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 (u,v)(u, v) with respect to (x,y)(x, y), we need to follow these steps:

Step 1: Definitions

We have: u=x+yu = x + y v=xyv = x - y 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.

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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)