Question

Paraboloidal coordinates \(u, v, \phi\) are defined in terms of Cartesian coordinates by $$ x=u v \cos \phi, \quad y=u v \sin \phi, \quad z=\frac{1}{2}\left(u^{2}-v^{2}\right) $$ Identify the coordinate surfaces in the \(u, v, \phi\) system. Verify that each coordinate surface \((u=\) constant, say) intersects every coordinate surface on which one of the other two coordinates \((v\), say \()\) is constant. Show further that the system of coordinates is an orthogonal one and determine its scale factors. Prove that the \(u\)-component of \(\nabla \times \mathbf{a}\) is given by $$ \frac{1}{\left(u^{2}+v^{2}\right)^{1 / 2}}\left(\frac{a_{\phi}}{v}+\frac{\partial a_{\phi}}{\partial v}\right)-\frac{1}{u v} \frac{\partial a_{v}}{\partial \phi} $$

Short answer

Identify coordinate surfaces, verify intersections at coordinate points, show gradients are orthogonal, find scale factors. Curl component: \frac{1}{(u^2 + v^2)^{1/2}} \left( \frac{a_\phi}{v} + \frac{\partial a_\phi}{\partial v} \right) - \frac{1}{uv} \frac{\partial a_v}{\partial \phi}

Step-by-step solution

Identify the Coordinate Surfaces

We start by interpreting the given parametric equations: \[x = uv \, \cos \phi, \, y = uv \, \sin \phi, \, z = \frac{1}{2} \left( u^2 - v^2 \right)\]A surface is defined when one of the coordinates is kept constant.

Surface for Constant u

Consider \(u = u_0\) constant: \[x = u_0 v \, \cos \phi, \, y = u_0 v \, \sin \phi, \, z = \frac{1}{2} \left( u_0^2 - v^2 \right)\]This is a paraboloid of revolution where \(u_0\) defines the paraboloid's scale and shape.

Surface for Constant v

Consider \(v = v_0\) constant: \[x = u v_0 \, \cos \phi, \, y = u v_0 \, \sin \phi, \, z = \frac{1}{2} \left( u^2 - v_0^2 \right)\]This is also a paraboloid of revolution where \(v_0\) defines the paraboloid's shape.

Surface for Constant \(\phi\)

Consider \(\phi = \phi_0\) constant: \[x = u v \, \cos \phi_0, \, y = u v \, \sin \phi_0, \, z = \frac{1}{2} \left( u^2 - v^2 \right)\]This represents a plane passing through the z-axis, inclined at angle \(\phi_0\) relative to the x-axis.

Verify Intersection

Verify that each coordinate surface intersects every coordinate surface. E.g., a surface \(u = u_0\) intersects the surface \(v = v_0\):For \(u = u_0\) and \(v = v_0\), we have:\[x = u_0 v_0 \, \cos \phi, \, y = u_0 v_0 \, \sin \phi, \, z = \frac{1}{2} (u_0^2 - v_0^2)\]This point belongs to both surfaces.

Show Orthogonality

To prove the coordinates are orthogonal, we check that the gradient vectors of the coordinate functions are orthogonal. Gradient vectors are: \[abla u = (\cos \phi v, \sin \phi v, u)^{T},abla v = (\cos \phi u, \sin \phi u, -v)^{T},abla \phi = (-uv \, \sin \phi, uv \, \cos \phi, 0 )^{T}\]Check that \(abla u \cdot abla v = 0\), \(abla u \cdot abla \phi = 0\), and \(abla v \cdot abla \phi = 0\).

Determine Scale Factors

Calculate the scale factors\(h_u, h_v\), and \(h_\phi \) from:\[h_u = | abla u | = \sqrt{\cos^2 \phi v^2 + \sin^2 \phi v^2 + u^2 } = \sqrt{u^2 + v^2} \]\[h_v = | abla v | = \sqrt{\cos^2 \phi u^2 + \sin^2 \phi u^2 + -v^2 } = \sqrt{u^2 + v^2} \]\[h_\phi = | abla \phi | = \sqrt{ (-uv \sin \phi)^2 + (uv \cos \phi)^2 + 0^2 } = uv\]

Find \(u\)-Component of \( abla \times \mathbf{a} \)

