Subject: Calculus

Jacobian Matrix

The Jacobian Matrix, more commonly referred as Jacobian, is an algebraic method for determining the probability distribution of a variable y that is a function of another variable x when we know the probability distribution for x. Basically, it can be thought of as an operator of a function. Jacobian also generalizes the gradient of a function of multiple variables which itself generalizes the derivative of a scalar function.

The Jacobian is named after the German mathematician Carl Gustav Jacob Jacobi (18041851). Although the French mathematician Cauchy first used these special determinants involving partial derivatives, Jacobi developed them into a method for evaluating multiple integrals.

Now, let us a consider a set of function y = f_1, f_2, f_3, , f_n where f_1 is a function of xs and is written as f_1(x_1, x_2, x_3, , x_n), f_2(x_1, x_2, x_3, , x_n) and so on. In mathematical notation we can simply write y as

y = \begin{pmatrix} f_1{x} \\ f_2{x} \\ f_3{x} \\ : \\ f_n{x} \end{pmatrix}

or in a more detailed form

\begin{Bmatrix} y_1 & = & f_1(x_1, x_2,...,x_n)\\ : & = & : \\ y_n & = & f_n(x_1, x_2,...,x_n) \end{Bmatrix}

The Jacobian of this function can be written as,

\mathbf{J}(x_1, x_2,...,x_n) = \begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \cdot \cdot \cdot & \frac{\partial y_1}{\partial x_n} \\ : & \cdot \cdot & : \\ \frac{\partial y_n}{\partial x_1} & \cdot \cdot \cdot & \frac{\partial y_n}{\partial x_n} \end{bmatrix}

The concept of Jacobian can be extended to two or more functions. If F(u, v) and G(u, v) are differentiable in a region, the Jacobian of F and G with respect to u and v is defined by

\mathbf{J}(u,v) = \frac{\partial (F, G)}{\partial(u, v)}= \begin{pmatrix} \frac{\partial F}{\partial u} & \frac{\partial F }{\partial v} \\ \frac{\partial G }{\partial u} & \frac{\partial G }{\partial v} \end{pmatrix}

Similarly, if we have three functions, the Jacobian can be described as,

\mathbf{J}(u,v, w) = \frac{\partial (F, G, H)}{\partial(u, v, w)}= \begin{bmatrix} \frac{\partial F}{\partial u} & \frac{\partial F }{\partial v} & \frac{\partial F }{\partial w} \\ \frac{\partial G }{\partial u} & \frac{\partial G }{\partial v} & \frac{\partial G }{\partial w} \\ \frac{\partial h }{\partial u} & \frac{\partial H }{\partial v} & \frac{\partial H }{\partial w}\end{bmatrix}

From here on, the extensions of Jacobian can be made easily.

Partial Derivatives Using Jacobians

Jacobians often prove useful in obtaining partial derivatives of implicit functions. Thus, for example, given the simultaneous equations

F(x, y, u, v) = 0 and G(x, y, u,v)=0

In general, consider u and v as functions of x and y. In this case, we have

\frac{\partial u}{\partial x}= -\frac{\frac{\partial (F, G)}{\partial (x, v)}}{\frac{\partial (F, G)}{\partial (u, v)}}
\frac{\partial u}{\partial y}= -\frac{\frac{\partial (F, G)}{\partial (y, v)}}{\frac{\partial (F, G)}{\partial (u, v)}}
\frac{\partial v}{\partial x}= \frac{\frac{\partial (F, G)}{\partial (u, x)}}{\frac{\partial (F, G)}{\partial (u, v)}}
\frac{\partial v}{\partial y}= \frac{\frac{\partial (F, G)}{\partial (u, y)}}{\frac{\partial (F, G)}{\partial (u, v)}}

The ideas are easily extended. Thus, if we consider the simultaneous equations

F(x, y, u, v)=0, G(x, y, u, v)=0, H(x, y, u, v)=0

For example, consider u, v and w as functions of x and y. In this case

\frac{\partial u}{\partial x}= -\frac{\frac{\partial (F, G, H)}{\partial (x, v, w)}}{\frac{\partial (F, G, H)}{\partial (u, v, w)}}, \frac{\partial u}{\partial x}= -\frac{\frac{\partial (F, G, H)}{\partial (u, v, y)}}{\frac{\partial (F, G, H)}{\partial (u, v, w)}}

with similar results for the remaining partial derivatives.

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