Exterior Derivative Without Local Coordinates?

Introduction

The exterior derivative is a fundamental concept in differential geometry and topology, used to extend the differential of a function to differential forms of higher degree. While it is often introduced in the context of local coordinates, it can also be defined in a coordinate-free manner. In this article, we will explore the exterior derivative without local coordinates, providing a deeper understanding of this important mathematical tool.

Coordinate-Free Definition

The exterior derivative is a linear map $d: \Omega^k(M) \to \Omega^{k+1}(M)$, where $\Omega^k(M)$ is the space of $k$-forms on a manifold $M$. In local coordinates, the exterior derivative is defined as

$$d\omega = \sum_{i=1}^n \frac{\partial \omega}{\partial x^i} dx^i \wedge dx^1 \wedge \cdots \wedge \widehat{dx^i} \wedge \cdots \wedge dx^n,$$

where $\omega$ is a $k$-form, $x^i$ are the local coordinates, and $\widehat{dx^i}$ indicates that the $i$-th term is omitted.

However, this definition relies on the choice of local coordinates, which may not be well-defined or even exist in certain situations. To overcome this limitation, we can define the exterior derivative in a coordinate-free manner.

Definition using the Cartan-Euler Formula

The Cartan-Euler formula states that for any $k$-form $\omega$ and any vector field $X$,

$$d\omega(X_1, \ldots, X_{k+1}) = \sum_{i=1}^{k+1} (-1)^{i+1} X_i(\omega(X_1, \ldots, \widehat{X_i}, \ldots, X_{k+1})).$$

This formula can be used to define the exterior derivative without local coordinates.

Definition using the Lie Derivative

The Lie derivative is a way to differentiate vector fields and tensor fields along a curve on a manifold. It can be used to define the exterior derivative in a coordinate-free manner.

Let $\omega$ be a $k$-form and $X$ be a vector field. The Lie derivative of $\omega$ with respect to $X$ is defined as

$$\mathcal{L}_X \omega = \lim_{t \to 0} \frac{1}{t} (\phi_t^* \omega - \omega),$$

where $\phi_t$ is the flow of $X$.

Using the Lie derivative, we can define the exterior derivative as

$$d\omega(X_1, \ldots, X_{k+1}) = \mathcal{L}_X \omega(X_1, \ldots, X_{k+1}).$$

Properties of the Exterior Derivative

The exterior derivative has several important properties, including:

• Linearity: The exterior derivative is a linear map, meaning that it preserves the addition and scalar multiplication of forms.
• Leibniz Rule: The exterior derivative satisfies the Leibniz rule, which states that for any forms $\omega$ and $\eta$,

$$d(\omegawedge \eta) = d\omega \wedge \eta + (-1)^k \omega \wedge d\eta,$$

where $k$ is the degree of $\omega$.

• Closed Forms: A form $\omega$ is said to be closed if $d\omega = 0$. The exterior derivative is a way to detect whether a form is closed or not.

Applications of the Exterior Derivative

The exterior derivative has numerous applications in mathematics and physics, including:

• Differential Geometry: The exterior derivative is used to define the curvature of a connection on a manifold.
• Topology: The exterior derivative is used to define the Euler characteristic of a manifold.
• Physics: The exterior derivative is used to describe the behavior of physical systems, such as the motion of particles and the propagation of electromagnetic waves.

Conclusion

In this article, we have explored the exterior derivative without local coordinates, providing a deeper understanding of this important mathematical tool. We have defined the exterior derivative using the Cartan-Euler formula and the Lie derivative, and discussed its properties and applications. The exterior derivative is a fundamental concept in differential geometry and topology, and its coordinate-free definition has far-reaching implications for our understanding of the world around us.

References

• Cartan, E. (1922). Leçons sur la géométrie des espaces de Riemann. Paris: Gauthier-Villars.
• Euler, L. (1760). Recherches sur la courbure des surfaces. Mémoires de l'Académie des Sciences de Berlin, 10, 175-194.
• Kobayashi, S., & Nomizu, K. (1963). Foundations of differential geometry. New York: Wiley.

