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

Transform the grid from \((u, v)\) space into \((x, y)\) space to sketch the \(u\) and \(v\) curves.

Step 1: Analyze the Transformation

We have the transformations given as two equations: \(x = 2u + v\) and \(y = v - u\). These will transform the grid lines defined in \((u, v)\) space into \((x, y)\) space.

Step 2: Define the u-curves

Considering the curves where \(u\) is constant (\(u = 2, 3, 4, 5\)), substitute in the first transformation equation, \(x = 2u + v\), which becomes \(x = 4 + v, 6 + v, 8 + v, 10 + v\). Use the second equation to get \(y = v - u\), resulting in curves \(y = v - 2, v - 3, v - 4, v - 5\).

Step 3: Sketching the u-curves for 1 \(\leq v \leq 3\)

For each value of \(u\), compute the line equations for \(v = 1\) to \(3\):- \(u = 2\): Points will be \((x, y): (5, -1), (6, 0), (7, 1)\).- \(u = 3\): Points will be \((7, -2), (8, -1), (9, 0)\).- \(u = 4\): Points will be \((9, -3), (10, -2), (11, -1)\).- \(u = 5\): Points will be \((11, -4), (12, -3), (13, -2)\).

Step 4: Define the v-curves

For curves where \(v\) is constant (\(v = 1, 2, 3\)), use the transformation equations \(x = 2u + v\), resulting in \(x = 2u + 1, 2u + 2, 2u + 3\). The second equation becomes \(y = v - u\), resulting in \(y = 1 - u, 2 - u, 3 - u\).

Step 5: Sketching the v-curves for 2 \(\leq u \leq 5\)

For each value of \(v\), compute the line equations:- \(v = 1\): Points will be \((5, -1), (7, -2), (9, -3), (11, -4)\) for each \(u\).- \(v = 2\): Points will be \((6, 0), (8, -1), (10, -2), (12, -3)\) for each \(u\).- \(v = 3\): Points will be \((7, 1), (9, 0), (11, -1), (13, -2)\) for each \(u\).

Step 6: Sketch the Grid in xy-plane

Draw the transformed grid on the \(x-y\) plane using the points derived from steps 3 and 5. Draw lines connecting the points derived from consecutive \(u\) and \(v\) values. This grid will fulfill both the conditions: \(u = 2, 3, 4, 5\) and \(1 \leq v \leq 3\), \(v = 1, 2, 3\) and \(2 \leq u \leq 5\).

Key Concepts

Grid Transformation

In coordinate geometry, a grid transformation refers to the process of changing a grid from one coordinate system to another. This is often performed using a set of transformation equations that map each point in the original coordinate system to a corresponding point in the new system.

• In our exercise, the transformation equations are given by:
  • \( x = 2u + v \)
  • \( y = v - u \)

These equations transform the points from \((u, v)\) space into \((x, y)\) space.The transformation is applied to a grid defined by specific values of \(u\) and \(v\). This means that when you change the \(u\) or \(v\) values, you can see how each line in the original grid (formed by constant \(u\) or \(v\)) is mapped to a line in the new coordinate system. By doing this, the grid patterns change, reflecting how each point's new position is determined by the transformation equations. Understanding grid transformations is key in scenarios such as translating graphs and reshaping data geometry in computational graphics.

Curve Sketching

Curve sketching involves drawing the curves and lines represented by equations in a coordinate plane. This technique is essential in understanding mathematical relationships in visual terms. By sketching the transformations, one can see how curves representing constant variables translate from one coordinate system to another. In the exercise, two types of curves are sketched: - **u-curves**: These result from holding \(u\) constant while varying \(v\). For each fixed \(u\) value (\(u = 2, 3, 4, 5\)), different lines in the \((x, y)\) space are sketched by calculating the endpoints of each line (based on \(1 \leq v \leq 3\)) using the transformation equations. - **v-curves**: Similarly, when \(v\) is constant (\(v = 1, 2, 3\)) and \(u\) varies (\(2 \leq u \leq 5\)), you sketch the resulting lines. This offers insight into how the relationships defined in \(u, v\) space reconfigure in \(x, y\) space.By examining the intersections and orientations of these curves, students can grasp the mapping characteristics provided by the transformation equations, enhancing their spatial intuition.

Parametric Equations

Parametric equations provide a method to describe curves using parameters, offering a flexible way to define complex curves as opposed to simple functions. In the given problem, the equations \(x = 2u + v\) and \(y = v - u\) serve as parametric equations, with \(u\) and \(v\) acting as parameters.

• Through these parameters, each point in the new \((x, y)\) space is expressed in terms of the parameters \(u\) and \(v\).
• Parametric equations allow for describing and understanding trajectories of objects by encapsulating the spatial relationship between coordinates in a dynamic manner. In visual terms, it enables tracing out shapes like paths or grids as seen in this transformation exercise.

In mathematical applications, parametric equations are frequently used in calculus and physics to describe motion paths and other scenarios where variables change over time. The ability to parameterize grid transformations, as shown, illustrates both the flexibility and power of parametric representation, particularly in higher-level mathematics and computer graphics.