What is Jacobian of change of variables?

Question: What is Jacobian of change of variables?

Answer:

In multivariable calculus, the Jacobian of a change of variables is a determinant that represents the local scaling factor of a transformation from one coordinate system to another. It's essential for evaluating multiple integrals after a change of variables.

Here's a breakdown of key concepts:

Change of Variables:

    It involves switching from one set of variables (e.g., x and y) to a new set (e.g., u and v) to simplify a problem.
    This often involves functions like x = g(u, v) and y = h(u, v).

Jacobian Matrix:

    A matrix containing the partial derivatives of the new variables with respect to the original ones: 
| ∂x/∂u ∂x/∂v | 
| ∂y/∂u ∂y/∂v |

Jacobian Determinant:

    The determinant of this matrix, denoted as J(u, v) or ∂(x, y)/∂(u, v).

Role in Integrals:

    When changing variables in a double integral, the Jacobian is used to adjust the area element: 
    ∬f(x, y) dx dy = ∬f(g(u, v), h(u, v)) |J(u, v)| du dv
    It ensures the integral's value remains consistent after the transformation.

Geometric Interpretation:

    The Jacobian's absolute value represents the factor by which a small area element in the u-v plane transforms into a corresponding area element in the x-y plane.

Key Points:

    The Jacobian is a 2x2 determinant for 2D transformations, 3x3 for 3D, and so on.
    If the Jacobian is zero at a point, the transformation might distort or fold space, potentially leading to issues in integration.

Example:

    For polar coordinates (x = r cosθ, y = r sinθ), the Jacobian is r. This means a small square in the r-θ plane transforms into a corresponding sector in the x-y plane with area r times larger.