How to Compute the Jacobian Matrix and Determinant

Math Problem Statement

. If then find the Jacobian .

To find the Jacobian of a transformation, let's break it into steps.

If $\mathbf{x}$ is transformed to $\mathbf{u}$ and $\mathbf{v}$ (in 2D, say $(x_1, x_2) \rightarrow (u, v)$ ), the Jacobian matrix is defined as:

\frac{\partial u}{\partial x_1} & \frac{\partial u}{\partial x_2} \\ \frac{\partial v}{\partial x_1} & \frac{\partial v}{\partial x_2} \end{bmatrix}$$ The **Jacobian determinant** is then: $$\det(J) = \left| \frac{\partial (u, v)}{\partial (x_1, x_2)} \right|$$ ### Steps: 1. Write down the transformation equations $$u(x_1, x_2)$$ and $$v(x_1, x_2)$$. 2. Compute the partial derivatives $$\frac{\partial u}{\partial x_1}$$, $$\frac{\partial u}{\partial x_2}$$, $$\frac{\partial v}{\partial x_1}$$, and $$\frac{\partial v}{\partial x_2}$$. 3. Arrange the partial derivatives in the Jacobian matrix. 4. Calculate the determinant. Please provide the transformation equations or any further details (e.g., $$u = u(x_1, x_2)$$ and $$v = v(x_1, x_2)$$), and I'll proceed to solve the Jacobian.

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Calculus
Multivariable Calculus
Jacobian Matrix

Jacobian Matrix: J = [[∂u/∂x1, ∂u/∂x2], [∂v/∂x1, ∂v/∂x2]]
Jacobian Determinant: det(J) = |∂(u, v)/∂(x1, x2)|

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