Question

If \(u\) and \(v\) are functions of two independent variables \(x\) and \(y\), then the determinant \[ \begin{vmatrix} \dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\ \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y} \end{vmatrix} \] is called:

• A. Jacobian of $x$ and $y$ with respect to $u$ and $v$
• B. Jacobian of $u$ and $x$ with respect to $v$ and $y$
• C. Jacobian of $u$ and $v$ with respect to $x$ and $y$
• D. Jacobian of $u$ and $y$ with respect to $v$ and $x$

Hint:

The numerator in the notation $\frac{\partial(u, v)}{\partial(x, y)}$ represents the output functions (rows), and the denominator represents the independent input variables (columns).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The Jacobian determinant is a functional determinant that represents the local scale factor of a multi-variable transformation.

Step 2: Detailed Explanation:
Let $u$ and $v$ be differentiable functions of $x$ and $y$:
\[ u = u(x, y), \quad v = v(x, y) \]
The matrix of all first-order partial derivatives of these functions is the Jacobian matrix.
The determinant of this matrix is the Jacobian of $u$ and $v$ with respect to $x$ and $y$, denoted as:
\[ J\left(\frac{u, v}{x, y}\right) = \frac{\partial(u, v)}{\partial(x, y)} = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix} \]
Therefore, this determinant is called the Jacobian of $u$ and $v$ with respect to $x$ and $y$.

Step 3: Final Answer
The correct option is (C).