Math Problem Statement
. If u = x+y x-y = tan x + tan' y then find the Jacobian a(u,v) d(x,y)
Solution
To compute the Jacobian of with respect to , we need to follow these steps:
Step 1: Definitions
We have: The Jacobian matrix is given by:
\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix}$$ ### Step 2: Partial Derivatives From the equations for $$u$$ and $$v$$: 1. $$\frac{\partial u}{\partial x} = 1$$, $$\frac{\partial u}{\partial y} = 1$$ 2. $$\frac{\partial v}{\partial x} = 1$$, $$\frac{\partial v}{\partial y} = -1$$ Thus, the Jacobian matrix becomes: $$J = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$$ ### Step 3: Determinant of the Jacobian Matrix The Jacobian determinant is: $$\text{Jacobian} = \det(J) = (1)(-1) - (1)(1) = -1 - 1 = -2$$ ### Final Answer: $$\boxed{-2}$$ Would you like a deeper explanation or any clarifications? Here are related questions for further exploration: 1. How are Jacobians used in coordinate transformations? 2. What is the geometric interpretation of a Jacobian determinant? 3. Can Jacobians be applied in polar to Cartesian transformations? 4. How do we compute higher-dimensional Jacobians? 5. What happens if the Jacobian determinant is zero? **Tip:** When working with Jacobians, carefully compute partial derivatives and verify the determinant calculation for accuracy.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Jacobian
Partial Derivatives
Linear Algebra
Formulas
Jacobian matrix formula: J = [[∂u/∂x, ∂u/∂y], [∂v/∂x, ∂v/∂y]]
Determinant of a 2x2 matrix: det(J) = ad - bc
Theorems
Jacobian Determinant for Coordinate Transformation
Suitable Grade Level
Undergraduate Mathematics (Calculus I/II)