Problem 5 - Calculus Of Variations, With Applications To Physics And Engineering

Question

By solving the equations of transformation to spherical coordinates $$ x-r \sin \theta \cos \phi, \quad y-r \sin \theta \sin \phi, \quad z=r \cos \theta $$ for \(r, \theta, \phi\), show that \(r\) represents the distance from \((x, y, z)\) to the origin, \(\theta\) the angle between the positive \(z\) axis and the line drawn to \((x, y, z)\) from the origin, \(\phi\) the angle betwecn the \(x\) plane (positive \(x\) ) and the half-plane bounded by the \(a\) axis and containing \((x, y, z)\). Thus describe the families of surfaces \(r=\) constant, \(\theta=\) constant, \(\phi=\) constant

Step-by-Step Solution

Answer

The variables \(r, θ, φ\) represent the distance from origin, angle with positive z-axis and angle with positive x-axis, respectively. For constant \(r, θ, φ\) they generate spheres, cones, and half-planes respectively.

Step 1: Isolate for 'r'

Start with the formula for the spherical radius \(r\). Extracting \(r\) from the equations we have \(r = \sqrt{x^2 + y^2 + z^2}\)

Step 2: Isolate for 'θ'

By using trigonometric relations and the definition of \(r\), we can isolate for \(θ\) and we find that \(θ = \cos^{-1} \frac{z}{r}\) or \(θ = \cos^{-1} \frac{z}{\sqrt{x^2 + y^2 + z^2}}\)

Step 3: Isolate for 'φ'

Isolating \(φ\) using trigonometry and spherical coordinates, we establish that \(φ = \tan^{-1} \frac{y}{x}\)

Step 4: Geometric interpretation

Geometrically, \(r\) can be perceived as the straight line distance from the origin to the point \((x, y, z)\). \(θ\) is the angle created by the point from the positive \(z \)-axis while \(φ\) is the angle from the positive \(x\)-axis to the \(z\)-plane. This means \(r\), \(θ\), and \(φ\) specify the position of the point in spherical coordinates

Step 5: Describe the surfaces

Surfaces of constant \(r\) are spheres centered at the origin. Surfaces of constant \(θ\) are cones centered around the \(z\)-axis. Surfaces where \(φ\) is constant are half-planes originating from the origin.

Key Concepts

Transformation Equations

Transformation equations are mathematical formulas that allow us to convert coordinates from one system to another. In the context of spherical coordinates, the equations relate Cartesian coordinates \( (x, y, z) \) to spherical ones \( (r, \theta, \)phi) \.

The equations for transformation to spherical coordinates are defined as:
\[ x = r \sin(\theta) \cos(\phi) \] \[ y = r \sin(\theta) \sin(\phi) \] \[ z = r \cos(\theta) \]
These equations compile the foundations upon which we build the understanding of the geometrical relationships between the two coordinate systems, enabling us to visualize and solve complex spatial problems easily.

Trigonometric Relations

Trigonometric relations are essential tools for understanding the angles and sides of triangles, and they become especially useful when dealing with spherical coordinates.

For instance, in our exercise, the relations allowed us to solve for \( r \) using the Pythagorean theorem, \( \theta \) using the inverse cosine function, and \( \phi \) using the inverse tangent function, based on the given Cartesian coordinates. These relations form a bridge linking angles to distances and are vital for calculating positions and movements in fields ranging from astronomy to engineering.

Key Trigonometric Functions:

• \( \sin(\theta) \) corresponds to the ratio of the opposite side over the hypotenuse in a right triangle.
• \( \cos(\theta) \) deals with the adjacent side over the hypotenuse.
• \( \tan(\theta) \) represents the opposite over the adjacent side.

These functions are what guide us to solve for and understand spherical coordinates in the given problem.

Geometric Interpretation

The geometric interpretation of spherical coordinates provides a visual way to represent the position of points in three-dimensional space.

In our exercise, \( r \) represents the radial distance from the origin to the point \( (x, y, z) \), and is often visualized as the length of a line segment radiating outward from the center of a sphere. The angle \( \theta \) indicates the elevation angle measured from the positive \( z \)-axis, offering a way to gauge the 'latitude' of the point. Lastly, \( \phi\) is the azimuthal angle measured from the positive \( x \) axis in the \( x \)y-plane, akin to the 'longitude'.

Visualizing Constant Values:

• A constant \( r \) yields a sphere.
• Constant \( \theta \) forms a cone around the \( z \) axis.
• Constant \( \phi \) results in a half-plane.

These visual cues help students not only solve mathematical problems but also lay the groundwork for greater comprehension in physical sciences and engineering disciplines.

Calculus of Variations

Calculus of variations is an exciting field of mathematics that deals with optimizing functions which depend on other functions. It extends the insight provided by calculus to understand variations and optimize multi-functional systems.

Although our specific exercise doesn't explicitly cover an optimization problem, the principles of the calculus of variations can be used to delve deeper into the properties of spherical coordinates. For example, one could use it to find the shortest path between two points on a sphere's surface—a problem known as the geodesic. This concept marries the elegance of mathematical theory with the practicality of real-world situations, encompassing a vast range of applications from physics to economics.