperature , pressure and volume of the gas. The coordinate representation of the heat exchange contact is given by: ${\displaystyle \delta Q}$ ${\displaystyle T}$ ${\displaystyle P}$ ${\displaystyle V}$

${\displaystyle {\overrightarrow {\delta Q}}=f_{P}*{\overrightarrow {dP}}+f_{V}*{\overrightarrow {dV}}+f_{T}*{\overrightarrow {dT}}}$.

The above integral is formed along a closed path in three-dimensional state space . A closed path in the state space is called a cycle in thermodynamics. The differential form has an antiderivative if and only if every cycle is reversible: ${\displaystyle c}$ ${\displaystyle (P,V,T)}$ ${\displaystyle c}$ ${\displaystyle {\tfrac {1}{T}}{\overrightarrow {\delta Q}}}$

${\displaystyle \oint _{c}{\frac {1}{T}}{\overrightarrow {\delta Q}}_{rev}=0}$

In this case, the Pfaff form has an antiderivative called entropy. The following applies to reversible cycle processes: ${\displaystyle {\frac {1}{T}}{\overrightarrow {\delta Q}}_{rev}}$ ${\displaystyle S}$

${\displaystyle {\overrightarrow {DS}}={\frac {1}{T}}{\overrightarrow {\delta Q}}_{rev}}$

${\displaystyle 1/T}$represents an integrating factor which generates a total differential from the differential form . ${\displaystyle {\vec {\delta Q}}_{rev}}$ ${\displaystyle {\vec {DS}}}$

From this follows the second law of thermodynamics:

${\displaystyle \int _{\vec {\varphi }}{\overrightarrow {DS}}\geq \int _{\vec {\varphi }}{\frac {1}{T}}{\overrightarrow {\delta Q}}}$

or

${\displaystyle \Delta S\geq {\frac {1}{T}}\Delta Q}$

In an isolated system there is no heat exchange with the environment, which is why . It follows from the second law that the entropy of an isolated system cannot decrease. ${\displaystyle \Delta S\geq 0}$

literature

• Otto Forster : Analysis. Volume 3: Measure and integration theory, integral theorems in R ⁿ and applications , 8th improved edition. Springer Spectrum, Wiesbaden, 2017, ISBN 978-3-658-16745-5 .
• Martin Schottenloher: Geometry and Symmetry in Physics. Leitmotiv of mathematical physics (= Vieweg textbook mathematical physics ). Vieweg, Braunschweig 1995, ISBN 3-528-06565-6 .

Individual evidence

1. ↑ Günther J. Wirsching: Ordinary differential equations. An introduction with examples, exercises and sample solutions. Teubner Verlag, Wiesbaden 2006, ISBN 3-519-00515-8 , p. 63, books.google.de