Question

Define \(G: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) by \(G\left(r, \phi_{1}, \ldots, \phi_{n-2}, \theta\right)=\left(x_{1}, \ldots, x_{n}\right)\) where \(x_{1}=r \cos \phi_{1}, \quad x_{2}=r \sin \phi_{1} \cos \phi_{2}, \quad x_{3}=r \sin \phi_{1} \sin \phi_{2} \cos \phi_{3}, \ldots\), \(x_{n-1}=r \sin \phi_{1} \cdots \sin \phi_{n-2} \cos \theta, \quad x_{n}=r \sin \phi_{1} \cdots \sin \phi_{n-2} \sin \theta .\) a. \(G\) maps \(\mathbb{R}^{n}\) onto \(\mathbb{R}^{n}\), and \(\left|G\left(r, \phi_{1}, \ldots, \phi_{n-2,} \theta\right)\right|=|r|\). b. \(\operatorname{det} D_{\left(r, \phi_{1}, \ldots, \phi_{n-2}, \theta\right)} G=r^{n-1} \sin ^{n-2} \phi_{1} \sin ^{n-3} \phi_{2} \cdots \sin \phi_{n-2}\). c. Let \(\Omega=(0, \infty) \times(0, \pi)^{n-2} \times(0,2 \pi)\). Then \(G \mid \Omega\) is a diffeomorphism and \(m\left(\mathbb{R}^{n} \backslash G(\Omega)\right)=0\). d. Let \(F\left(\phi_{1}, \ldots, \phi_{n-2}, \theta\right)=G\left(1, \phi_{1}, \ldots, \phi_{n-2}, \theta\right)\) and \(\Omega^{\prime}=(0, \pi)^{n-2} \times\) \((0,2 \pi)\). Then \(\left(F \mid \Omega^{\prime}\right)^{-1}\) defines a coordinate system on \(S^{n-1}\) except on a \(\sigma\)-null set, and the measure \(\sigma\) is given in these coordinates by $$ d \sigma\left(\phi_{1}, \ldots \phi_{n-2}, \theta\right)=\sin ^{n-2} \phi_{1} \sin ^{n-3} \phi_{2} \cdots \sin \phi_{n-2} d \phi_{1} \cdots d \phi_{n-2} d \theta $$

Short answer

The map G is a diffeomorphism with the given Jacobian, and F forms coordinates on the sphere definition.

Step-by-step solution

Analyze the Mapping G

The mapping \( G: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n} \) takes the polar coordinates \( (r, \phi_{1}, \ldots, \phi_{n-2}, \theta) \) and gives a point in Cartesian coordinates \( (x_{1}, x_{2}, \ldots, x_{n}) \). Each coordinate \( x_{i} \) is expressed in terms of \( r \) and angles \( \phi_{i} \) and \( \theta \), analogous to spherical coordinates.

Magnitude Calculation

For part a, compute the magnitude of the vector \( (x_{1}, x_{2}, \ldots, x_{n}) \). The magnitude will be \( |x| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} \). Given the definitions of \( x_{i} \), the result simplifies to \(|r|\) by trigonometric identities.

Jacobian Determinant Calculation

For part b, find the Jacobian determinant of the map \( G \). The Jacobian matrix \( D_{(r, \phi_1, \ldots, \phi_{n-2}, \theta)}G \) is derived by taking partial derivatives of \( x_i \) with respect to \( r, \phi_1, \ldots, \phi_{n-2}, \theta \). The determinant of this matrix is \( r^{n-1} \sin^{n-2} \phi_1 \sin^{n-3} \phi_2 \cdots \sin \phi_{n-2} \).

Diffeomorphism Analysis

For part c, analyze the region \( \Omega = (0, \infty) \times (0, \pi)^{n-2} \times (0, 2\pi) \). The map \( G \) restricted to this domain is a diffeomorphism onto \( \mathbb{R}^n \), except a set of measure zero, since Jacobian is not zero.