Using the given scale factors, the expression is re-written from the curl formula: \[abla \times \mathbf{a} = \frac{1}{h_u h_v h_\phi} \begin{vmatrix} \, \mathbf{e}_u & h_u \mathbf{e}_v & h_\phi \mathbf{e}_\phi \, \ \partial_u & \partial_v & \partial_\phi \, \ h_u a_u & h_v a_v & h_\phi a_\phi \end{vmatrix} \]The \( \mathbf{e}_u)\) component contributes: \[ \frac{1}{(u^2 + v^2)^{1/2}} \left ( \frac{a_\phi}{v} + \frac{\partial a_\phi}{\partial v} \right ) - \frac{1}{uv} \frac{\partial a_v}{\partial \phi} \]

Key concepts

Coordinate Systems

Coordinate systems help us describe locations in space using numbers or parameters. In the case of paraboloidal coordinates, we use three parameters: \(u\), \(v\), and \(\phi\). These parameters describe points in 3D space relative to the Cartesian coordinates \(x\), \(y\), and \(z\).

• \(u\) and \(v\) represent distances along axes that form a paraboloid shape.
• \(\phi\) represents the angle around the z-axis, much like in cylindrical coordinates.

Coordinate surfaces are formed by keeping one parameter constant. For example:

• If \(u\) is constant, we get a surface that forms a particular paraboloid shape.
• If \(v\) is constant, another paraboloid shape is formed.
• If \(\phi\) is constant, we obtain vertical planes passing through the z-axis.

By understanding how these parameters relate to Cartesian coordinates, we can more easily visualize the location and geometry of points in space.

Orthogonal Coordinates

Orthogonal coordinates are those in which the coordinate surfaces intersect at right angles. In our paraboloidal coordinate system, this means that the gradients of the coordinate functions \(u\), \(v\), and \(\phi\) are orthogonal to each other.
The gradient vectors are computed as follows:

• \( abla u = (\cos \phi v, \sin \phi v, u)^{T} \)
• \( abla v = (\cos \phi u, \sin \phi u, -v)^{T} \)
• \( abla \phi = (-uv \sin \phi, uv \cos \phi, 0)^{T} \)

To confirm orthogonality, we check that the dot products of these gradient vectors are zero:

• \( abla u \cdot abla v = 0 \)
• \( abla u \cdot abla \phi = 0 \)
• \( abla v \cdot abla \phi = 0 \)

These dot products being zero proves that the coordinate system is orthogonal, meaning each coordinate direction is perpendicular to the others, simplifying many mathematical analyses and visualizations.

Gradient Vectors

The gradient vector of a function gives the direction and rate of fastest increase of the function. For coordinate functions in our paraboloidal system, gradients are essential in determining orthogonality and scale factors.
The gradient vectors for the coordinates \(u, v, \phi\) are:

• \( abla u = (\cos \phi v, \sin \phi v, u)^{T} \)
• \( abla v = (\cos \phi u, \sin \phi u, -v)^{T} \)
• \( abla \phi = (-uv \sin \phi, uv \cos \phi, 0)^{T} \)

These vectors help us calculate important properties like scale factors. The scale factor \(h\) of a coordinate is the magnitude of its gradient vector.
For paraboloidal coordinates, the scale factors are:

• \( h_u = | abla u | = \sqrt{u^2 + v^2} \)
• \( h_v = | abla v | = \sqrt{u^2 + v^2} \)
• \( h_{\phi} = | abla \phi | = uv \)

These scale factors are crucial for transforming vector operations, like the curl and divergence, between different coordinate systems.

Curl of a Vector

In vector calculus, the curl of a vector field measures the rotation or swirling of the field. In our paraboloidal coordinates, computing the curl requires using the scale factors \( h_u, h_v, \) and \( h_{\phi} \).
The curl in terms of these scale factors is given by: \[ abla \times \mathbf{a} = \frac{1}{h_u h_v h_{\phi}} \begin{vmatrix} \mathbf{e}_u & h_u \mathbf{e}_v & h_{\phi} \mathbf{e}_{\phi} \ \partial_u & \partial_v & \partial_{\phi} \ h_u a_u & h_v a_v & h_{\phi} a_{\phi} \ end{vmatrix} \] The \(u\)-component of the curl can be found using these steps, and is provided in the problem as: \[ \frac{1}{(u^2 + v^2)^{1/2}} \left( \frac{a_{\phi}}{v} + \frac{\partial a_{\phi}}{\partial v} \right) - \frac{1}{uv} \frac{\partial a_v}{\partial \phi} \] This expression uses the interplay between the scale factors and the partial derivatives of the vector components, illustrating the complexity of vector calculus in non-Cartesian coordinate systems.