91Ó°ÊÓ

In mathematics, partial derivatives are used to measure how a function changes in response to small changes in its input variables. Unlike ordinary derivatives, which apply to functions of a single variable, partial derivatives are applied to functions with multiple variables, like our functions for \(x(u,v)\) and \(y(u,v)\). For these functions, we have two input variables: \(u\) and \(v\).

To find partial derivatives, select one variable to vary while keeping the others constant. For instance, in our solution, when calculating \(\frac{\partial x}{\partial u}\), we keep \(v\) constant and observe how \(x\) changes. Similarly, \(\frac{\partial x}{\partial v}\) involves keeping \(u\) constant.

Partial derivatives help us construct the Jacobian matrix, which provides information about how functions transform input spaces to output spaces. In our solution, these derivatives are:

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

A linear system of equations consists of multiple linear equations involving the same set of variables. In our scenario, we have a system of two equations:

\[ \begin{cases} u = x+y \ v = 2x-y \end{cases} \]

Often, solving such a system requires methods like substitution or elimination, which simplify the system to find explicit solutions for each unknown variable. Our solution employs the elimination method. By adding and manipulating the given equations, we isolate individual variables.

The elimination method helps us efficiently find expressions for \(x\) and \(y\) in terms of \(u\) and \(v\):

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

These solutions form the foundation for evaluating the Jacobian matrix and its determinant.

The determinant of a matrix is a special number that gives important insights into the properties of the matrix. Specifically, for a 2x2 matrix (like our Jacobian matrix), the determinant is calculated using the formula:

\[ \det J = ad - bc \]

where \(a\), \(b\), \(c\), and \(d\) are the elements of the matrix:

\[ J = \begin{bmatrix} a & b \ c & d \ \end{bmatrix} \]

In our particular case, the Jacobian matrix is:

\[ J(u, v) = \begin{bmatrix} \frac{1}{3} & \frac{1}{3} \ \frac{2}{3} & -\frac{1}{3} \end{bmatrix} \]

Calculating the determinant using the provided formula gives:

\[ \det J(u, v) = \left(\frac{1}{3} \times -\frac{1}{3}\right) - \left(\frac{1}{3} \times \frac{2}{3}\right) = -\frac{1}{9} - \frac{2}{9} = -\frac{3}{9} \]

The determinant value, \(-\frac{3}{9}\), signifies the scaling factor of the transformation represented by the Jacobian matrix, influencing how areas and volumes change under the mapping.