Jacobian Matrix and Determinant
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High Quality Content by WIKIPEDIA articles! The Jacobian of a function describes the orientation of a tangent plane to the function at a given point. In this way, the Jacobian generalizes the gradient of a scalar valued function of multiple variables which itself generalizes the derivative of a scalar-valued function of a scalar. Likewise, the Jacobian can also be thought of as describing the amount of "stretching" that a transformation imposes. For example, if (x2,y2) = f(x1,y1) is used to transform an image, the Jacobian of f, J(x1,y1) describes how much the image in the neighborhood of (x1,y1) is stretched in the x, y, and xy directions.…mehr

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High Quality Content by WIKIPEDIA articles! The Jacobian of a function describes the orientation of a tangent plane to the function at a given point. In this way, the Jacobian generalizes the gradient of a scalar valued function of multiple variables which itself generalizes the derivative of a scalar-valued function of a scalar. Likewise, the Jacobian can also be thought of as describing the amount of "stretching" that a transformation imposes. For example, if (x2,y2) = f(x1,y1) is used to transform an image, the Jacobian of f, J(x1,y1) describes how much the image in the neighborhood of (x1,y1) is stretched in the x, y, and xy directions.