Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

For the transformation \(x=u+v, y=v-u\), sketch the \(u\) -curves and \(v\) -curves for the grid \(\\{(u, v):(u=2,3,4,5\) and \(1 \leq v \leq 3)\) or \((v=1,2,3\) and \(2 \leq u \leq 5)\\}\)

Step-by-Step Solution

Answer

Sketch lines for each fixed \(u\) and \(v\) with slopes ±1, using transformations.

Step 1: Understand the Transformation Equations

Given the transformation equations are \(x = u + v\) and \(y = v - u\). We need to interpret these equations to transform the curves defined in the \((u, v)\) coordinate system into the \((x, y)\) coordinate system.

Step 2: Sketch u-curves for Fixed u

For the given values of \(u = 2, 3, 4, 5\), we'll express \(x\) and \(y\) in terms of \(v\). Since \(x = u + v\), for fixed \(u\), we have: \(x = constant + v\), and \(y = v - constant\). This gives us lines in the \((x, y)\) plane with a slope of 1, because as \(v\) changes, \(x\) and \(y\) both increase linearly.

Step 3: Sketch v-curves for Fixed v

For the fixed values of \(v = 1, 2, 3\), substitute each into the equations \(x = u + v\) and \(y = v - u\). For a fixed \(v\), \(x = u + constant\) results in a line with a positive slope, while \(y = constant - u\) results in a line with a negative slope, effectively mirroring the u-curves but with different intercepts.

Step 4: Plot each u-curve and v-curve

Draw the u-curves, which are lines of the form \(x = u + v\), and the v-curves, which are of the form \(y = v - u\), on the same grid. Match the starting and ending points of these lines based on the conditions \(1 \leq v \leq 3\) and \(2 \leq u \leq 5\). The u-curves should be parallel to each other and have a positive slope in the \((x, y)\) plane, while the v-curves mirror the behavior of the u-curves.

Key Concepts

Transformation Equations

Coordinate transformation is a powerful concept that allows us to represent points or curves from one coordinate system in another. In this particular case, we are dealing with the transformation equations:

• \( x = u + v \)
• \( y = v - u \)

These equations transform the \((u, v)\) coordinate system into the \((x, y)\) coordinate system.
Understanding these equations is critical as they define how each point in the source system (\(u, v\)) corresponds to a point in the target system (\(x, y\)).
With transformation, a change in one coordinate system will reflect as a different change in another, keeping the geometric relationships the same but altering their appearance.
In practice, transformation equations help connect different frames of observation, often simplifying complex geometry by changing its representation.

Coordinate System

The coordinate system is the backbone of any spatial representation.
Typical examples include Cartesian, Polar, and in this case, a custom transformation defined by our transformation equations.
In the Cartesian coordinate system, each point is represented by an ordered pair \((x, y)\).

• For our task, the transformation connects the original \((u, v)\) system to this Cartesian framework.
• This dictates how we move from lines defined by \(u\) and \(v\) in the source system to lines in \((x, y)\).

With this transformation, what was previously described using \(u\) and \(v\) now gets a new representation through \(x\) and \(y\).
This understanding helps sketch the grid by accurately measuring and plotting points or lines based on the new system's rules.

Grid Sketching

Grid sketching involves drawing plots based on a set of conditions or equations on a coordinate plane.
In this exercise, we draw \(u\)-curves and \(v\)-curves, representing specific lines in the \((x, y)\) coordinate system.
For grid sketching, we:

• Start by choosing fixed values of \(u\) or \(v\) from the original grid parameters.
• Substitute these into the transformation equations to find corresponding \(x\) and \(y\) values.
• Plot lines for each fixed parameter across the range given in the exercise.

Grid sketching effectively visualizes how each coordinate line behaves under transformation.
The lines for fixed \(u\) (say \(u = 2\) to \(5\)) will show varying behavior from the lines for fixed \(v\) (say \(v = 1\) to \(3\)), according to their specific transformations.

Curve Plotting

Curve plotting is the process of drawing the curves that result from our grid sketching, turning the mathematical definitions into visual aids.
In our scenario, both \(u\)-curves and \(v\)-curves are plotted on the \((x, y)\) coordinate system:

• For \(u\)-curves: Each is represented by the equation \(x = u + v\).
• For \(v\)-curves: Each is represented by \(y = v - u\).

These curves intersect and create a grid pattern which is essential for understanding the transformation.
The lines' slopes demonstrate how input variables \(u\) and \(v\) change across the grid, leading to a clearer visualization of the coordinate transformation.
Curve plotting transforms abstract equations into tangible patterns that can be analyzed for insights, aiding in the comprehension of spatial transformations.