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 (1804�1851). 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 x�s 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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