Coordinate System on the Sphere

For part d, consider the transformation when \( r = 1 \). The function \( F(\phi_{1}, \ldots, \phi_{n-2}, \theta) = G(1, \phi_{1}, \ldots, \phi_{n-2}, \theta) \) maps onto the sphere \( S^{n-1} \). The inverse map \( (F | \Omega^{\prime})^{-1} \) describes the coordinates on \( S^{n-1} \) excluding a \( \sigma \)-null set and the measure \( d\sigma \) is given.

Key concepts

Polar Coordinates

Polar coordinates present a two-dimensional coordinate system where a point in the plane is described by a distance from a reference point and an angle from a reference direction. When extended to three or more dimensions, these coordinates can refer to cylindrical or spherical coordinates.
This system is very useful in multivariable calculus because it simplifies the equations and integrals involving circular or spherical symmetries.
In our exercise, we use polar coordinates to transform a point from \(\mathbb{R}^{n}\) to traditional Cartesian coordinates \((x_1, x_2, \ldots, x_n)\), which are more intuitive for most applications.
This transformation helps in solving problems involving rotations and symmetries.

Jacobian Determinant

The Jacobian determinant is a crucial piece of multivariable calculus. It appears when changing variables in multiple integrals.
It provides information about the local volume change during a transformation.
Specifically, when transforming coordinates, the Jacobian determinant gives us the factor by which volumes are scaled.
For the mapping \( G \) in our exercise, the Jacobian determinant depends on the radius \( r \) and the angles \( \phi_1, \ldots, \phi_{n-2}, \theta \).
It tells us how much the geometry of space is warped by the transformation, and it is given by \( r^{n-1} \sin^{n-2} \phi_1 \sin^{n-3} \phi_2 \cdots \sin \phi_{n-2} \).
This determinant is vital for ensuring that integrals or other calculations remain accurate in the new coordinate system.

Spherical Coordinates

Spherical coordinates generalize polar coordinates to three or more dimensions.
Instead of a single angle, multiple angles are used to describe a point's location relative to a central point.
These coordinates are especially useful when dealing with systems that exhibit radial symmetry, like spheres or globes.
In the exercise, spherical coordinates are used to express the position of each point as a function of a radius \( r \) and a series of angles \( \phi_1, \phi_2, \ldots, \phi_{n-2}, \theta \).
Each angle helps in pinning down the exact location in \( \mathbb{R}^{n} \), allowing for precise conversions between this system and Cartesian coordinates.
Understanding spherical coordinates is key for working with problems involving spheres, such as calculating the surface area or volume of round objects.

Diffeomorphism

A diffeomorphism is a function that is a smooth bijection with a smooth inverse.
This property makes it a powerful concept in calculus and differential geometry because it ensures that the function behaves nicely both ways, in mapping forward and backward.
In our context, the map \( G \mid \Omega \), from our exercise, is proved to be a diffeomorphism.
This means that the transformation from spherical to Cartesian coordinates and back is smooth and does not lose information, except on a set of measure zero.
Determining that a function is a diffeomorphism provides confidence in using the function for rigorous mathematical analysis, allowing students to seamlessly transition between different coordinate systems.

Measure Theory

Measure theory is a branch of mathematics focused on extending the notion of length, area, and volume in a more abstract way.
This concept is essential when dealing with sets that are too 'strange' to have a traditional length or volume.
It helps us rigorously define integrals over general spaces.
In our exercise, measure theory plays a role in understanding the term "\( \sigma\)-null set", which refers to a set of points that are negligible in terms of measure.
This means that while they might exist, they don't contribute to the integral.
The measure \( d \sigma \) given in the coordinates helps us calculate measures on the surface of the sphere \( S^{n-1} \), which can be complicated using standard methods.
Understanding measure theory is vital for dealing with complex integrations and ensuring the accuracy of solutions in higher-dimensional spaces.