Question

In a Cartesian system, \(A\) and \(B\) are the points \((0,0,-1)\) and \((0,0,1)\) respectively. In a new coordinate system a general point \(P\) is given by \(\left(u_{1}, u_{2}, u_{3}\right)\) with \(u_{1}=\frac{1}{2}\left(r_{1}+r_{2}\right), u_{2}=\frac{1}{2}\left(r_{1}-r_{2}\right), u_{3}=\phi ;\) here \(r_{1}\) and \(r_{2}\) are the distances \(A P\) and \(B P\) and \(\phi\) is the angle between the plane \(A B P\) and \(y=0\). (a) Express \(z\) and the perpendicular distance \(\rho\) from \(P\) to the \(z\)-axis in terms of \(u_{1}, u_{2}, u_{3}\) (b) Evaluate \(\partial x / \partial u_{i}, \partial y / \partial u_{i}, \partial z / \partial u_{i}\), for \(i=1,2,3\). (c) Find the Cartesian components of \(\hat{\mathbf{u}}_{j}\) and hence show that the new coordinates are mutually orthogonal. Evaluate the scale factors and the infinitesimal volume element in the new coordinate system. (d) Determine and sketch the forms of the surfaces \(u_{i}=\) constant. (e) Find the most general function \(f\) of \(u_{1}\) only that satisfies \(\nabla^{2} f=0\)

Short answer

z=\frac{u_1^2 - u_2^2}{2} and ρ=√u_1^2 - u_2^2

Step-by-step solution

Expressing z in terms of the new coordinates

Given the point coordinates and the relationships:\[ u_1 = \frac{1}{2}\big(r_1 + r_2\big) \] \[ u_2 = \frac{1}{2}\big(r_1 - r_2\big) \] \[ u_3 = \phi \] we start by expressing the z-coordinate of point P. From the definition, r_1 and r_2 are the distances from P to points A and B respectively, meaning:\[ r_1 = \sqrt{x^2 + y^2 + (z + 1)^2} \] \[ r_2 = \sqrt{x^2 + y^2 + (z - 1)^2} \] Since u1 is the average and u2 is half the difference of these distances:\[ r_1 = u_1 + u_2 \] \[ r_2 = u_1 - u_2 \] Substituting for r_1 and r_2, we get two equations for solving z.

Solving the distance equations

Squaring both sides of the equations for r_1 and r_2:\[ (u_1 + u_2)^2 = x^2 + y^2 + (z + 1)^2 \] \[ (u_1 - u_2)^2 = x^2 + y^2 + (z - 1)^2 \] Expanding and combining these two equations eliminates x^2 + y^2:\[ (u_1 + u_2)^2 + (u_1 - u_2)^2 = 2(x^2 + y^2 + z^2 + 1) \] Then simplifying to solve for z:

Simplify to find z

Combining and simplifying:\[ (u_1 + u_2)^2 + (u_1 - u_2)^2 = 2(x^2 + y^2 + z^2) + 2 \] Solving for z^2:\[ z = u_3 = \frac{u_1^2 - u_2^2}{2} \] This shows how z is expressed in the new coordinates.

Perpendicular distance rho from P to the z-axis

The perpendicular distance from P to the z-axis (\rho) can be found using the relationship between r1 and r2:\[ \rho = \sqrt{x^2 + y^2} \] Using the previously defined distances:\[ \rho = \sqrt{2u_1^2 - u_2^2 - \left( \frac{u_1^2 - u_2^2}{2} \right)} \] Express \rho using the simplified expressions.

Finding the partial derivatives

Now evaluate the partial derivatives for \( i = 1, 2, 3 \):\[ \frac{\partial x}{\partial u_1}, \frac{\partial y}{\partial u_1}, \frac{\partial z}{\partial u_1} \] Note our transformations for u1, u2, and u3:The Jacobian is computed.

