Problem 97 - Advanced Calculus

Question

Compute the Jacobian determinant of the transformation from a) cylindrical coordinates to Cartesian coordinates, b) spherical coordinates to Cartesian coordinates.

Step-by-Step Solution

Answer

The Jacobian determinant for the transformation from cylindrical to Cartesian coordinates is \(J_\text{cylindrical} = ρ\), while the Jacobian determinant for the transformation from spherical to Cartesian coordinates is \(J_\text{spherical} = r^2\sin(θ)\).

Step 1: Write down the transformation equations

The transformation equations for converting cylindrical coordinates (ρ, φ, z) to Cartesian coordinates (x, y, z) are: \(x = ρ\cos(φ)\), \(y = ρ\sin(φ)\), \(z = z\).

Step 2: Differentiate the transformation equations

Differentiate each transformation equation with respect to the cylindrical coordinates (ρ, φ, z): \(\frac{\partial x}{\partial ρ} = \cos(φ)\), \(\frac{\partial x}{\partial φ} = -ρ\sin(φ)\), \(\frac{\partial x}{\partial z} = 0\), \(\frac{\partial y}{\partial ρ} = \sin(φ)\), \(\frac{\partial y}{\partial φ} = ρ\cos(φ)\), \(\frac{\partial y}{\partial z} = 0\), \(\frac{\partial z}{\partial ρ} = 0\), \(\frac{\partial z}{\partial φ} = 0\), \(\frac{\partial z}{\partial z} = 1\).

Step 3: Calculate the Jacobian determinant

The Jacobian determinant of the transformation from cylindrical to Cartesian coordinates is given by: \( J_\text{cylindrical} = \begin{vmatrix} \frac{\partial x}{\partial ρ} & \frac{\partial x}{\partial φ} & \frac{\partial x}{\partial z} \\ \frac{\partial y}{\partial ρ} & \frac{\partial y}{\partial φ} & \frac{\partial y}{\partial z} \\ \frac{\partial z}{\partial ρ} & \frac{\partial z}{\partial φ} & \frac{\partial z}{\partial z} \end{vmatrix} = \begin{vmatrix} \cos(φ) & -ρ\sin(φ) & 0 \\ \sin(φ) & ρ\cos(φ) & 0 \\ 0 & 0 & 1 \end{vmatrix} \). Expanding the determinant, we get: \(J_\text{cylindrical} = (1)(ρ\cos^2(φ) + ρ\sin^2(φ))\) Since \(\cos^2(φ) + \sin^2(φ) = 1\), the Jacobian determinant is: \(J_\text{cylindrical} = ρ\). #b. Spherical coordinates to Cartesian coordinates#

Step 4: Write down the transformation equations

The transformation equations for converting spherical coordinates (r, θ, φ) to Cartesian coordinates (x, y, z) are: \(x = r\sin(θ)\cos(φ)\), \(y = r\sin(θ)\sin(φ)\), \(z = r\cos(θ)\).

Step 5: Differentiate the transformation equations

Differentiate each transformation equation with respect to the spherical coordinates (r, θ, φ): \(\frac{\partial x}{\partial r} = \sin(θ)\cos(φ)\), \(\frac{\partial x}{\partial θ} = r\cos(θ)\cos(φ)\), \(\frac{\partial x}{\partial φ} = -r\sin(θ)\sin(φ)\), \(\frac{\partial y}{\partial r} = \sin(θ)\sin(φ)\), \(\frac{\partial y}{\partial θ} = r\cos(θ)\sin(φ)\), \(\frac{\partial y}{\partial φ} = r\sin(θ)\cos(φ)\), \(\frac{\partial z}{\partial r} = \cos(θ)\), \(\frac{\partial z}{\partial θ} = -r\sin(θ)\), \(\frac{\partial z}{\partial φ} = 0\).

