Jacobian Matrix
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 is used to transform an image, the Jacobian of, describes how much the image in the neighborhood of is stretched in the x and y directions.
If a function is differentiable at a point, its derivative is given in coordinates by the Jacobian, but a function doesn't need to be differentiable for the Jacobian to be defined, since only the partial derivatives are required to exist.
The importance of the Jacobian lies in the fact that it is a factor in one term of the best linear approximation to a differentiable function near a given point. In this sense, the Jacobian is the derivative of a multivariate function.
If p is a point in Rn and F is differentiable at p, then its derivative is given by JF(p). In this case, the linear map described by JF(p) is the best linear approximation of F near the point p, in the sense that
for x close to p and where o is the little o-notation (for ) and is the distance between x and p.
Compare this to a Taylor series for a scalar function of a scalar argument, truncated to first order:
In a sense, both the gradient and Jacobian are "first derivatives" — the former the first derivative of a scalar function of several variables, the latter the first derivative of a vector function of several variables. In general, the gradient can be regarded as a special version of the Jacobian: it is the Jacobian of a scalar function of several variables.
The Jacobian of the gradient has a special name: the Hessian matrix, which in a sense is the "second derivative" of the scalar function of several variables in question.
Read more about this topic: Jacobian Matrix And Determinant
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