Problem 47 - Calculus Early Transcendentals

Question

Spherical coordinates Evaluate the Jacobian for the transformation from spherical to rectangular coordinates: \(x=\rho \sin \varphi \cos \theta, y=\rho \sin \varphi \sin \theta, z=\rho \cos \varphi .\) Show that \(J(\rho, \varphi, \theta)=\rho^{2} \sin \varphi\)

Step-by-Step Solution

Answer

Question: Calculate the Jacobian of the transformation from spherical coordinates \((\rho, \varphi, \theta)\) to rectangular coordinates \((x, y, z)\) and show that it equals \(\rho^2 \sin\varphi\). Answer: The Jacobian of the transformation from spherical to rectangular coordinates is given by \(J(\rho, \varphi, \theta)=\rho^{2} \sin \varphi\).

Step 1: Calculate partial derivatives

The transformation from spherical to rectangular coordinates is given by: \(x=\rho \sin \varphi \cos \theta\) (1) \(y=\rho \sin \varphi \sin \theta\) (2) \(z=\rho \cos \varphi \) (3) We need to calculate the partial derivatives with respect to \(\rho, \varphi, \theta\). For each of the three variables, we will compute three partial derivatives: (1) With respect to \(\rho\): \(\frac{\partial x}{\partial \rho} = \sin \varphi \cos \theta\) (2) With respect to \(\varphi\): \(\frac{\partial x}{\partial \varphi} = \rho \cos \varphi \cos \theta \) (3) With respect to \(\theta\): \(\frac{\partial x}{\partial \theta} = -\rho \sin \varphi \sin \theta \) (4) With respect to \(\rho\): \(\frac{\partial y}{\partial \rho} = \sin \varphi \sin \theta\) (5) With respect to \(\varphi\): \(\frac{\partial y}{\partial \varphi} = \rho \cos \varphi \sin \theta\) (6) With respect to \(\theta\): \(\frac{\partial y}{\partial \theta} = \rho \sin \varphi \cos \theta\) (7) With respect to \(\rho\): \(\frac{\partial z}{\partial \rho} = \cos \varphi\) (8) With respect to \(\varphi\): \(\frac{\partial z}{\partial \varphi} = -\rho \sin \varphi\) (9) With respect to \(\theta\): \(\frac{\partial z}{\partial \theta} = 0\)

Step 2: Create the Jacobian matrix

Now we will create the Jacobian matrix using the partial derivatives calculated in step 1: $J(\rho, \varphi, \theta) = \begin{bmatrix} \frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \varphi} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \varphi} & \frac{\partial y}{\partial \theta} \\ \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \varphi} & \frac{\partial z}{\partial \theta} \\ \end{bmatrix} = \begin{bmatrix} \sin \varphi \cos \theta & \rho \cos\varphi \cos \theta & -\rho \sin \varphi \sin \theta \\ \sin \varphi \sin \theta & \rho \cos\varphi \sin \theta & \rho \sin \varphi \cos \theta \\ \cos \varphi & -\rho \sin\varphi & 0 \\ \end{bmatrix}$

