Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

For the transformation \(x=u /\left(u^{2}+v^{2}\right), y=\) \(-v /\left(u^{2}+v^{2}\right)\), sketch the \(u\)-curves and \(v\)-curves for the grid \(\\{(u, v):(u=0,1,2,3\) and \(1 \leq v \leq 3)\) or \((v=1,2,3\) and \(0 \leq u \leq 3)\\}\).

Step-by-Step Solution

Answer

Sketch the curves based on calculated transformation points for given ranges of \(u\) and \(v\).

Step 1: Understanding the Exercise

The exercise involves sketching transformed coordinate curves using given transformations and grid. The transformation formulas are given for \(x\) and \(y\), where \(x = \frac{u}{{u^2 + v^2}}\) and \(y = -\frac{v}{{u^2 + v^2}}\). The task is to sketch \(u\)-curves (where \(u\) is constant) and \(v\)-curves (where \(v\) is constant) over specified intervals.

Step 2: Setting Up the u-Curves

For the given transformation, when \(u\) is constant, \(u\) takes values 0, 1, 2, 3. We need to vary \(v\) from 1 to 3 and for each constant \(u\), plot \(x = \frac{u}{{u^2 + v^2}}\) and \(y = -\frac{v}{{u^2 + v^2}}\).

Step 3: Calculating Points for u=0

When \(u = 0\), the transformation simplifies to \(x = 0\) and \(y = -\frac{v}{{v^2}} = -\frac{1}{v}\). As \(v\) ranges from 1 to 3, points will be \((0, -1)\), \((0, -\frac{1}{2})\), and \((0, -\frac{1}{3})\). These are vertical points along the negative y-axis.

Step 4: Calculating Points for u=1

For \(u = 1\), the calculations become \(x = \frac{1}{{1 + v^2}}\) and \(y = -\frac{v}{{1 + v^2}}\). Varying \(v\) from 1 to 3, the points are \((\frac{1}{2}, -\frac{1}{2})\), \((\frac{1}{5}, -\frac{2}{5})\), and \((\frac{1}{10}, -\frac{3}{10})\). These points form a curve in the negative coordinate space.

Step 5: Calculating Points for u=2

When \(u = 2\), the transformations yield \(x = \frac{2}{{4 + v^2}}\) and \(y = -\frac{v}{{4 + v^2}}\). Proceeding with \(v\) values 1 to 3, the points are \((\frac{2}{5}, -\frac{1}{5})\), \((\frac{2}{8}, -\frac{2}{8})\), and \((\frac{2}{13}, -\frac{3}{13})\).

Step 6: Calculating Points for u=3

For \(u = 3\), compute \(x = \frac{3}{{9 + v^2}}\) and \(y = -\frac{v}{{9 + v^2}}\). With \(v\) values 1, 2, 3, the points become \((\frac{3}{10}, -\frac{1}{10})\), \((\frac{3}{13}, -\frac{2}{13})\), and \((\frac{3}{18}, -\frac{3}{18})\).

Step 7: Setting Up the v-Curves

Next, we consider constant \(v\) values and vary \(u\) from 0 to 3. Specifically, for each \(v\), the curves need to be plotted and form transformations \(x = \frac{u}{{u^2 + v^2}}\) and \(y = -\frac{v}{{u^2 + v^2}}\).

Step 8: Calculating Points for v=1

With \(v = 1\), the points are derived for \(u = 0, 1, 2, 3\) resulting in \((0, -1)\), \((\frac{1}{2}, -\frac{1}{2})\), \((\frac{2}{5}, -\frac{1}{5})\), and \((\frac{3}{10}, -\frac{1}{10})\).

Step 9: Calculating Points for v=2

For \(v = 2\), let \(u\) vary resulting in points \((0, -\frac{1}{2})\), \((\frac{1}{5}, -\frac{2}{5})\), \((\frac{2}{8}, -\frac{2}{8})\), \((\frac{3}{13}, -\frac{2}{13})\).

Step 10: Calculating Points for v=3

For \(v = 3\), with varying \(u\), the points are \((0, -\frac{1}{3})\), \((\frac{1}{10}, -\frac{3}{10})\), \((\frac{2}{13}, -\frac{3}{13})\), and \((\frac{3}{18}, -\frac{3}{18})\).

Step 11: Sketching the Curves

Using all the calculated points for constant \(u\) and \(v\), sketch the curves reflecting transformed behavior. The \(u\)-curves display as distinct groups of points spaced by their \(u\) values, while the \(v\)-curves show horizontal transformations for each constant \(v\) value.

Key Concepts

Understanding U-Curves

U-curves are a fundamental part of understanding coordinate transformations, particularly in two-dimensional spaces. In this context, a u-curve is a set of points plotted on a graph where the parameter \(u\) remains constant while \(v\) varies.
These curves can be visually represented through points determined using transformation equations.
For instance, in the given transformation, the equation for \(x\) and \(y\) is:

• \(x = \frac{u}{{u^2 + v^2}}\)
• \(y = -\frac{v}{{u^2 + v^2}}\)

To sketch u-curves:- Choose certain constant values for \(u\), such as 0, 1, 2, and 3.- Substitute these constants and vary \(v\). This defines the curve's shape and form.As an example, when \(u = 0\), the calculations simplify the x-coordinate to zero, meaning the points are strictly aligned on the y-axis.

Comprehending V-Curves

Similar to u-curves, v-curves represent another aspect of coordinate transformations. These curves maintain the parameter \(v\) constant, allowing \(u\) to vary.
They help understand how a transformation affects shapes in a grid system.
For v-curves, the transformation equations are the same:

• \(x = \frac{u}{{u^2 + v^2}}\)
• \(y = -\frac{v}{{u^2 + v^2}}\)

When sketching v-curves:- Set \(v\) at constant values, such as 1, 2, and 3.- Adjust \(u\) within a range and observe the transformation.For example, with \(v = 1\), each point on the v-curve represents a specific horizontal arrangement and illustrates how transformations shift positions horizontally.

Exploring Sketching Transformations

Sketching transformations involves creating visual representations of mathematical transformations like those specified for u-curves and v-curves. This practice is invaluable in calculus and geometry as it offers a tangible way to comprehend abstract equations.
To effectively sketch transformation:

• Identify points using transformation formulas.
• Plot these points for varied values of the variable parameters.
• Connect these points to visualize the curve.

This process not only helps in understanding transformations but also aids in visualizing how coordinate systems and geometric entities are altered in different planes.

Role of Calculus in Coordinate Transformation

Calculus plays a crucial role in understanding coordinate transformations, providing tools to analyze changes in our equations. In transformations involving u-curves and v-curves, derivatives can offer insights into the slope or rate of change of these curves.
Key applications include:

• Examining the rate of change as the variables \(u\) and \(v\) vary.
• Utilizing integrals to determine the area under these transformed curves.
• Applying calculus concepts to optimize and analyze the behaviors of transformations over continuous ranges.

Such tools are fundamental in fields that require precise modeling and solving of dynamic problems, making calculus indispensable to the study of transformations.