Further Reading

• Bott, R., & Tu, L. W. (1982). Differential forms in algebraic topology. New York: Springer-Verlag.
• Spivak, M. (1979). A comprehensive introduction to differential geometry. Boston: Publish or Perish.
  Exterior Derivative without Local Coordinates: Q&A =====================================================

Introduction

In our previous article, we explored the exterior derivative without local coordinates, providing a deeper understanding of this important mathematical tool. In this article, we will answer some frequently asked questions about the exterior derivative, its properties, and its applications.

Q: What is the exterior derivative?

A: The exterior derivative is a linear map $d: \Omega^k(M) \to \Omega^{k+1}(M)$, where $\Omega^k(M)$ is the space of $k$-forms on a manifold $M$. It is a way to extend the differential of a function to differential forms of higher degree.

Q: How is the exterior derivative defined?

A: The exterior derivative can be defined using the Cartan-Euler formula or the Lie derivative. The Cartan-Euler formula states that for any $k$-form $\omega$ and any vector field $X$,

$$d\omega(X_1, \ldots, X_{k+1}) = \sum_{i=1}^{k+1} (-1)^{i+1} X_i(\omega(X_1, \ldots, \widehat{X_i}, \ldots, X_{k+1})).$$

The Lie derivative is a way to differentiate vector fields and tensor fields along a curve on a manifold. It can be used to define the exterior derivative as

$$d\omega(X_1, \ldots, X_{k+1}) = \mathcal{L}_X \omega(X_1, \ldots, X_{k+1}).$$

Q: What are the properties of the exterior derivative?

A: The exterior derivative has several important properties, including:

• Linearity: The exterior derivative is a linear map, meaning that it preserves the addition and scalar multiplication of forms.
• Leibniz Rule: The exterior derivative satisfies the Leibniz rule, which states that for any forms $\omega$ and $\eta$,

$$d(\omega \wedge \eta) = d\omega \wedge \eta + (-1)^k \omega \wedge d\eta,$$

where $k$ is the degree of $\omega$.

• Closed Forms: A form $\omega$ is said to be closed if $d\omega = 0$. The exterior derivative is a way to detect whether a form is closed or not.

Q: What are the applications of the exterior derivative?

A: The exterior derivative has numerous applications in mathematics and physics, including:

• Differential Geometry: The exterior derivative is used to define the curvature of a connection on a manifold.
• Topology: The exterior derivative is used to define the Euler characteristic of a manifold.
• Physics: The exterior derivative is used to describe the behavior of physical systems, such as the motion of particles and the propagation of electromagnetic waves.

Q: Can the exterior derivative be used to solve problems in differential geometry?

A: Yes, the exterior derivative can be used to solve problems in differential geometry. For example, it can be used to compute the curvature of a connection on a manifold, which is an important concept in differential geometry.

Q: Can the derivative be used to solve problems in topology?

A: Yes, the exterior derivative can be used to solve problems in topology. For example, it can be used to compute the Euler characteristic of a manifold, which is an important concept in topology.

Q: Can the exterior derivative be used to solve problems in physics?

A: Yes, the exterior derivative can be used to solve problems in physics. For example, it can be used to describe the behavior of physical systems, such as the motion of particles and the propagation of electromagnetic waves.

Conclusion

In this article, we have answered some frequently asked questions about the exterior derivative, its properties, and its applications. The exterior derivative is a fundamental concept in differential geometry and topology, and its coordinate-free definition has far-reaching implications for our understanding of the world around us.

References

• Cartan, E. (1922). Leçons sur la géométrie des espaces de Riemann. Paris: Gauthier-Villars.
• Euler, L. (1760). Recherches sur la courbure des surfaces. Mémoires de l'Académie des Sciences de Berlin, 10, 175-194.
• Kobayashi, S., & Nomizu, K. (1963). Foundations of differential geometry. New York: Wiley.

Further Reading

• Bott, R., & Tu, L. W. (1982). Differential forms in algebraic topology. New York: Springer-Verlag.
• Spivak, M. (1979). A comprehensive introduction to differential geometry. Boston: Publish or Perish.