Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

Sketch both \(u\)-curves and \(v\)-curves using the transformation equations and plot them.

Step 1: Understand the Transformation

The transformation given is defined by the equations \(x = u + \frac{u}{u^2 + v^2}\) and \(y = v - \frac{v}{u^2 + v^2}\). These equations map the variables \(u\) and \(v\) to \(x\) and \(y\) in the coordinate plane.

Step 2: Sketch the u-Curves for Fixed v Values

For each fixed value of \(v\) (1, 2, 3), substitute \(v\) into the transformation equations and vary \(u\) from -2 to 2. Plot the resulting curves on the \(xy\)-plane.

Step 3: Sketch the v-Curves for Fixed u Values

For each fixed value of \(u\) (-2, -1, 0, 1, 2), substitute \(u\) into the transformation equations and vary \(v\) from 1 to 3. Plot the resulting curves on the \(xy\)-plane.

Step 4: Plot the Full Grid

Combine the sketches of \(u\)-curves and \(v\)-curves on the same set of axes. This will form a grid that illustrates the transformation's effect on the \((u, v)\) space.

Key Concepts

Parametric Equations

Parametric equations are powerful tools in mathematics. They express a set of related quantities as functions of one or more independent parameters.

• In our transformation, we have two parametric equations:
  • For the x-coordinate: \(x = u + \frac{u}{u^2 + v^2}\)
  • For the y-coordinate: \(y = v - \frac{v}{u^2 + v^2}\)
• The parameters here are \(u\) and \(v\). These parameters help describe how points in the \((u, v)\)-plane are mapped to the \((x, y)\)-plane.

These equations allow us to create complex shapes and curves that would be difficult to describe using simple Cartesian coordinates. This transformation equation is a perfect example of how parametric equations can illustrate such relationships by using parameters to trace paths or motions in a plane.By substituting different values of \(u\) and \(v\), we can explore how each combination affects the final position on the \((x, y)\) plane. This method is essential in many areas of mathematics, physics, and engineering, highlighting its practical and theoretical significance.

Sketching Curves

Sketching curves involves plotting points on a graph to represent equations visually. This is particularly useful for parametric equations, as it shows the relationship between different parameters and helps in understanding their influence on the curve shapes.In the provided exercise:

• First, sketch curves for fixed \(v\) values (1, 2, 3):
  • Substitute each \(v\) value into the parametric equations.
  • Vary \(u\) from -2 to 2 and calculate corresponding \(x\) and \(y\). Plot these points to form the curve.
• Then, sketch curves for fixed \(u\) values (-2, -1, 0, 1, 2):
  • Substitute each \(u\) value into the equations.
  • Vary \(v\) from 1 to 3, computing and plotting the resulting \(x\) and \(y\).
• Combine all curves to visualize the full transformation grid.

This process can be thought of as drawing a map, where curves from different parameter sets intersect to form a grid pattern. This graphical approach makes it much easier to understand how changes in \((u, v)\) affect the \((x, y)\) coordinates.

Mathematical Mapping

Mathematical mapping involves a set of rules or functions that translate coordinates from one system to another. This translation is crucial in understanding transformations between different coordinate systems.The transformation here

• Maps coordinates from the parameter plane (\(u, v\)) to the Cartesian plane (\(x, y\)).
• Involves non-linear equations for both \(x\) and \(y\), showcasing a more complex mapping than simple linear transformations.

In practice, mathematical mappings allow us to visualize how different regions or points in the original plane become altered in the target plane. When applied, these transformations can significantly shift and morph the original shapes, leading to various graphical and experimental applications.By utilizing mathematical mappings, we gain insights into dynamic systems, transforming coordinates for graphic creation, simulations, or solving geometric problems. The exercise's mapping conversion demonstrates the practical application and reasonings behind using such transformations, emphasizing their critical role in bridging worlds between different mathematical or real-world scenarios.