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Chapter 21: Problem 8

Find a number \(a\) so that the change of coordinates \(s=x+a y, t=y\) transforms the integral \(\iint_{R} d x d y\) over the parallelogram \(R\) in the \(x y\) -plane into an integral $$\iint_{T}\left|\frac{\partial(x, y)}{\partial(s, t)}\right| d s d t$$ over a rectangle \(T\) in the \(s t\) -plane. \(R\) has vertices (0,0),(10,0),(12,3),(22,3)

Short Answer

Expert verified
The number \(a\) is \(\frac{2}{3}\).

Step by step solution

01

Understand the Transformation Matrix

The change of variables from \((x, y)\) to \((s, t)\) is given by \(s = x + ay\) and \(t = y\). We need to express \(x\) and \(y\) in terms of \(s\) and \(t\), which leads to \(x = s - at\) and \(y = t\). This transformation corresponds to the following matrix: \[\begin{pmatrix}\frac{\partial x}{\partial s} & \frac{\partial x}{\partial t} \\frac{\partial y}{\partial s} & \frac{\partial y}{\partial t}\end{pmatrix} = \begin{pmatrix}1 & -a \0 & 1\end{pmatrix}.\]
02

Calculate the Jacobian Determinant

The Jacobian determinant of the transformation is determined by the matrix of partial derivatives from Step 1:\[\left| \frac{\partial(x,y)}{\partial(s,t)} \right| = \begin{vmatrix}1 & -a \0 & 1\end{vmatrix} = 1 \times 1 - (-a) \times 0 = 1.\]The determinant equals 1, which indicates a volume-preserving transformation.
03

Verify the Transformation Results in a Rectangle

To ensure the transformation maps the parallelogram \(R\) into a rectangle \(T\), we'll examine how the vertices of \(R\) transform. By substituting \(s = x + ay\) and \(t = y\) for each vertex:- Vertex \((0,0)\) maps to \((0,0)\).- Vertex \((10,0)\) maps to \((10,0)\).- Vertex \((12,3)\) maps to \((12 + 3a, 3)\).- Vertex \((22,3)\) maps to \((22 + 3a, 3)\).The condition for these points to form a rectangle is that the sides parallel to the \(t\)-axis should be equal; thus, \((12 + 3a) - 10 = (22 + 3a) - 20\), which implies \(3a = 2\) or \(a = \frac{2}{3}\).
04

Conclusion

The parallel sides of the transformed quadrilateral are perpendicular to the \(s\)-axis and equal in length when \(a = \frac{2}{3}\), successfully transforming the parallelogram into a rectangle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Transformation
The process of a coordinate transformation involves changing from one coordinate system to another. This change is crucial in many areas of mathematics and physics, allowing us to simplify problems or transform them into more understandable forms. In our exercise, we're performing a transformation from the coordinates \((x, y)\) to \((s, t)\) using the equations \(s = x + a y\) and \(t = y\).
  • Purpose: This change of variables can simplify integrals by transforming complex geometries into more manageable shapes, like a rectangle.
  • Functional Aspect: To move between these systems, we derive the reverse relations, namely \(x = s - at\) and \(y = t\), crucial for calculating the Jacobian in further steps.
  • Linear Transformation: The specific linear transformation applied affects the mapping from the original shape to the new geometric configuration.
Coordinate transformations form the groundwork of linear algebra applications in physics and engineering, especially when dealing with non-standard geometries or coordinate systems.
Volume-Preserving Transformation
One key property we check for is whether a transformation is volume-preserving. This means that the transformation does not alter the volume in the new coordinate system. To determine this, we calculate the Jacobian determinant.
  • Jacobian Determinant Calculation: It is the determinant of the matrix formed by partial derivatives of the transformation equations and provides a measure of how the "volume" is scaled during the transformation.
  • Example: In our example, the matrix is \(\begin{pmatrix}1 & -a \ 0 & 1\end{pmatrix}\). Its determinant is \(1\), indicating no change in volume during the transformation.
  • Implications: Since the determinant is \(1\), the transformation from \(R\) to \(T\) preserves area, meaning no compression or expansion occurs.
Volume-preserving transformations are especially significant in fields such as fluid dynamics and systems described by constant density.
Rectangle Mapping
Mapping complex shapes into simpler ones, like rectangles, is a common technique in calculus to simplify integrals. In our task, we transform the parallelogram into a rectangle through careful calculation of parameters.
  • Vertex Transformation: Each vertex of the original shape (parallelogram) is transformed using the new coordinate equations, mapping them into the rectangle's corners in the new system.
  • Condition for Rectangular Shape: To ensure that the shape becomes a rectangle, side lengths perpendicular to one axis must remain equal after transformation. This is why we needed to compute the correct \(a = \frac{2}{3}\).
  • Simplification of Integration: Mapping to a rectangle allows for the straightforward application of integration techniques, as rectangles conform to the standard limits easier than most other shapes.
This process of rectangle mapping leverages transformation properties to achieve more efficient and simpler computational models, facilitating easier solutions of integral-based problems.

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Most popular questions from this chapter

Give a parameterization of the circle of radius \(a\) centered at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) and in the plane parallel to two given unit vectors \(\vec{u}\) and \(\vec{v}\) such that \(\vec{u} \cdot \vec{v}=0\).

Are the statements true or false? Give reasons for your answer. If \(\vec{r}=\vec{r}(s, t)\) parameterizes the upper hemisphere \(x^{2}+y^{2}+z^{2}=1, z \geq 0,\) then \(\vec{r}=-\vec{r}(s, t)\) parameterizes the lower hemisphere \(x^{2}+y^{2}+z^{2}=1, z \leq 0\).

Describe the families of parameter curves with \(s=s_{0}\) and \(t=t_{0}\) for the parameterized surface. $$\begin{aligned}&x=s, \quad y=\cos t, \quad z=\sin t \text { for }-\infty

For a sphere parameterized using the spherical coordinates \(\theta\) and \(\phi,\) describe in words the part of the sphere given by the restrictions. $$\pi \leq \theta<2 \pi, \quad 0 \leq \phi \leq \pi$$

Are the statements true or false? Give reasons for your answer. If \(\vec{r}_{1}(s, t)\) parameterizes a plane then \(\vec{r}_{2}(s, t)=\) \(\vec{r}_{1}(s, t)+2 \vec{i}-3 \vec{j}+\vec{k}\) parameterizes a parallel plane.
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