91Ó°ÊÓ

In mathematics, transformations allow us to convert coordinates from one system to another. For the given transformation:

• \( x = u + 2v \)
• \( y = -u + v \)

The variables \( x \) and \( y \) are expressed in terms of \( u \) and \( v \). This mapping can help simplify complex problems by transforming them into easier forms to work with.
In this exercise, understanding the change from \( (u, v) \) to \( (x, y) \) is crucial for calculating the Jacobian, which tells us how this transformation affects size and shape.
To get the full picture, identifying how each new variable is derived from the old ones helps us set up the next steps in deriving the Jacobian matrix.

Partial derivatives measure how a function changes as one of its variables changes while keeping other variables constant. For transformations, they help us understand how each new variable changes with respect to the original variables.
In our transformation:

• The partial derivative of \( x \) with respect to \( u \) is \( \frac{\partial x}{\partial u} = 1 \)
• The partial derivative of \( x \) with respect to \( v \) is \( \frac{\partial x}{\partial v} = 2 \)
• The partial derivative of \( y \) with respect to \( u \) is \( \frac{\partial y}{\partial u} = -1 \)
• The partial derivative of \( y \) with respect to \( v \) is \( \frac{\partial y}{\partial v} = 1 \)

These derivatives gather into a matrix known as the Jacobian matrix, essential for understanding how moving in the \( uv \) plane translates into movements in the \( xy \) plane.

The determinant of a matrix provides important information about linear transformations, such as whether transformations preserve area and orientation. To calculate it for a 2x2 matrix:
Given a matrix \( J = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated as \( ad - bc \).
For our Jacobian matrix:\[J = \begin{bmatrix} 1 & 2 \ -1 & 1 \end{bmatrix}\]The determinant is:\[\text{det}(J) = (1)(1) - (2)(-1) = 1 + 2 = 3\]This determinant value, 3, illustrates that the transformation scales areas by a factor of 3. Therefore, any shape will stretch, expanding its area three times compared to its original size in the \( uv \) plane. This enlargement factor is crucial in applications involving changes of coordinate systems.