Problem 57 - Calculus Early Transcendentals

Question

Parabolic coordinates Let \(T\) be the transformation \(x=u^{2}-v^{2}\) \(y=2 u v\) a. Show that the lines \(u=a\) in the \(u v\) -plane map to parabolas in the \(x y\) -plane that open in the negative \(x\) -direction with vertices on the positive \(x\) -axis. b. Show that the lines \(v=b\) in the \(u v\) -plane map to parabolas in the \(x y\) -plane that open in the positive \(x\) -direction with vertices on the negative \(x\) -axis. c. Evaluate \(J(u, v)\) d. Use a change of variables to find the area of the region bounded by \(x=4-y^{2} / 16\) and \(x=y^{2} / 4-1\) e. Use a change of variables to find the area of the curved rectangle above the \(x\) -axis bounded by \(x=4-y^{2} / 16\) \(x=9-y^{2} / 36, x=y^{2} / 4-1,\) and \(x=y^{2} / 64-16\) f. Describe the effect of the transformation \(x=2 u v\) \(y=u^{2}-v^{2}\) on horizontal and vertical lines in the \(u v\) -plane.

Step-by-Step Solution

Answer

Answer: Lines \(u=a\) in the \(uv\)-plane map to parabolas in the \(xy\)-plane that open in the negative \(x\)-direction with vertices on the positive \(x\)-axis. Lines \(v=b\) in the \(uv\)-plane map to parabolas in the \(xy\)-plane that open in the positive \(x\)-direction with vertices on the negative \(x\)-axis.

Step 1: Write the transformation equations for x and y

We are given the transformation equations: \(x = u^2 - v^2\) \(y = 2uv\)

Step 2: Replace u with a constant a in the transformation equations

Since the lines are \(u = a\), we can replace \(u\) with a constant value, say \(a\), in the transformation equations: \(x = a^2 - v^2\) \(y = 2av\)

Step 3: Rearrange the equations for a parabola

To show that the lines in the \(uv\)-plane map to parabolas in the \(xy\)-plane, we can rearrange the equations in the form of a parabola in either \(x\) or \(y\). Here, we will do it for \(y\): $$y = 2av \implies y^2 = 4a^2v^2$$ $$x = a^2 - v^2$$ Add \(v^2\) to both sides of the equation for \(x\): $$x + v^2 = a^2$$ Now, divide the equation for \(y^2\) by 4 and substitute it into the equation for \(x + v^2\): $$x + \frac{y^2}{4a^2} = a^2$$

Step 4: Analyze the obtained equation

The obtained equation is of the form of a parabola equation opening in the negative \(x\) direction with its vertex on the positive x-axis: $$x = a^2 - \frac{y^2}{4a^2}$$ Thus, the lines \(u=a\) in the \(uv\)-plane map to parabolas in the \(xy\)-plane that open in the negative \(x\)-direction with vertices on the positive \(x\)-axis. #b. Lines v=b to parabolas in the xy plane (positive x-direction)#

Step 1: Replace v with a constant b in the transformation equations

Since the lines are \(v = b\), we can replace \(v\) with a constant value, say \(b\), in the transformation equations: \(x = u^2 - b^2\) \(y = 2ub\)

Step 2: Rearrange the equations for a parabola

To show that the lines in the \(uv\)-plane map to parabolas in the \(xy\)-plane, we can rearrange the equations in the form of a parabola in either \(x\) or \(y\). Here, we will do it for \(y\): $$y = 2ub \implies y^2 = 4u^2b^2$$ $$x = u^2 - b^2$$ Add \(b^2\) to both sides of the equation for \(x\): $$x + b^2 = u^2$$ Now, divide the equation for \(y^2\) by 4 and substitute it into the equation for \(x + b^2\): $$x - \frac{y^2}{4b^2} = -b^2$$

Step 3: Analyze the obtained equation

The obtained equation is of the form of a parabola equation opening in the positive \(x\) direction with its vertex on the negative x-axis: $$x = \frac{y^2}{4b^2} - b^2$$ Thus, the lines \(v=b\) in the \(uv\)-plane map to parabolas in the \(xy\)-plane that open in the positive \(x\)-direction with vertices on the negative \(x\)-axis.

Key Concepts

Coordinate Transformation

Coordinate transformation is a fundamental concept in mathematics and physics where one set of coordinates is converted into another set to simplify the problem or to describe a system more conveniently. In the context of parabolic coordinates, the transformation given by

\( x = u^2 - v^2 \)
\( y = 2uv \)

maps points from the UV plane to the XY plane. It helps us analyze complex shapes and regions, such as parabolas, in a more manageable coordinate system.

Understanding Transformation

Through this conversion, equations of lines or curves in the UV plane can be expressed differently in the XY plane, revealing geometric shapes like parabolas not immediately obvious in their original form.

Parabola Vertex

The vertex of a parabola is a significant point where the curve turns; it is the maximum or minimum point of the parabola, depending on the direction it opens. In the context of parabolic coordinates involving the equations

\( x = u^2 - v^2 \)
\( y = 2uv \)

we utilize a set value for either ‘u’ or ‘v’ to derive parabolas with vertices lying along the X-axis. When we hold ‘u’ constant and vary ‘v’, the resulting parabolas open negatively along the X-axis with vertices on the positive side. Conversely, holding ‘v’ constant maps to parabolas that open positively, with their vertices located on the negative X-axis.

Jacobian Determinant

The Jacobian determinant is a critical value when performing coordinate transformations, particularly for calculating areas, volumes, and other higher-dimensional integrals. It measures the rate at which the area (or volume) changes during the transformation process.

Importance of the Jacobian

For parabolic coordinates, evaluating the Jacobian determinant, often denoted as \( J(u, v) \), helps us transition between the UV plane and the XY plane and is necessary to calculate the area using a change of variables. The determinant provides the factor by which the area in one coordinate system expands or contracts to yield the area in the other system.

Area Using Change of Variables

Calculating the area of a region by using a change of variables is a powerful technique in multiple-variable calculus. It involves transforming a complex region in one coordinate system to a simpler form in another system wherein the areas can be computed more easily.

Change of Variables Technique

In the exercise given, we convert the region bounded by certain curves into parabolic coordinates, making use of the Jacobian determinant to account for how space is stretched or compressed. By using this method, the challenging process of finding the areas bounded by parabolas in the XY plane becomes a more standard operation.

Effects of Transformation on Lines

Transformations can have varying effects on different types of lines or curves in a coordinate plane. In our context, transforming lines in the UV-plane to the XY-plane via parabolic coordinates does not yield lines but rather curves—specifically parabolas.

Identifying Transformations

Horizontal and vertical lines in the UV plane are transformed into parabolas in the XY plane. Horizontal lines correspond to setting ‘v’ as constant, resulting in parabolas that open in the positive X-direction. Meanwhile, vertical lines correspond to setting ‘u’ as constant, leading to parabolas that open negatively with respect to the X-axis. Thus, simple lines in one coordinate system can become complex curves in another.