Question

Let \(x^{1}=x, x^{2}=y, x^{3}=z\) be coorduates on the manifold \(\mathbb{R}^{3}\). Write out the components \(\alpha_{u}\) and \((\mathrm{d} \alpha)_{n k}\), etc. for each of the following 2 -forms: $$ \begin{aligned} &\alpha=\mathrm{d} y \wedge \mathrm{d} z+\mathrm{d} x \wedge \mathrm{d} y \\ &\beta=x \mathrm{~d} z \wedge \mathrm{d} y+y \mathrm{~d} x \wedge \mathrm{d} z+z \mathrm{\phi} y \wedge \mathrm{d} x \\ &\gamma=\mathrm{d}\left(r^{2}(x \mathrm{~d} x+y \mathrm{~d} y+z \mathrm{~d} z)\right) \text { where } r^{2}=x^{2}+y^{2}+z^{2}. \end{aligned} $$

Short answer

The non-zero components for the forms are: \(\alpha_{23} = \alpha_{31} = 1\), \(\alpha_{12} = -1\); \(\beta_{32} = x\), \(\beta_{13} = y\), \(\beta_{21} = z\); \(\gamma_{32} = \gamma_{13} = \gamma_{21} = 2x^2 + 2y^2 + 2z^2\).

Step-by-step solution

Find the components of \(\alpha\)

In this differential form, \(\alpha = \mathrm{d} y \wedge \mathrm{d} z + \mathrm{d} x \wedge \mathrm{d} y\). The non-zero components of \(\alpha\) are directly given from the 2-form itself, i.e., \(\alpha_{23} = \alpha_{31} = 1\), and \(\alpha_{12} = -1\) (this comes from reversing the order of the wedge product).

Find the components of \(\beta\)

In this differential form, \(\beta = x\, \mathrm{d} z \wedge \mathrm{d} y + y\, \mathrm{d} x \wedge \mathrm{d} z + z\, \mathrm{d} y \wedge \mathrm{d} x\). We have \(\beta_{32} = x\), \(\beta_{13} = y\), and \(\beta_{21} = z\).

Find the components of \(\gamma\)

In this differential form, \(\gamma = \mathrm{d}(r^{2}(x\, \mathrm{d} x + y\, \mathrm{d} y + z\, \mathrm{d} z)\) where \(r^{2}=x^{2}+y^{2}+z^{2}\). Notice that \(\gamma\) is actually the exterior derivative of another 1-form, which is \(r^{2}(x\, \mathrm{d} x + y\, \mathrm{d} y + z\, \mathrm{d} z)\). Expand the exterior derivative, and we get \(\gamma_{32} = 2x^2 + 2y^2 + 2z^2\), \(\gamma_{13} = 2x^2 + 2y^2 + 2z^2\), and \(\gamma_{21} = 2x^2 + 2y^2 + 2z^2\).

Key concepts

Exterior Derivative

The exterior derivative is a fundamental concept in differential geometry. It generalizes the idea of taking a derivative to functions on manifolds. In simple terms, it lets you "differentiate" differential forms. If you have a smooth function, say \( f \), its exterior derivative \( df \) is a 1-form. For a 1-form \( \alpha = g \, dx + h \, dy + i \, dz \), its exterior derivative \( d\alpha \) would be a 2-form. Its computation involves taking partial derivatives and using the wedge product, which is anti-commutative.

Here's how it works: given a 1-form \( \alpha \), the exterior derivative \( d\alpha \) is calculated by organizing the partial derivatives of the components, followed by pairing them using the wedge product. For a 2-form \( \beta \), \( d\beta \) results in a 3-form, and the process continues similarly for higher degree forms. The beauty of the exterior derivative is that it is coordinate independent, making it an invaluable tool in manifold theory and Riemannian geometry. Remember, the exterior derivative of a 0-form (a function) is a 1-form, the exterior derivative of a 1-form is a 2-form, and so on.

Manifold

Manifolds can be thought of as multi-dimensional surfaces that locally plane-like, or Euclidean. They are mathematical spaces that, at a small enough scale, resemble the Euclidean space \( \mathbb{R}^n \). This means that you can apply calculus on manifolds because they're, under a magnifying glass, just like the spaces you're used to working with.

Manifolds come with charts and atlases, which are sets of coordinates that help navigate the manifold just like a grid system helps with a map of Earth. While the surface of a sphere might at first seem complex, you can make sense of it by working within small, simple patches using coordinate systems. Using these coordinate systems, the properties of manifolds are described in terms of functions and forms.

The study of manifolds bridges global topology with local geometry, leading to many results in fields like physics, where Einstein's theory of general relativity perceives the universe as a four-dimensional manifold.

Coordinate System

A coordinate system on a manifold is a tool that assigns a set of numbers to every point on the manifold. On \( \mathbb{R}^3 \), commonly used coordinates are \( (x, y, z) \). This work as a map helping to navigate the mathematical 'terrain' of the manifold<.br>
These coordinates help define how a manifold appears and are crucial in working with differential forms. They also allow various operations like the calculation of derivatives to be performed as if on simple Cartesian space, facilitated by charts (regions) covering the manifold.

The choice of a coordinate system can largely simplify complex operations, making some calculations intuitive and straightforward. This flexibility is advantageous in fields like differential geometry, where transformations and morphisms between different manifolds or coordinate systems often occur.

Riemannian Geometry

Riemannian geometry introduces the concept of a Riemannian metric, enabling the calculation of distances, angles, and volumes on manifolds. It builds upon Euclidean geometry but in the context of curved spaces. This is crucial in understanding physical theories where the concept of space isn't flat, such as in general relativity.

A Riemannian manifold is equipped with a metric tensor \( g \), a function that measures infinitesimal distances, allowing for geodesic computations (paths of shortest distance between points).

In the realm of differential forms, Riemannian geometry offers a robust framework for using integrals over manifolds. It introduces notions of curvature and provides the tools to study surfaces and their properties, thereby deepening our understanding of space itself. This plays a pivotal role in transforming abstract mathematical concepts into something that describes the tangible universe.