Jacobian Matrix for Multivariable System of Equations

Math Problem Statement

Find the Jacobian matrix for the given system of equations: u = x + y + z, u^2v = y + z, u^3w = z.

Solution

The problem asks to find the Jacobian matrix $J \left( \frac{u, v, w}{x, y, z} \right)$ for the system of equations provided. The Jacobian matrix represents the partial derivatives of the functions $u$ , $v$ , and $w$ with respect to $x$ , $y$ , and $z$ .

Given equations:

$u = x + y + z$
$u^2 v = y + z$
$u^3 w = z$

Step-by-step approach to solving for the Jacobian:

The Jacobian matrix $J$ is a $3 \times 3$ matrix of the partial derivatives of $u$ , $v$ , and $w$ with respect to $x$ , $y$ , and $z$ . The general form of the Jacobian is:

\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} & \frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} & \frac{\partial w}{\partial z} \end{pmatrix}$$ 1. **Partial derivatives for $$u = x + y + z$$:** $$\frac{\partial u}{\partial x} = 1, \quad \frac{\partial u}{\partial y} = 1, \quad \frac{\partial u}{\partial z} = 1$$ 2. **For the equation $$u^2 v = y + z$$:** Differentiate this equation to find partial derivatives of $$v$$. First, rearrange for $$v$$: $$v = \frac{y + z}{u^2}$$ Now compute the partial derivatives: $$\frac{\partial v}{\partial x} = -\frac{2(y + z)}{u^3}, \quad \frac{\partial v}{\partial y} = \frac{1}{u^2}, \quad \frac{\partial v}{\partial z} = \frac{1}{u^2}$$ 3. **For the equation $$u^3 w = z$$:** Rearrange for $$w$$: $$w = \frac{z}{u^3}$$ Now compute the partial derivatives: $$\frac{\partial w}{\partial x} = -\frac{3z}{u^4}, \quad \frac{\partial w}{\partial y} = -\frac{3z}{u^4}, \quad \frac{\partial w}{\partial z} = \frac{1}{u^3}$$ ### Jacobian Matrix: $$J = \begin{pmatrix} 1 & 1 & 1 \\ -\frac{2(y + z)}{u^3} & \frac{1}{u^2} & \frac{1}{u^2} \\ -\frac{3z}{u^4} & -\frac{3z}{u^4} & \frac{1}{u^3} \end{pmatrix}$$ This is the Jacobian matrix for the system of equations. Would you like further clarification on any step? Here are some related questions for better understanding: 1. What is the significance of the Jacobian matrix in multivariable calculus? 2. How can the Jacobian be used to transform coordinates in a multivariable system? 3. Can you explain how the chain rule is applied when deriving Jacobians? 4. What are the conditions for a Jacobian to be invertible? 5. How is the Jacobian used in optimization problems? **Tip**: The Jacobian is often used in nonlinear systems to approximate small changes in multivariable functions, so it's very useful in numerical methods and differential equations.

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Jacobian Matrix
Partial Derivatives

Formulas

Jacobian matrix: J = [∂u/∂x ∂u/∂y ∂u/∂z; ∂v/∂x ∂v/∂y ∂v/∂z; ∂w/∂x ∂w/∂y ∂w/∂z]
u = x + y + z
v = (y + z) / u^2
w = z / u^3

Theorems

Jacobian Determinant

Suitable Grade Level

University-level Calculus

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