https://file.aiquickdraw.com/mathbot/file/1726385330663b5oxx2s4.jpg

Solve 2(b) . Solve it full and do the proof step by step 

Let's solve the question 2(b), which asks to prove the following:

Given:
\[
u = \frac{x}{\sqrt{1 - r^2}}, \quad v = \frac{y}{\sqrt{1 - r^2}}, \quad w = \frac{z}{\sqrt{1 - r^2}},
\]
where \( r^2 = x^2 + y^2 + z^2 \), show that the Jacobian determinant:
\[
\frac{\partial(x, y, z)}{\partial(u, v, w)} = \frac{1}{\sqrt{(1 - r^2)^5}}.
\]

### Step-by-Step Solution:

#### Step 1: Jacobian Matrix
The Jacobian matrix for the transformation from \((x, y, z)\) to \((u, v, w)\) is given by:
\[
J = \frac{\partial(x, y, z)}{\partial(u, v, w)} =
\begin{bmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\
\frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}
\end{bmatrix}.
\]
We need to find each of these partial derivatives.

#### Step 2: Express \( x, y, z \) in terms of \( u, v, w \)

Given the relations:
\[
u = \frac{x}{\sqrt{1 - r^2}}, \quad v = \frac{y}{\sqrt{1 - r^2}}, \quad w = \frac{z}{\sqrt{1 - r^2}},
\]
we can invert these equations to express \(x\), \(y\), and \(z\) in terms of \(u\), \(v\), and \(w\). Solving for \(x\

Solve 2(b) from the given image: Given u = x / sqrt(1 - r^2), v = y / sqrt(1 - r^2), w = z / sqrt(1 - r^2) where r^2 = x^2 + y^2 + z^2, show that the Jacobian determinant ∂(x, y, z) / ∂(u, v, w) = 1 / sqrt((1 - r^2)^5).

This page provides a detailed solution to a Jacobian determinant problem involving a transformation of coordinates. It shows how to compute the Jacobian for u = x / sqrt(1 - r^2), v = y / sqrt(1 - r^2), w = z / sqrt(1 - r^2), with r^2 = x^2 + y^2 + z^2. Step-by-step instructions and formulas are included, suitable for university-level calculus students.

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