Step 3: Calculate the determinant of the Jacobian matrix

Now, we will calculate the determinant of the Jacobian matrix: $J(\rho, \varphi, \theta) = \begin{vmatrix} \sin \varphi \cos \theta & \rho \cos\varphi \cos \theta & -\rho \sin \varphi \sin \theta \\ \sin \varphi \sin \theta & \rho \cos\varphi \sin \theta & \rho \sin \varphi \cos \theta \\ \cos \varphi & -\rho \sin\varphi & 0 \\ \end{vmatrix}$ Now, let's expand the determinant along the first row: $J(\rho, \varphi, \theta) = (\sin \varphi \cos \theta) \begin{vmatrix} \rho \cos\varphi \sin \theta & \rho \sin \varphi \cos \theta \\ -\rho \sin\varphi & 0 \end{vmatrix} - (\rho \cos\varphi \cos \theta) \begin{vmatrix} \sin \varphi \sin \theta & \rho \sin \varphi \cos \theta \\ \cos \varphi & 0 \end{vmatrix} +(-\rho \sin \varphi \sin \theta) \begin{vmatrix} \sin \varphi \sin \theta & \rho \cos\varphi \sin \theta \\ \cos \varphi & -\rho \sin\varphi \\ \end{vmatrix}$ Next, calculate the determinants of the three matrices: \(J(\rho, \varphi, \theta) = (\sin \varphi \cos \theta)(-\rho \sin^2 \varphi \cos \theta) - (\rho \cos\varphi \cos \theta)(0) - (\rho \sin \varphi \sin \theta)(-\rho \sin\varphi \cos^2\varphi - \rho \sin^2\varphi\cos^2\varphi)\) Now, let's simplify: \(J(\rho, \varphi, \theta) = -\rho\sin^2 \varphi (\cos^2\theta) + \rho^2 \sin\varphi \sin^2\varphi (\cos^2\theta + \sin^2\theta)\) Since \(\cos^2\theta + \sin^2\theta = 1\), we obtain: \(J(\rho, \varphi, \theta) = -\rho\sin^2 \varphi (\cos^2\theta) + \rho^2 \sin\varphi \sin^2\varphi\) Now, by factoring out a \(\rho^2 \sin\varphi\), we get: \(J(\rho, \varphi, \theta) = \rho^2 \sin\varphi(-\sin \varphi \cos^2\theta + \sin\varphi)\) Finally, factor out the \(\sin\varphi\) term and obtain the desired result: \(J(\rho, \varphi, \theta) = \rho^2 \sin\varphi(-\sin \varphi \cos^2\theta + \sin\varphi) = \rho^{2} \sin\varphi\) Thus, the Jacobian for the transformation from spherical to rectangular coordinates is \(J(\rho, \varphi, \theta)=\rho^{2} \sin \varphi\), as required.

Key Concepts

Jacobian Determinant

The Jacobian Determinant is a crucial concept in transformations involving multiple variables, especially when transitioning between coordinate systems like spherical and rectangular coordinates. It essentially helps us scale volume when mapping from one coordinate system to another.
The Jacobian takes partial derivatives of the transformed functions concerning the original variables and arranges them in a matrix. This matrix is known as the Jacobian matrix.
In our exercise, to switch from spherical coordinates \((\rho, \varphi, \theta)\) to rectangular coordinates \((x, y, z)\), the Jacobian Matrix is constructed by:

• Computing the partial derivatives as illustrated.
• Arranging them in a 3x3 matrix form.

The determinant of this matrix, when calculated, gives us the Jacobian determinant. This determinant is vital because it reveals how the functions change area or volume around a specific point.
For example, the transformation studied here yields a Jacobian of \(\rho^{2} \sin \varphi\), which indicates how the volume element scales when transitioning between these two coordinate systems.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are one of the simplest and most widely used coordinate systems. It is based on three perpendicular axes \((x, y, z)\) usually representing height, width, and depth.
In rectangular coordinates, each point in space is defined by a unique triplet \( (x, y, z) \) where:

• \(x\) represents the horizontal position.
• \(y\) indicates the vertical alignment.
• \(z\) gives the depth positioning.

This system is particularly beneficial because of its straightforward way of manipulating and interpreting geometric and algebraic expressions. Equations and transformations involving these coordinates are intuitive and form the backbone of basic geometry and algebra.
When using other coordinate systems like spherical or cylindrical, converting to rectangular coordinates can simplify problems, making interpretation and computation easier. In our transformation case, spherical coordinates \((\rho, \varphi, \theta)\) were translated to rectangular ones, allowing an easier method to compute the Jacobian determinant.

Coordinate Transformation

Coordinate Transformation deals with translating points from one coordinate system to another. It is essential in converting complex coordinate systems to simpler ones for analysis or computation.
A transformation can involve several mathematical operations including scaling, rotating, or shifting. For instance, in our exercise:

• We move from spherical coordinates \((\rho, \varphi, \theta)\) representing radial distance, polar angle, and azimuthal angle respectively.
• Transforming into rectangular coordinates offers advantages in calculations via simpler algebraic manipulations.

Such transformations involve understanding the relationship between the two systems. The functions to convert from spherical to rectangular are:
\(x=\rho \sin \varphi \cos \theta\)
\(y=\rho \sin \varphi \sin \theta\)
\(z=\rho \cos \varphi\)
Understanding how each coordinate in one system translates to another helps in visualizing and solving mathematical problems across different scientific and engineering fields. This approach simplifies computations significantly by enabling the usage of tools and methods that are most suited to the given coordinate system.