Question

Let \((x, y)\) denote the coordinates in a rectangular Cartesian coordinate system \( C \). Let \((x', y')\) denote the coordinates in another coordinate system \( C' \) defined by
\[ x' = 2x + 3y, \quad y' = -3x + 4y \] The area element in \( C' \) is:

• A. \( \frac{1}{17} dx'dy' \)
• B. \( 12dx'dy' \)
• C. \( dx'dy' \)
• D. \( x'dx'dy' \)

Hint:

When transforming coordinates, use the Jacobian determinant to find how area or volume elements scale between coordinate systems.

Answer 1

The Correct Option is A

Step 1: Find the Jacobian of transformation. 
The area element transforms as \[ dx\,dy = |J|\,dx'dy', \] where \[ J = \frac{\partial(x, y)}{\partial(x', y')} = \frac{1}{\frac{\partial(x', y')}{\partial(x, y)}}. \] Step 2: Compute the determinant. 

Step 3: Therefore, 
\[ dx\,dy = \frac{1}{17}dx'dy'. \] Step 4: Final Answer. 
Hence, the area element in \( C' \) is \( \frac{1}{17}dx'dy' \).