Chapter 10: Problem 6

As in Problem $1,$ find $d s^{2},$ the scale factors, the vector $d \mathbf{s},$ the volume (or area) element, the a vectors, and the e vectors for each of the following coordinate systems. Parabolic cylinder coordinates $u, v, z:$ $$\begin{aligned} &x=\frac{1}{2}\left(u^{2}-v^{2}\right),\\\ &y=u v,\\\ &z=z. \end{aligned}$$

Short Answer

Expert verified

The scale factors are $ h_u = h_v = \sqrt{u^2 + v^2} $ and $ h_z = 1 $. The infinitesimal distance squared is $ ds^2 = (u^2 + v^2) du^2 + (u^2 + v^2) dv^2 + dz^2 $. The volume element is $ dV = (u^2 + v^2) du dv dz $.

Step by step solution

Express Cartesian Coordinates

Given the parabolic cylinder coordinates are defined as $ x = \frac{1}{2}(u^2 - v^2) $ $ y = uv $ $ z = z $.

Compute Differential Elements

The next step involves calculating the differentials of the coordinates: $ dx = u du - v dv $ $ dy = v du + u dv $ $ dz = dz $ These can be derived using partial derivatives.

Calculate Infinitesimal Distance

Now, calculate the infinitesimal distance squared $ ds^2 $: \[ ds^2 = dx^2 + dy^2 + dz^2 \] \[ ds^2 = (u du - v dv)^2 + (v du + u dv)^2 + dz^2 \] Expanding and simplifying: \[ ds^2 = u^2 du^2 + v^2 dv^2 - 2uv du dv + v^2 du^2 + u^2 dv^2 + 2uv du dv + dz^2 \] \[ ds^2 = (u^2 + v^2) du^2 + (u^2 + v^2) dv^2 + dz^2 \]

Extract Scale Factors

Identify the scale factors from the expression for $ ds^2 $. In parabolic cylinder coordinates, we get the scale factors as: \[ h_u = h_v = \sqrt{u^2 + v^2}, \] \[ h_z = 1 \]

Compute the Vector $ d\mathbf{s} $

The vector $ d\mathbf{s} $ is given by: \[ d\mathbf{s} = h_u du \hat{u} + h_v dv \hat{v} + h_z dz \hat{z} \] Substituting our scale factors: \[ d\mathbf{s} = \sqrt{u^2 + v^2} du \hat{u} + \sqrt{u^2 + v^2} dv \hat{v} + dz \hat{z} \]

Calculate the Volume (or Area) Element

The volume element in parabolic cylinder coordinates is given by: \[ dV = h_u h_v h_z du dv dz \] Substituting the scale factors: \[ dV = (u^2 + v^2) du dv dz \]

Find the a and e Vectors

For the vectors: \[ \mathbf{a}_u = h_u \hat{u} = \sqrt{u^2 + v^2} \hat{u} \] \[ \mathbf{a}_v = h_v \hat{v} = \sqrt{u^2 + v^2} \hat{v} \] \[ \mathbf{a}_z = h_z \hat{z} = \hat{z} \] The unit vectors $ \mathbf{e}_i $ correspond to the directions of the $ u, v, z $-axes and do not scale by any factors: \[ \mathbf{e}_u = \hat{u}, \mathbf{e}_v = \hat{v}, \mathbf{e}_z = \hat{z}. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Elements

In any coordinate system, differential elements represent very small changes in each coordinate. For parabolic cylinder coordinates, we start with the transformations:
\[ x = \frac{1}{2}(u^2 - v^2) \] \[ y = uv \] \[ z = z \]
To get the differential elements for these coordinates, we differentiate each expression:
\[ dx = u du - v dv \] \[ dy = v du + u dv \] \[ dz = dz \]
These differential elements $dx$, $dy$, and $dz$ tell us how small changes in $u$, $v$, and $z$ affect the coordinates $x$, $y$, and $z$, respectively. This is an important step for understanding how distances and volumes are calculated in this coordinate system.

Scale Factors

Scale factors are crucial for converting differential elements in a curvilinear coordinate system to real distances. They are derived from the transformed forms of $dx$, $dy$, and $dz$:
\[ ds^2 = dx^2 + dy^2 + dz^2 \]
Using the differential elements and expanding, we find:
\[ ds^2 = (u du - v dv)^2 + (v du + u dv)^2 + dz^2 \]
\[ ds^2 = (u^2 + v^2) du^2 + (u^2 + v^2) dv^2 + dz^2 \]
From this, we extract the scale factors:
\[ h_u = \sqrt{u^2 + v^2} \]
\[ h_v = \sqrt{u^2 + v^2} \]
\[ h_z = 1 \]
These scale factors convert differentials $du$, $dv$, and $dz$ into actual lengths in the coordinate system.

Vector d\mathbf{s}

The vector $d\mathbf{s}$ represents an infinitesimal displacement in the coordinate system. It combines differentials from all coordinates scaled by their respective scale factors. For parabolic cylinder coordinates:
\[ d\mathbf{s} = h_u du \hat{u} + h_v dv \hat{v} + h_z dz \hat{z} \]
Substituting the scale factors, we get:
\[ d\mathbf{s} = \sqrt{u^2 + v^2} du \hat{u} + \sqrt{u^2 + v^2} dv \hat{v} + dz \hat{z} \]
This vector encompasses directional changes in $u$, $v$, and $z$ and gives a complete picture of infinitesimal displacement in this coordinate system.

Volume Element

The volume element $dV$ represents the tiny volume enclosed by infinitesimal changes in all coordinates. It is the product of scale factors and differentials:
\[ dV = h_u h_v h_z du dv dz \]
For parabolic cylinder coordinates:
\[ dV = (u^2 + v^2) du dv dz \]
This volume element helps in calculating integrals over the volume of any region defined in this coordinate system. It's particularly useful in physics problems involving flux, mass, or charge distributions.

Coordinate Systems

Coordinate systems are frameworks that allow us to describe locations in space. Unlike Cartesian coordinates, which use straight lines and right angles for reference, parabolic cylinder coordinates use parabolas and a straight axis.
The transformation relationships:
\[ x = \frac{1}{2}(u^2 - v^2) \]
\[ y = uv \]
\[ z = z \]
show how regular Cartesian coordinates $(x, y, z)$ are mapped to parabolic cylinder coordinates $(u, v, z)$. This is especially useful for problems with symmetry related to parabolas, such as those in electromagnetism or fluid dynamics.