Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

Plot circles for each u and straight lines for each v on the xy-plane.

Step 1: Understand the Transformation

The transformation given is a change of variables from \((x, y)\) to \((u,v)\) where \(x = u \cos v\) and \(y = u \sin v\). This represents a polar coordinate system where \(u\) gives the radial distance from the origin and \(v\) represents the angle from the x-axis.

Step 2: Plot u-Curves

The \(u\)-curves are plotted for constant values of \(u\). For \(u=0\), all points are at the origin \((0,0)\). For \(u=1\), \(u=2\), and \(u=3\), the curves are circles centered at the origin with radii equal to the value of \(u\). Thus, draw circles of radii 1, 2, and 3.

Step 3: Plot v-Curves

The \(v\)-curves are plotted for constant values of \(v\). For each of \(v=0\), \(v=\pi\), and \(v=2\pi\), these curves represent lines. For \(v=0\) and \(v=2\pi\), the line coincides with the positive x-axis; for \(v=\pi\), the line coincides with the negative x-axis.

Step 4: Combine the Curves

Overlay the \(u\)-curves (circles) and \(v\)-curves (lines) onto the same coordinate plane. Each circle should intersect each line at one or two points based on their definitions. This creates a grid pattern overlaid on the Cartesian plane.

Key Concepts

Transformation in Polar Coordinates

In mathematics, a transformation is a way to convert one set of coordinates to another. Here, we're working with a transformation that takes us from Cartesian coordinates \(x, y\) to polar coordinates \(u, v\). In this specific transformation, \(x = u \cos v\) and \(y = u \sin v\). This represents a point's position in terms of a radial distance and an angle. Think of each data point \(u, v\) as describing a point on a circle. \(u\) denotes how far the point is from the center (origin), while \(v\) describes its angle from the positive x-axis. You might notice that this echoes how a circle is described, providing a neat and organized method to map out the grid of points effectively on a plane. Such transformations are critical in applications like physics and engineering, where varying systems of measurement and reference frames are encountered.

Understanding the X-Y Plane

The x-y plane, also known as the Cartesian plane, is a flat surface defined by two axes: the horizontal x-axis and the vertical y-axis. In this exercise, we transform polar coordinate data onto this plane. \(u\) and \(v\) are converted into \(x\) and \(y\) respectively, letting us plot this data within the familiar x-y framework.

• The x-axis runs left to right.
• The y-axis runs bottom to top.
• The origin is the point (0,0), where both axes intersect.

The beauty of using this plane is that it allows us to visually analyze geometrical shapes and relationships—like in this exercise, where circles and lines intersect to form a grid. Recognizing these intersections helps one understand the relationship between polar and Cartesian coordinates, highlighting how angles and radii translate into linear and curvilinear points on this plane.

Grid Plotting in Transformations

Grid plotting is a systematic approach to visualizing data points on a coordinate plane through the intersection of curves or lines. In this scenario, we are interested in the interplay between u-curves and v-curves.

• U-Curves: These are circles of varying radii originating from the origin. For \(u = 1, 2, 3\), we draw circles with those radii, creating concentric circles centered at (0,0).
• V-Curves: For angles \(v=0, \pi, 2\pi\), we draw straight lines. When \(v=0\) or \(v=2\pi\), the line aligns with the positive x-axis; for \(v=\pi\), the line aligns with the negative x-axis.

When these curves intersect on the x-y plane, they form an intricate grid pattern. Grid plotting this transformation lets us see how values from polar coordinates translate to specific locations on the x-y plane. Each intersection provides insights into coordinate transformation, highlighting the seamless blend between radial and linear systems.