# Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations

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## Abstract

**:**

## 1. Introduction

## 2. Vector Fields, One-Form Fields, and Their Evolutionary Form

#### Evolutionary Vector Fields and One-Form Fields

## 3. Geometric Formulation of Symmetries and Adjoint-Symmetries

**Theorem**

**1.**

**Proposition**

**1.**

#### Examples of Adjoint-Symmetries

## 4. Some Applications

**Remark**

**1.**

#### 4.1. Conservation Laws from Symmetries and Adjoint-Symmetries

**Theorem**

**2.**

#### 4.2. Action of Symmetries on Adjoint-Symmetries

**Proposition**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. Geometrical Adjoint-Symmetries of Evolution Equations

**Proposition**

**3.**

**Theorem**

**4.**

#### Evolution Equations with Spatial Constraints

**Theorem**

**5.**

**Theorem**

**6.**

**Theorem**

**7.**

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Anco, S.C.; Wang, B.
Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations. *Symmetry* **2020**, *12*, 1547.
https://doi.org/10.3390/sym12091547

**AMA Style**

Anco SC, Wang B.
Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations. *Symmetry*. 2020; 12(9):1547.
https://doi.org/10.3390/sym12091547

**Chicago/Turabian Style**

Anco, Stephen C., and Bao Wang.
2020. "Geometrical Formulation for Adjoint-Symmetries of Partial Differential Equations" *Symmetry* 12, no. 9: 1547.
https://doi.org/10.3390/sym12091547