Problem 42 - Thomas Calculus

Question

A field of tangent vectors a. Find a field \(G=P(x, y) \mathbf{i}+Q(x, y)\) in the \(x y\) -plane with the property that at any point \((a, b) \neq(0,0), G\) is a unit vector tangent to the circle \(x^{2}+y^{2}=a^{2}+b^{2}\) and pointing in the clockwise direction. b. How is G related to the spin field \(F\) in Figure 16.12\(?\)

Step-by-Step Solution

Answer

G is \( \frac{y}{\sqrt{x^2 + y^2}} \mathbf{i} - \frac{x}{\sqrt{x^2 + y^2}} \mathbf{j} \) and is a scaled, reversed version of the spin field F.

Step 1: Establish the Condition for Tangency

To determine the field \( G \) as a unit vector tangent to the circle \( x^2 + y^2 = a^2 + b^2 \), first consider the derivative of the implicit function \( x^2 + y^2 = r^2 \), which is \( 2x + 2y \left( \frac{dy}{dx} \right) = 0 \). This yields the slope \( \frac{dy}{dx} = -\frac{x}{y} \). Hence, any tangent vector at a point \((a, b)\) must have direction \((b, -a)\) to be tangent.

Step 2: Create the Unit Tangent Vector

To ensure \( G \) is a unit vector, compute its magnitude and divide by it. For the vector \( (b, -a) \), this magnitude is \( \sqrt{b^2 + (-a)^2} = \sqrt{a^2 + b^2} \). Thus, the unit tangent vector is: \[ \frac{(b, -a)}{\sqrt{a^2 + b^2}} = \left( \frac{b}{\sqrt{a^2 + b^2}}, \frac{-a}{\sqrt{a^2 + b^2}} \right) \].

Step 3: Express the Field G

Now express \( G \) in terms of \( x \) and \( y \) as \( G = \frac{(y, -x)}{\sqrt{x^2 + y^2}} = \frac{y}{\sqrt{x^2 + y^2}} \mathbf{i} - \frac{x}{\sqrt{x^2 + y^2}} \mathbf{j} \). This confirms \( G \) has components: \( P(x, y) = \frac{y}{\sqrt{x^2 + y^2}} \) and \( Q(x, y) = -\frac{x}{\sqrt{x^2 + y^2}} \).

Step 4: Relate G to the Spin Field F

The standard spin field \( F \) is typically given by \( F = -y \mathbf{i} + x \mathbf{j} \), which rotates counterclockwise. Comparing with \( G \), notice that \( G \) reverses the direction of \( F \) and scales it to unit length, maintaining tangency to the circle while ensuring a clockwise direction.

Key Concepts

Tangent Vectors

Tangent vectors are essential in vector fields, providing direction along surfaces or curves. Imagine a circle centered at the origin. A tangent vector at any point on this circle lies along the tangent line that just "touches" the circle without crossing it.
For a more mathematical approach, consider a unit circle with the equation \(x^2 + y^2 = r^2\). The derivative, or instantaneous rate of change, helps define tangent vectors at various points. By differentiating the circle's equation, we find that the slope is given by \(\frac{dy}{dx} = -\frac{x}{y}\). The tangent vector perpendicular to the radius thus has the coordinates \((b, -a)\). This means that for a point \((a, b)\), the tangent vector \(T\) to the circle has components \((b, -a)\).
This vector shows the direction of the curve at that point without indicating the magnitude or the speed of the tangent field.

Unit Vector

A unit vector is simply a vector with a length, or magnitude, of 1. It retains direction but scales its length to exactly 1, making it useful in various applications, such as defining direction without concern for magnitude.
Unit vectors help to simplify tangential directions. When given a vector, one can convert it into a unit vector by dividing each component by the vector’s total length. Let's say we have a vector \((b, -a)\).
We compute its magnitude using the Pythagorean theorem: \( \sqrt{b^2 + (-a)^2} = \sqrt{a^2 + b^2} \). Then divide each component of the original vector by this magnitude.
The resultant unit vector, tangent to a circle, becomes:

• \( \frac{b}{\sqrt{a^2 + b^2}} \)
• \( \frac{-a}{\sqrt{a^2 + b^2}} \)

This ensures that the vector keeps its direction while scaling its length to 1.

Spin Field

A spin field is a type of vector field where vectors align to create a swirling motion around a point or a center. These fields describe rotational characteristics in physics and engineering while maintaining an elegant application in mathematics.
A spin field, like \(F = -y \mathbf{i} + x \mathbf{j}\), often assumes a rotation around the origin. It usually rotates counterclockwise by default.
In the study of vector fields, a standard spin field displays vectors always perpendicular to a positioned radial vector. This establishes a swirling effect characteristic of rotational fields.
In our exercise, the field \( G = \frac{(y, -x)}{\sqrt{x^2 + y^2}} \) represents a transformation of the standard spin field \( F \). While \( F \) naturally spins counterclockwise, \( G \) applies a scaling to unit length but orients the vectors so the spin occurs in a clockwise direction.

Clockwise Direction

Rotational directions like clockwise further describe vector fields, aiding in the visualization and understanding of rotation within these fields. Clockwise describes the direction taken by the hands of a clock, from top to right, bottom, left, and back to top.
In vector terms, the counterclockwise direction is often standard, as seen in spin fields such as \(F = -y \mathbf{i} + x \mathbf{j}\).
However, turning it inward in a clockwise fashion like our vector field \(G\) allows for a distinct interpretation. Comparing spin fields \(F\) and \(G\) highlights rotation. While \(F\) naturally feels expansive and outward, the field \(G\), through vector inversion and scaling, contracts inward, following a clockwise path.
Understanding these directions ensures a comprehensive grasp of vector properties in planar fields, whether in physics, computer graphics, or engineering applications.