I am a math student and now studying Calculus 4, I came across the subject of conservative vector field today.

The definition is that

A vector field F is called a conservative vector field if it is the gradient of some scalar function, that is, if there exists a function f such that $\mathbf{F}= ∇ f$ . In this situation f is called a potential function for F.

This just made me think of the association from what I learned from Linear Algebra. So, please give me some direction to start digging deeper to understand this concept. If there exists a connection.

  • $\begingroup$ What is it exactly about the definition of a conservative vector field that reminds you of eigenvalues and eigenvectors? $\endgroup$ – EuYu Mar 20 '17 at 15:52
  • $\begingroup$ It's really not clear to me what you have in mind, exactly. The gradient map $\nabla$ is linear over $\Bbb R$, but its domain (say, the space of differentiable functions on $\Bbb R^n$) and codomain (say, the space of vector fields on $\Bbb R^n$) are different, so there is no notion of eigenvector available. $\endgroup$ – Travis Mar 20 '17 at 15:57

One area of tangential relation that comes to mind, relating gradients, vector fields, potentials, and linear algebra, is non-linear dynamics. (Perhaps you're already familiar with it).

For instance, consider the following system: $$ \frac{\partial \vec{x}}{\partial t} = f(\vec{x},t) $$ where $\vec{x}(t)\in\mathbb{R}^n$ and $f(\vec{x},t)=(f_1(\vec{x},t),\ldots,f_n(\vec{x},t))$. The Jacobian is then: $$ J_p(t) = \begin{bmatrix}\nabla f_1\\\vdots\\\nabla f_n\end{bmatrix}_{\vec{x}=p} $$ where the gradients are treated as row vectors. The Jacobian matrix allows one to compute the stability of fixed points. (Also here). The eigenvalues and eigenvectors essentially determine the behaviour of the system around those points (e.g. see here).

Here, again, we find the notion of conservative systems. For instance, if $V$ is a potential function, the system: $$ \frac{\partial}{\partial t}x(t)=−\frac{\partial V}{\partial x} $$ is a conservative one. (Notice the analogy to your example: $f(\vec{x},t)=-\nabla V$.) See Strogatz's book on non-linear dynamics and chaos. One example from there is $$ m\frac{\partial^2}{\partial t^2}x(t) = -\frac{\partial V}{\partial x} $$ for which one can show that the total energy $$ E=\frac{1}{2}m\left(\frac{\partial x}{\partial t}\right)^2 + V(x) $$ is always conserved over time. (Notice term 1 is kinetic energy and term 2 is potential energy.) As before, in such systems, the stability behaviour of the system is determined by the spectrum of the Jacobian!


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