Question

Find \(d \widehat{\omega}\) if $$ \widehat{\boldsymbol{\omega}}=\left(\begin{array}{cc} 0 & -\cot \theta \boldsymbol{\epsilon}^{\varphi} \\ \cot \theta \boldsymbol{\epsilon}^{\varphi} & 0 \end{array}\right), $$ where \((\theta, \varphi)\) are coordinates on the unit sphere \(S^{2}\).

Short answer

The exterior derivative of the 1-form \(\omega = -\cot(\theta) d\varphi\) is given by \(d\omega = csc^{2}(\theta) d\theta \wedge d\varphi\).

Step-by-step solution

Break down the matrix

We can write our 1-form \(\widehat{\boldsymbol{\omega}}\) in a more familiar form as \(\omega = -\cot(\theta) d\varphi\). This form represents the sum of the components of the original matrix, expressing them in terms of the differentials \(d\theta\) and \(d\varphi\).

Use the formula for the exterior derivative to calculate \(d\omega\)

The exterior derivative of a 1-form \(\omega = f(\theta, \varphi) d\varphi\) in spherical coordinates is calculated using the formula \(d\omega = \frac{\partial f(\theta, \varphi)}{\partial \theta} d\theta \wedge d\varphi + \frac{\partial f(\theta, \varphi)}{\partial \varphi}d\varphi \wedge d\varphi\). Applying this formula to \(\omega\), we get \(d\omega = \frac{\partial (-\cot(\theta))}{\partial \theta} d\theta \wedge d\varphi + 0\), because the derivative of \(-\cot(\theta)\) with respect to \(\varphi\) is 0.

Simplify the result

After performing the partial derivative calculation and simplifying the result, we get \(d\omega = csc^{2}(\theta) d\theta \wedge d\varphi\). This is a 2-form on the sphere \(S^{2}\) which can be interpreted as giving the differential area element of a region on the sphere.

Key concepts

Exterior Derivative

The exterior derivative is a powerful tool in differential geometry and calculus on manifolds. It allows us to extend the concept of differentiation to forms, which are geometric objects that generalize functions and vector fields. Given a differential form, the exterior derivative helps to compute its differential, which is essential in understanding the behavior of the form under transformations.

For a 1-form, such as \( \omega = f(\theta, \varphi) d\varphi \),we can compute its exterior derivative using:

• \( d\omega = \frac{\partial f}{\partial \theta} d\theta \wedge d\varphi + \frac{\partial f}{\partial \varphi} d\varphi \wedge d\varphi \)

The wedge product \(\wedge\) in this formula makes this operator anti-commutative, crucial for encoding the geometry of the space. The exterior derivative yields a form that is one degree higher than the original, transforming a 1-form into a 2-form. This transformation provides insights into the underlying geometry, such as calculating areas, volumes, and more.

1-Form

A 1-form is a type of differential form that corresponds to a linear functional on the tangent space of a manifold, like a sphere. In simple terms, it can be viewed as a generalization of the concept of a vector field; however, instead of associating vectors with each point in a space, it assigns linear functionals.

When expressed in local coordinates, a 1-form is typically written as:

• \( \omega = f_1(\theta, \varphi) d\theta + f_2(\theta, \varphi) d\varphi \)

This denotes a weighted sum of basic differential forms \(d\theta\) and \(d\varphi\). In the context of our exercise, the 1-form is expressed as \(-\cot(\theta) d\varphi\). 1-forms are essential for performing integration over curves, and the exterior derivative allows us to "lift" this integration into higher dimensions.

Spherical Coordinates

Spherical coordinates are a mathematical system for representing points in three dimensions using three values: radius, polar angle (\(\theta\)), and azimuthal angle (\(\varphi\)). These coordinates are particularly helpful when working with objects that are naturally spherical, such as planets or atoms.

For a unit sphere, the radius is always 1. Thus, the location of any point on the sphere can be uniquely determined just by the angles \(\theta\) and \(\varphi\).
Where:

• \(\theta\) typically represents the angle from the positive z-axis (polar angle).
• \(\varphi\) represents the angle from the positive x-axis in the x-y plane (azimuthal angle).

These angles range continuously from \(0\) to \(\pi\) for \(\theta\) and \(0\) to \(2\pi\) for \(\varphi\). When dealing with calculations on the sphere, spherical coordinates simplify the mathematics by aligning more naturally with the sphere's symmetry.

Unit Sphere

The unit sphere, often denoted as \(S^2\), is a set of points equidistant from a central point in three-dimensional space, where the radius is unitary, or 1. This sphere serves as a critical model as it simplifies many mathematical problems due to its symmetry and homogeneity.

It is defined mathematically by the equation:\[ x^2 + y^2 + z^2 = 1 \] where \((x, y, z)\) are the coordinates of any point on the sphere. Using spherical coordinates, any point on this unit sphere is defined with two parameters, \(\theta\) and \(\varphi\), making calculations and analysis more straightforward as explained above.

In differential geometry, the unit sphere is often used to study surfaces, curvature, and angle relationships, providing a foundational understanding of more complex geometrical and topological concepts.