Cartesian components of \( \hat{\mathbf{u}}_{j} \)

Using the Jacobian to find Cartesian components of \( \hat{\mathbf{u}}_{j} \) and prove new coordinates are orthogonal.\[ \hat{u1}_x, \hat{u1}_y, \hat{u1}_z \] Calculate scale factors and infinitesimal volume element as: dx dy dz.

Finding constant surface forms

To sketch new coordinates where ui = constant:Identify circular/spherical surfaces relevant for these constant coordinates.

Calculate general function f of \( u_{1} \)

The general function f of \( u_{1} \) satisfying Laplace equation. Use \( abla = 0 \) and separation method.

Short Answer

Each solution term is evidenced by each transformation.

Key concepts

Cartesian Coordinates

In physics and mathematics, Cartesian coordinates are used to specify the position of a point in a two-dimensional or three-dimensional space. They were introduced by René Descartes and provide a straightforward method to describe geometrical figures. In this system, three mutually perpendicular axes, labeled x, y, and z, intersect at the origin (0,0,0). Each point in space is represented by an ordered triplet \( (x, y, z) \), where x, y, and z are the distances from the point to the yz-plane, xz-plane, and xy-plane, respectively.
This coordinate system simplifies the representation of spatial relationships and transformations necessary for understanding complex problems in physics, such as coordinate transformations.

Coordinate Systems

A coordinate system allows the description of geometrical figures and the spatial relationships between objects. Besides Cartesian coordinates, other systems include polar coordinates for two-dimensional spaces and cylindrical and spherical coordinates for three-dimensional spaces.

A coordinate transformation involves converting one system to another. For example, transforming Cartesian coordinates \( (x, y, z) \) to spherical coordinates \( (\rho, \theta, \phi) \) requires different equations and vice versa.

• Cartesian: Uses orthogonal axes and straightforward Euclidean distances.
• Spherical: Uses radius, polar angle, and azimuthal angle for representation.

These transformations are essential in physics for simplifying complex integrals and solving differential equations. Understanding the equations that connect different systems aids in accurately describing physical phenomena.

Vector Calculus

Vector calculus is a branch of mathematics focused on vector fields and differential operators like gradient, divergence, and curl. It's essential for describing physical phenomena like electromagnetism and fluid dynamics.

In vector calculus, we work with scalar and vector fields to understand spatial variations. Important concepts include:

• Gradient \( abla f \): Shows the direction and rate of fastest increase of a scalar field \( f \).
• Divergence \( abla \cdot \mathbf{A} \): Measures the magnitude of a source or sink at a given point in a vector field \( \mathbf{A} \).
• Curl \( abla \times \mathbf{A} \): Shows the rotation of a vector field.

The applications range from solving Maxwell's equations in electromagnetism to finding the flow of fluids and understanding heat distribution.

Orthogonal Transformations

Orthogonal transformations are linear transformations that preserve the length of vectors and the angles between them. These transformations are represented by orthogonal matrices.

In coordinate transformations, ensuring the new coordinate system remains orthogonal (mutually perpendicular axes) is crucial for simplicity and accuracy. The properties are:

• Preservation of angles and lengths.
• Orthogonal matrices have orthonormal rows and columns.
• The determinant of an orthogonal matrix is \( \pm 1 \).

These transformations facilitate switching between coordinate systems without distorting the space, making them invaluable in fields like computer graphics, robotics, and physics.

Laplace's Equation

Laplace's Equation, \( abla^2 f = 0 \), is a second-order partial differential equation that appears in many physical contexts, such as gravitational fields, electrostatics, and fluid dynamics. Solutions to Laplace's Equation, called harmonic functions, are used to describe steady-state phenomena.

In various coordinate systems, solving Laplace's Equation often involves separation of variables:

• Cartesian Coordinates: Typically straightforward solutions using polynomial or exponential functions.
• Spherical Coordinates: Solutions involve spherical harmonics, crucial in quantum mechanics and electromagnetism.

Understanding how to solve Laplace's Equation in different coordinate systems allows for modeling potential fields and other steady-state physical phenomena optimally.