$$ \begin{aligned} &x=\frac{1}{2}\left(u^{2}-v^{2}\right) \\ &y=u v \end{aligned} $$ ( \(u\) and \(v\) are called parabolic cylinder coordinates).

A coordinate transformation is a way to convert coordinates from one system to another. Here, we transform parabolic cylinder coordinates \((u, v)\) into Cartesian coordinates \((x, y)\). This is done using the equations: \ x = \frac{1}{2}(u^2 - v^2) \ and \ y = uv \. This process helps in solving complex geometrical problems by simplifying the coordinate system.

Coordinate transformations are common in fields like physics and engineering. They help in understanding various phenomena by choosing a more convenient coordinate system.

To perform the transformation, we first need to express the new coordinates \((x, y)\) in terms of the old coordinates \((u, v)\). Then, we solve these equations to find values in the new coordinate system.

Cartesian coordinates are the standard way of defining positions on a plane using two perpendicular axes: the x-axis and the y-axis. These coordinates are written as \((x, y)\), where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance.

Transforming parabolic cylinder coordinates to Cartesian coordinates involves using given equations to find \((x, y)\). In our exercise, the Cartesian coordinates are derived using the formulas given:

• \ x = \frac{1}{2}(u^2 - v^2) \
• \ y = uv \

These transformations help in visualizing and solving problems more effectively.

Understanding how to switch between different coordinate systems is crucial for analyzing mathematical and physical problems accurately.

Quadratic equations are mathematical expressions of the form ax^2 + bx + c = 0. They are called ‘quadratic’ because they involve the square of the unknown variable. In our exercise, we deal with a transformed quadratic equation to find the values of \(u\) and \(v\).

To solve for \(u\), we convert the given transformation equations to a standard quadratic equation:

• \ 2xu^2 = u^4 - y^2 \ becomes \ u^4 - 2xu^2 - y^2 = 0 \

We then use the quadratic formula, which is:

\[ u^2 = \frac{2x \pm \sqrt{4x^2 + 4y^2}}{2} = x \pm \sqrt{x^2 + y^2} \]

This approach efficiently finds the required parabolic cylinder coordinates.

Coordinate systems are frameworks used to define positions in a space. Common systems include Cartesian, polar, cylindrical, and spherical coordinates. Each system has unique properties that make it suitable for different types of problems.

In this exercise, we use parabolic cylinder coordinates \((u, v)\) and transform them into Cartesian coordinates \((x, y)\). The parabolic cylinder coordinates are particularly useful for problems involving parabolic shapes.

Using appropriate coordinate systems simplifies complex problems. It provides better geometric insights and aids in analytical and computational solutions.

Understanding different coordinate systems and their transformations is fundamental for higher-level mathematics and physics. It allows seamless transition and analysis across different geometrical contexts.