Math Problem Statement

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

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).

Solution

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

Given: u=x1−r2,v=y1−r2,w=z1−r2,u = \frac{x}{\sqrt{1 - r^2}}, \quad v = \frac{y}{\sqrt{1 - r^2}}, \quad w = \frac{z}{\sqrt{1 - r^2}}, where r2=x2+y2+z2r^2 = x^2 + y^2 + z^2, show that the Jacobian determinant: ∂(x,y,z)∂(u,v,w)=1(1−r2)5.\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)(x, y, z) to (u,v,w)(u, v, w) is given by:

\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\

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Jacobian Determinant
Coordinate Transformation

Formulas

Jacobian Matrix: J = ∂(x, y, z) / ∂(u, v, w)
r^2 = x^2 + y^2 + z^2
Transformation formulas: u = x / sqrt(1 - r^2), v = y / sqrt(1 - r^2), w = z / sqrt(1 - r^2)

Theorems

Jacobian Determinant

Suitable Grade Level

College/University Level

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