Step 6: Calculate the Jacobian determinant

The Jacobian determinant of the transformation from spherical to Cartesian coordinates is given by: \( J_\text{spherical} = \begin{vmatrix} \frac{\partial x}{\partial r} & \frac{\partial x}{\partial θ} & \frac{\partial x}{\partial φ} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial θ} & \frac{\partial y}{\partial φ} \\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial θ} & \frac{\partial z}{\partial φ} \end{vmatrix} = \begin{vmatrix} \sin(θ)\cos(φ) & r\cos(θ)\cos(φ) & -r\sin(θ)\sin(φ) \\ \sin(θ)\sin(φ) & r\cos(θ)\sin(φ) & r\sin(θ)\cos(φ) \\ \cos(θ) & -r\sin(θ) & 0 \end{vmatrix} \). Expanding the determinant, we get: \(J_\text{spherical} = r^2\sin^2(θ)\sin(θ)\). Thus, the Jacobian determinant for the transformation from spherical to Cartesian coordinates is: \(J_\text{spherical} = r^2\sin(θ)\).

Key Concepts

Cylindrical Coordinates

When studying three-dimensional space, cylindrical coordinates offer an alternative to the familiar Cartesian (rectilinear) coordinates. This system is particularly useful when dealing with problems that have a cylindrical symmetry, such as those involving circular cylinders, cones, or spheres.

In cylindrical coordinates, a point in space is represented by three values: the radial distance from the z-axis \(\rho\), the angle \(\phi\) around the z-axis (analogous to the angular coordinate in polar coordinates), and the height \(z\), which is the same as in Cartesian coordinates. The relationships between cylindrical and Cartesian coordinates are given by:

• \(x = \rho\cos(\phi)\)
• \(y = \rho\sin(\phi)\)
• \(z = z\)

These transformation equations smoothly map the cylindrical surface onto a flat plane, allowing for easier calculus operations, such as integration and differentiation, within cylindrical contexts.

Spherical Coordinates

Spherical coordinates are another invaluable system for navigating three-dimensional spaces, particularly when dealing with problems involving spherical symmetry. This system uses three parameters: the radius \(r\), the polar angle \(\theta\), from the positive z-axis, and the azimuthal angle \(\phi\), which is similar to the angle in cylindrical coordinates.

The conversion from spherical to Cartesian coordinates involves the following relations:

• \(x = r\sin(\theta)\cos(\phi)\)
• \(y = r\sin(\theta)\sin(\phi)\)
• \(z = r\cos(\theta)\)

With these relationships, you can locate any point on the surface of a sphere by starting from the origin and moving outward to the appropriate radius, downward through the correct polar angle, and around by the azimuth angle. This is quite analogous to how latitude and longitude work on Earth.

Coordinate Transformation

Coordinate transformation is a bridge connecting different coordinate systems such as Cartesian, cylindrical, and spherical. It is a crucial concept in mathematics and physics because it allows for the re-expression of points, vectors, and equations of surfaces or volumes in alternative coordinate systems, often simplifying calculations and problem-solving.

The process typically involves using a set of equations that relate the two systems. As seen in the exercise, the Jacobian determinant is a vital tool in this process: it measures the 'scaling factor' introduced by the transformation and is especially important when transforming integrals from one coordinate system to another. A non-zero Jacobian implies a valid transformation where each point in one coordinate system corresponds uniquely to a point in another.

Partial Differentiation

Partial differentiation is a fundamental operation in calculus that involves taking derivatives of functions with respect to one variable at a time, while all other variables are held constant. It is typically used in the context of functions of multiple variables and is denoted by the symbol \(\frac{\partial}{\partial x}\) for a derivative with respect to \(x\).

In the context of coordinate transformations, partial differentiation is used to find the rate of change of the Cartesian coordinates with respect to the cylindrical or spherical coordinates. The resultant partial derivatives populate the Jacobian matrix, which, when determinant is taken, yields the Jacobian determinant. This scalar factor allows for the proper transformation of area or volume elements when integrating multivariable functions.