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Also, in algebraic geometry the Jacobian of a curve means the Jacobian variety: a group structure, which can be imposed on the curve.
They are all named after the mathematician Carl Gustav Jacobi; the term "Jacobian" may be pronounced as "yah-KO-bee-un".
The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function. Its importance lies in the fact that it represents the best linear approximation to a differentiable function near a given point. In this sense, the Jacobian is akin to a derivative of a multivariate function.
Suppose F : Rn → Rm is a function from Euclidean n-space to Euclidean m-space. Such a function is given by m real-valued component functions, y1(x1,...,xn), ..., ym(x1,...,xn). The partial derivatives of all these functions (if they exist) can be organized in an m-by-n matrix, the Jacobian matrix of F, as follows:
This matrix is denoted by
The i-th row of this matrix is given by the gradient of the function yi for i=1,...,m.
If p is a point in Rn and F is differentiable at p, then its derivative is given by JF(p) (and this is the easiest way to compute said derivative). 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.
The Jacobian matrix of the function F : R3 → R4 with components:
is:
If m = n, then F is a function from n-space to n-space and the Jacobi matrix is a square matrix. We can then form its determinant, known as the Jacobian determinant.
The Jacobian determinant at a given point gives important information about the behavior of F near that point. For instance, the continuously differentiable function F is invertible near p if and only if the Jacobian determinant at p is non-zero. This is the inverse function theoremIn mathematics, the inverse function theorem gives sufficient conditions for a vector-valued function to be invertible on an open region containing a point in its domain. The theorem states that if at a point P a function f R n → R n has a Jacobian d. Furthermore, if the Jacobian determinant at p is positive, then F preserves orientation near p; if it is negative, F reverses orientation. The absolute valueIn mathematics, the absolute value (or modulus of a number is that number without a negative sign. So, for example, 3 is the absolute value of both 3 and −3. Definition It can be defined as follows: For any real number a the absolute value of a deno of the Jacobian determinant at p gives us the factor by which the function F expands or shrinks volumes near p; this is why it occurs in the general substitution ruleIn calculus, the substitution rule is an important tool for finding antiderivatives and integrals. It is the counterpart to the chain rule of differentiation. Suppose f ''x is an integrable function, and φ t is a continuously differentiable function w.