Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The \(u\)-curves are semicircles, and the \(v\)-curves are radial lines.

Step 1: Understanding the Transformation

The transformation equations are given as \(x = u \sin v\) and \(y = u \cos v\). This set of equations represent polar coordinates where \(u\) acts like the radius and \(v\) acts like the angle.

Step 2: Sketching \(u\)-Curves

For \(u = 0, 1, 2, 3\), the parameter \(v\) varies from 0 to \(\pi\). The curve obtained when \(u\) is constant and \(v\) varies describes a circle with radius \(u\), moving from angle 0 to \(\pi\). Thus, for each value of \(u\), we draw a semicircle in the xy-plane with radius \(u\).

Step 3: Sketching \(v\)-Curves

For \(v = 0, \pi/2, \pi\), the parameter \(u\) varies from 0 to 3. Each \(v\)-curve will be a radial line starting at the origin, with different angles relative to the positive x-axis: \(v = 0\) is the x-axis, \(v = \pi/2\) is the y-axis, and \(v = \pi\) is the negative x-axis. These lines extend from the origin outward to a distance of 3.

Key Concepts

Coordinate Transformation

When dealing with transformations in mathematics, it's all about converting one kind of coordinate system into another. In the given problem, the transformation is defined by the equations \(x = u \sin v\) and \(y = u \cos v\). This setup is a type of coordinate transformation often linked to polar coordinates.
This transformation is named after its ability to translate between polar and Cartesian coordinates. In polar coordinates, you have a distance \(u\), similar to a radius, and an angle \(v\). In the Cartesian system, coordinates are defined by \(x\) and \(y\) positions in the plane. This transformation takes points defined in the \((u, v)\) system and places them in the \((x, y)\) system.
It helps to think of \(u\) as how far out you are from the origin, or center point, and \(v\) as the direction you're pointing from that center. So, by using trigonometric functions — sine for \(y\) and cosine for \(x\) — we see how this transformation converts circular motion into a more linear and flat plane.

U-Curves and V-Curves

In mathematical sketching using transformation equations, \(u\)-curves and \(v\)-curves are tools to visualize movements in coordinate systems.
**U-Curves:** These curves are determined by setting \(u\) constant and varying \(v\). In our problem, \(u\) takes values 0, 1, 2, and 3 while \(v\) ranges from 0 to \(\pi\). The \(u\)-curves represent semicircles. Each semicircle's radius matches the particular value of \(u\). When \(v\) sweeps from 0 to \(\pi\), it traces a semicircle starting at the positive \(x\)-axis and ending at the negative \(x\)-axis.
**V-Curves:** These occur when \(v\) is constant, and \(u\) varies from 0 to 3. Here, \(v\) is essentially the angle. In our example, \(v\) is 0, \(\pi/2\), or \(\pi\), making these curves straight lines radiating from the origin. \(v = 0\) corresponds to the positive \(x\)-axis, \(v = \pi/2\) to the \(y\)-axis, and \(v = \pi\) to the negative \(x\)-axis. Each radial line extends outward to the maximal \(u\) value of 3.

XY-Plane Sketching

Sketching in the \(xy\)-plane allows us to visually represent solutions and transformations, making complex mathematical concepts more intuitive.
When dealing with the transformation \(x = u \sin v\) and \(y = u \cos v\), sketching helps to clarify how each component, \(u\)-curves, and \(v\)-curves, interplay in the Cartesian plane. By visualizing different combinations of \(u\) and \(v\), you see both circular and radial symmetries materializing.
For effective sketching:

• Start by identifying zones: semicircles for \(u\)-curves and radial lines for \(v\)-curves.
• Plot the end points and extend \(v\)-curves from the center out to establish their direction.
• Draft \(u\)-curves as arcs between these points, verifying that the radii comply with the corresponding constant \(u\) values.

This kind of sketching outlines how transformations modify and morph grid systems from one form to another, bridging the intuitive with the analytical.