Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

For the transformation \(x=2 u+v, y=v-u\), 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 based on linear equations for fixed \(u\) and fixed \(v\) for given intervals.

Step 1: Understanding the Transformation

The transformation is given by the equations \(x=2u+v\) and \(y=v-u\). This means we need to evaluate how varying \(u\) or \(v\) affects the values of \(x\) and \(y\) through these equations.

Step 2: Plotting u-Curves

For each fixed \(u\) (i.e., \(u=2, 3, 4, 5\)) and varying \(v\) from 1 to 3, compute the corresponding \((x, y)\) coordinates. The equations for the curves are:- **\(u=2\):** \(x = 4+v\) and \(y = v-2\)- **\(u=3\):** \(x = 6+v\) and \(y = v-3\)- **\(u=4\):** \(x = 8+v\) and \(y = v-4\)- **\(u=5\):** \(x = 10+v\) and \(y = v-5\)These linear equations show lines of constant slope and varied intercept for each \(u\).

Step 3: Plotting v-Curves

For each fixed \(v\) (i.e., \(v=1, 2, 3\)) and varying \(u\) from 2 to 5, compute the corresponding \((x, y)\) coordinates. The equations for these curves are:- **\(v=1\):** \(x = 2u + 1\) and \(y = 1-u\)- **\(v=2\):** \(x = 2u + 2\) and \(y = 2-u\)- **\(v=3\):** \(x = 2u + 3\) and \(y = 3-u\)These are also linear, and the direction of the gradient changes as we fix \(v\).

Step 4: Sketching the Grid

Combine the results from Steps 2 and 3 onto a single coordinate system:- For each \(u\)-curve, draw a straight line using its respective equation.- For each \(v\)-curve, do the same, identifying where these lines intersect with \(u\)-curves.- The grid is constructed by intersections of these lines, within the limits specified for \((u, v)\).

Key Concepts

u-Curves

Understanding **u-curves** is essential when studying transformations of coordinate systems through different mappings. In our exercise, each u-curve is generated by holding the parameter \( u \) constant and varying \( v \) over its permitted range. This creates a series of linear equations because the relationship between \( x \) and \( u,v \) is linear.

For instance, when \( u=2 \), the transformation through the formulas gives us \( x=4+v \) and \( y=v-2 \). This represents a straight line with a slope of one in the Cartesian plane, showing the direct outcome of our equations.

• Each u-curve has a unique slope depending on the constant \( u \).
• u-curves form parallel lines as they change linearly with respect to \( v \).

Learning how u-curves form helps us understand how variations in one parameter affect the x-y coordinate outputs.

v-Curves

Much like u-curves, **v-curves** depict another aspect of the coordinate transformation but from the perspective of holding \( v \) constant. Each v-curve results from fixing \( v \) and varying \( u \). These also produce linear equations illustrating how different constraints on \( v \) alter the line's trajectory.

Consider the equation when \( v=1 \); it provides \( x=2u+1 \) and \( y=1-u \). Here, with a fixed \( v \), the line changes slope compared to when \( u \) was fixed. The result is a series of lines, with each line distinctive based on its constant \( v \).

• The slope of v-curves remains consistent within their set range.
• They display how combinations of \( u \) provide variations in \( x, y \).

V-curves show a complementary facet to understanding transformations in a coordinate grid.

Linear Equations

In the context of coordinate transformations, **linear equations** are pivotal. Both u-curves and v-curves in this exercise are represented by linear equations. But what makes an equation linear in this use?

Linear equations like \( y = mx + c \) specify a straight line. The \( m \) represents the slope, while \( c \) represents the y-intercept. In our transformation, both \( u \) and \( v \) alter these parameters, creating a collection of lines, each indicative of variations in coordinate inputs.

• Linear equations facilitate direct proportional outcomes between variables.
• They provide predictable pathways to solving transformations.

By analyzing the respective slopes and intercepts, these linear equations offer a direct method to understand how a transformation distorts the original coordinate system.

Coordinate System

A **coordinate system** provides the basic layout for plotting and understanding the interplay between transformed variables. The transformation problem at hand uses a transformation from \( (u,v) \) to \( (x,y) \) coordinates. This requires understanding how both sets of coordinates are situated in the standard Cartesian plane.

In our scenario, both u and v coordinates translate into new positions on the x-y plane. Familiar concepts like perpendicular axes and numerically symmetrical spaces form the basis of navigating this transformation.

• Understanding changes between coordinate systems requires a strong geometric sense.
• Helps visualize the interaction between u, v, and x, y spaces.

Proper grasp of coordinate systems ensures successful interpretation and sketching of transformed curves and intersections.

Grid Sketching

The art of **grid sketching** provides a visual method of understanding complex transformations. After deriving the linear equations, sketching these onto a grid offers clarity to theoretical solutions.

In the exercise, mapping the u-curves and v-curves onto a grid elucidates the intersections of these lines. It's an effective way to represent how differing values of \( u \) and \( v \) enclose unspecified regions within the coordinate system.

• Sketching helps solidify theoretical concepts into tangible outcomes.
• Visual representation aids spatial comprehension.

Through grid sketching, learners can appreciate both algebraic and geometric aspects of transformations, creating a holistic means of understanding.