Math Problem Statement
Find the Jacobian of uxyz, vxy + yz + zx, w=x+y+z
Solution
To compute the Jacobian matrix of the given functions:
with respect to the variables , we calculate the partial derivatives of each function with respect to and .
The Jacobian matrix is defined as:
\begin{bmatrix} \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{bmatrix}.$$ ### Step-by-Step Calculations: 1. **Partial derivatives of $$u = xyz$$:** - $$\frac{\partial u}{\partial x} = yz$$ - $$\frac{\partial u}{\partial y} = xz$$ - $$\frac{\partial u}{\partial z} = xy$$ 2. **Partial derivatives of $$v = xy + yz + zx$$:** - $$\frac{\partial v}{\partial x} = y + z$$ - $$\frac{\partial v}{\partial y} = x + z$$ - $$\frac{\partial v}{\partial z} = x + y$$ 3. **Partial derivatives of $$w = x + y + z$$:** - $$\frac{\partial w}{\partial x} = 1$$ - $$\frac{\partial w}{\partial y} = 1$$ - $$\frac{\partial w}{\partial z} = 1$$ ### Forming the Jacobian matrix: $$J = \begin{bmatrix} yz & xz & xy \\ y+z & x+z & x+y \\ 1 & 1 & 1 \end{bmatrix}.$$ This is the Jacobian matrix for the given functions. --- Do you have any questions, or would you like further explanation? ### Follow-up Questions: 1. How is the Jacobian used in real-world applications, like in transformations or optimization problems? 2. What happens if one of the functions depends on other parameters beyond $$x, y, z$$? 3. Could you derive the determinant of the Jacobian, and what would its significance be in this case? 4. How do the functions' nonlinearities affect the Jacobian interpretation? 5. Can this Jacobian matrix be used to analyze the behavior of the system near a specific point? ### Tip: When interpreting the Jacobian, remember that it captures the rate of change of each output variable with respect to each input variable, making it critical in multivariate calculus applications like linearization and change-of-variable techniques in integrals.Ask a new question for Free
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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]]
Theorems
Chain Rule in Multivariable Calculus
Suitable Grade Level
Undergraduate - Calculus II or III