Given that $u$ and $v$ are functions of $r$ and $s$, and $r$ and $s$ are themselves functions of $x$ and $y$, we are asked to find the Jacobian matrix $J(u, v / x, y)$, which represents the determinant of the matrix of partial derivatives.

Step 1: Recognize the structure

You have:

• $u = u(r(s(x, y), t(x, y)))$
• $v = v(r(s(x, y), t(x, y)))$

Now we will calculate the Jacobian matrix using the chain rule.

Step 2: Jacobian Matrix Expression

The Jacobian $J(u, v / x, y)$ can be written as:

$J(u, v / x, y) = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix}$

Using the multivariable chain rule:

• For $\frac{\partial u}{\partial x}$:

  $\frac{\partial u}{\partial x} = \frac{\partial u}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial u}{\partial s} \frac{\partial s}{\partial x}$

• For $\frac{\partial u}{\partial y}$:

  $\frac{\partial u}{\partial y} = \frac{\partial u}{\partial r} \frac{\partial r}{\partial y} + \frac{\partial u}{\partial s} \frac{\partial s}{\partial y}$

• Similarly for $\frac{\partial v}{\partial x}$:

  $\frac{\partial v}{\partial x} = \frac{\partial v}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial v}{\partial s} \frac{\partial s}{\partial x}$

• And for $\frac{\partial v}{\partial y}$:

  $\frac{\partial v}{\partial y} = \frac{\partial v}{\partial r} \frac{\partial r}{\partial y} + \frac{\partial v}{\partial s} \frac{\partial s}{\partial y}$

Step 3: Jacobian Determinant

Now, the determinant of the Jacobian matrix will be:

$\text{Det} = \begin{vmatrix} \frac{\partial u}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial u}{\partial s} \frac{\partial s}{\partial x} & \frac{\partial u}{\partial r} \frac{\partial r}{\partial y} + \frac{\partial u}{\partial s} \frac{\partial s}{\partial y} \\ \frac{\partial v}{\partial r} \frac{\partial r}{\partial x} + \frac{\partial v}{\partial s} \frac{\partial s}{\partial x} & \frac{\partial v}{\partial r} \frac{\partial r}{\partial y} + \frac{\partial v}{\partial s} \frac{\partial s}{\partial y} \end{vmatrix}$

This determinant, known as the Jacobian of $u$ and $v$ with respect to $x$ and $y$, encapsulates how $u$ and $v$ change with respect to changes in $x$ and $y$.

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Would you like a more detailed breakdown of any specific part, or further examples?

Here are five related questions:

1. How do you compute the Jacobian for more complex transformations between variables?
2. What is the physical significance of the Jacobian determinant in a multivariable context?
3. How does the Jacobian relate to inverse transformations?
4. What happens if the Jacobian determinant is zero?
5. How can you use the Jacobian to solve for transformations in multivariable calculus?

Tip: The Jacobian determinant is key in changing variables in integrals, especially in multivariable integration, as it adjusts for the scaling or distortion of areas or volumes during the transformation.