. If u = x+y x-y = tan x + tan' y then find the Jacobian a(u,v) d(x,y)

To compute the Jacobian of \((u, v)\) with respect to \((x, y)\), we need to follow these steps:

### Step 1: Definitions
We have:
\[
u = x + y
\]
\[
v = x - y
\]
The Jacobian matrix is given by:
\[
J = \begin{bmatrix}
\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\
\frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}
\end{bmatrix}
\]

### 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.

If u = x + y, v = x - y, find the Jacobian ∂(u,v)/∂(x,y).

Learn how to compute the Jacobian determinant for u = x + y and v = x - y with detailed steps. This guide explains partial derivatives, matrix formulation, and determinant calculation, ideal for Calculus students.

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