Question

Find the scale factors and hence the volume element for the coordinate system \((u, v, \theta)\) defined by $$ x_{1}=u v \cos \theta, \quad x_{2}=u v \sin \theta, \quad x_{3}=\left(u^{2}-v^{2}\right) / 2 $$ in which \(u\) and \(v\) are positive and \(0 \leq \theta<2 \pi\). Hence find the volume of the region enclosed by the curved surfaces \(u=1\) and \(v=1\).

Short answer

The volume of the region enclosed by the curved surfaces \(u=1\) and \(v=1\) is calculated by integrating the volume element obtained from the determinant of the Jacobian over the specified limits.

Step-by-step solution

Calculate the Jacobian Matrix

The first step to find the scale factors and the volume element is to calculate the Jacobian matrix of the transformation from \(u, v, \theta\) to \((x_{1}, x_{2}, x_{3})\). This is the matrix of all the first-order partial derivatives of \(x_{1}\), \(x_{2}\), and \(x_{3}\) with respect to \(u\), \(v\), and \(\theta\).

Compute the Determinant of the Jacobian Matrix

Next, we need to compute the determinant of the Jacobian matrix, which will give us the scale factor for the volume element in the new coordinate system \(u, v, \theta\). The determinant of the Jacobian is also known as the Jacobian determinant.

Calculate the Volume Element

The volume element in the new coordinate system is found by taking the absolute value of the Jacobian determinant. This gives us the scale factors for \(u\), \(v\), and \(\theta\) coordinates.

Integrate the Volume Element

To find the volume of the region enclosed by the surfaces \(u=1\) and \(v=1\), we need to integrate the volume element over the appropriate bounds of \(u\), \(v\), and \(\theta\).

Evaluate the Integral

The last step is to evaluate the integral using the bounds \(u\) from 0 to 1, \(v\) from 0 to 1, and \(\theta\) from 0 to \(2\pi\). This will give us the volume of the region enclosed by the given surfaces.

Key concepts

Jacobian Matrix

In multivariable calculus, the Jacobian matrix is a powerful tool that arises from the necessity to understand how differentiable functions behave under coordinate transformations. Think of it as a bridge between two worlds: one described by a set of coordinates \( (u, v, \theta) \) and another by \( (x_1, x_2, x_3) \).

This bridge is constructed by taking the partial derivatives of each 'new world' coordinate with respect to each 'original world' coordinate. For instance, in our exercise, we create a matrix where each element is a first-order partial derivative, such as \( \frac{\partial x_1}{\partial u} \) or \( \frac{\partial x_2}{\partial \theta} \), and so on. Each row in this matrix represents the gradient of one transformed coordinate with respect to all the original coordinates. This matrix not only directs us toward the Jacobian determinant, which we'll discuss next, but also helps us understand how changes in the original coordinates create ripple effects in the transformed space.

Jacobian Determinant

The Jacobian determinant, often simply called 'the Jacobian,' emerges from the Jacobian matrix as a single value encapsulating the 'density' of our bridge between coordinate systems. By 'density,' I mean how much the space is stretched or compressed when we move from our original to our transformed coordinates.

To calculate the Jacobian determinant, we find the determinant of the Jacobian matrix we established earlier. The determinant gives us a real number that represents the volume scaling factor between the two coordinate systems. When we change from \( (u, v, \theta) \) to \( (x_1, x_2, x_3) \), the volume of a small region around any point is multiplied by the value of this determinant. This scaling factor is inevitably intertwined with the volume element, which is our next concept to unwrap.

Multivariable Calculus

At the heart of our problem lies multivariable calculus, a field that extends single-variable calculus to higher dimensions. No longer are we looking at functions of a single variable like \( y = f(x) \); we've now got functions that depend on multiple variables—just like the transformation in our exercise that depends on \( u \), \( v \), and \( \theta \).

Within this broader scope, we encounter tasks such as finding volume elements, which are infinitesimally small 'chunks' of space that we can use to build up a volume through integration. This calculus of small pieces lets us compute volumes—even of regions with complex, curvy boundaries—by adding up the contributions of each little piece, scaled appropriately using concepts like the Jacobian determinant.

Coordinate Transformations

Coordinate transformations are navigational maneuvers that allow us to switch between different coordinate systems. Whether we're working with spherical coordinates, polar coordinates, or the bespoke system in our problem, transformations enable us to adapt the mathematics to the symmetry or geometry of a problem.

Through a transformation, we're effectively restating the problem in a new language that might make it easier to solve. This is precisely what we do when we calculate volume elements for the coordinate system \( (u, v, \theta) \): we start with a commonplace Cartesian system and morph it into one that suits our specific problem better. We perform this transformation by using the equations given and derive the factors by which our original unit volumes are stretched or squashed. These factors then guide how we integrate to find the volumes of regions—in this case, the region bounded by \( u=1 \) and \( v=1 \).