91Ó°ÊÓ

Coordinate transformation is a fundamental concept in mathematics and physics, allowing us to move from one coordinate system to another. In the context of the given exercise, we are transforming coordinates from the xy-plane to the uv-plane using specific equations: \(u = x - y\) and \(v = 2x + y\). These equations describe how every point

• \((x, y)\) in the original coordinate system is uniquely mapped to a point \((u, v)\) in the new coordinate system.
• The transformation allows for a different representation, which can simplify the problem-solving process depending on the context.

One practical application of such transformations is to analyze systems more effectively by choosing a coordinate system where the equations become easier to handle. By understanding the mapping characteristics and utilizing tools like the Jacobian determinant, we ensure our transformations are valid and maintain properties such as scale.

A linear system of equations is a system of equations in which each equation is a linear equation. In the exercise, we are given two linear equations:

• \(u = x - y\), representing a relationship between variables \(x\), \(y\), and \(u\).
• \(v = 2x + y\), further connecting these variables alongside another, \(v\).

The objective is to solve for \(x\) and \(y\) in terms of \(u\) and \(v\). This involves substituting and rearranging the equations:- By expressing one variable in terms of another and substituting, we isolate each to express them solely in terms of the given variables.The solution provides expressions for \(x\) and \(y\):

• \(x = \frac{v + u}{3}\)
• \(y = \frac{v - 2u}{3}\)

These expressions allow us to trace back from the \(uv\)-plane to the original \(xy\)-coordinates, further aiding diverse analyses such as curve sketching or region transformations.

The determinant is a value computed from the elements of a square matrix. It provides critical information about the matrix, such as whether it is invertible and, in geometric transformations, how the function behaves locally. In this exercise, the determinant of the Jacobian matrix is calculated to find the rate of change of variables.The Jacobian matrix comprises the partial derivatives of the transformed coordinates \((x, y)\) with respect to \((u, v)\):

• \( \frac{\partial x}{\partial u} = \frac{1}{3} \)
• \( \frac{\partial x}{\partial v} = \frac{1}{3} \)
• \( \frac{\partial y}{\partial u} = -\frac{2}{3} \)
• \( \frac{\partial y}{\partial v} = \frac{1}{3} \)

The determinant of the Jacobian matrix \(J\) is then calculated as:\[ \det(J) = \left(\frac{1}{3}\right)\left(\frac{1}{3}\right) - \left(-\frac{2}{3}\right)\left(\frac{1}{3}\right) = \frac{1}{3} \]The result tells us how the area element changes under transformation. A determinant of \(\frac{1}{3}\) indicates that the area in the \((uv)\)-plane is \(\frac{1}{3}\) of the area in the \((xy)\)-plane, reflecting the scaling effect due to the transformation.