5 Minute Math Lesson: What is a Jacobian and what is it good for?

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In Tensor Calculus, a Jacobian is a matrix defined asJ_{mn} = \frac{dy^m}{dx^n}  where the y^i 's are a new coordinate system defined in terms of the original coordinate system, the x^i 's. (Note that we are using subscripts, x^2 denotes the second element of the \mathbf{x} vector.) The determinant of this matrix J=det([J_{mn}]) is important because this represents the constant of proportionality between volumes in the old coordinates and in the new coordinates. As a simple example polar coordinates is the transformation  (x,y)= (r \cos {\theta} , r \sin {\theta} ) . When we want to perform an integration in cartesian coordinates by transforming into polar coordinates, we write \int f(x,y) dx dy = \int f(r,\theta) r dr d\theta  , the factor of r is precisely the Jacobian. So in general, to convert an integral in the x^i coordinate system to the y^i coordinate system we have \int f(\mathbf{x}) d\mathbf{x} = \int    f(\mathbf{y}) J(\mathbf{y}) d\mathbf{y}  where \mathbf{x}=(x^1,x^2,\cdots,x^n)  and \mathbf{y}=(y^1,y^2,\cdots,y^n) .

In differential equation theory, the Jacobian matrix plays a key role in defining the stability of solutions. As a simple example, consider the matrix ordinary differential equation \dot{\mathbf{x}} = A \mathbf{x}  where A = \left(\begin{array}{cc}a&b\\c&d\end{array} 
ight) . Because this is a linear system, the solution will always be a linear combination of exponentials \mathbf{x} = \mathbf{v_1} e^{\lambda_1 t} + \mathbf{v_2} e^{\lambda_2 t } where \mathbf{v_{1,2}} are the eigenvectors and \lambda_{1,2} are the eigenvalues, and t is time, which is positive. Since t > 0 , we must have  Re(\lambda_{1,2}) < 0 for the solution to decay to a steady state. If this is violated for any \lambda_i then that exponential "blows up" as t 
ightarrow \infty , since it is an exponential with an ever-increasing postive argument. Note that if the eigenvalues are complex then the imaginary part of the eigenvalue is related to the oscillatory part of the solution.

A  \lambda_i  would be called an unstable eigenvalue of the system and also forms part of the vector space called the unstable manifold. Instability is defined as the tendency for a system to shoot away from a certain state when it is slightly disturbed (perturbed) from that state. Stability is the feature of a system to come back to a certain state if it is slightly perturbed from that state.

Stability has important consequences for applications, because it can determine if your chemical/combustion reaction will go out of control, of if the species in your mathematical model go extinct, etc, depending on what the differential equations describe. Whatever your equations describe, knowing if the solutions are stable is pretty important and usually the first step of an analysis. If you have stable solutions, then you can reasonably trust numerics but if you are trying to numerically simulate an unstable solution of an equation, you must be much more careful. In these situations you want to use specialized integration algorithms that preserve certain properties of your solutions, like if it is symplectic. More about this later!

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This page contains a single entry by Jonathan Leto published on November 30, 2008 7:02 PM.

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