Vector fields
Definition A vector field \(\mathbf{v}\) on the space \(\RR^n\) is an assignment of a tangent vector \(\mathbf{v}_a\in T_a(\RR^n)\) to each point \(a\in\RR^n\). Since each tangent space \(T_a(\RR^n)\) has a coordinate basis \({\partial/\partial x^i|_a}\), at each point \(a\) we can write
\begin{equation}
\mathbf{v}_a=\sum_{i=1}^nv^i(a)\left.\frac{\partial}{\partial x^i}\right|_a
\end{equation}
or,
\begin{equation}
\mathbf{v}=\sum_{i=1}^nv^i\frac{\partial}{\partial x^i}
\end{equation}
where the \(v^i\) are functions \(v^i:\RR^n\mapto\RR\). The vector field is said to be smooth if the functions \(v^i\) are smooth.
Example On the space \(\RR^2-{0}\) we can visualise the vector field defined by,
\begin{equation}
\mathbf{v}=\frac{-y}{\sqrt{x^2+y^2}}\frac{\partial}{\partial x}+\frac{x}{\sqrt{x^2+y^2}}\frac{\partial}{\partial y},
\end{equation}
as,
Example On the space \(\RR^2\) the vector field defined as
\begin{equation}
\mathbf{v}=x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}
\end{equation}
can be visualised as
The vector fields on \(\RR^n\) and the derivations of the (algebra of) smooth functions on \(\RR^n\) are isomorphic as vector spaces. Note that a derivation, \(X\), of \(C^\infty(\RR^n)\) is a linear map \(X:C^\infty(\RR^n)\mapto C^\infty(\RR^n)\) such that the Leibniz rule,
\begin{equation}
X(fg)=(Xf)g+f(Xg)
\end{equation}
is satisfied for all \(f,g\in C^\infty(\RR^n)\).
Vector fields and ODEs — integral curves
Consider a fluid in motion such that its “flow” is independent of time. The path of a single particle would trace out a path in space — a curve, \(\gamma(t)\) say, parameterised by time. The velocity of such a particle, say at \(\gamma(0)\), is the tangent vector \(d\gamma(t)/dt|_0\). The “flow” of the whole system could be modelled by a 1-parameter family of maps \(\phi_t:\RR^3\mapto\RR^3\) such that \(\phi_t(a)\) is the location of a particle a time \(t\) after it was located at the point \(a\), in other words, \(\phi_t(a)\) is the curve \(\gamma\) such that \(\gamma(0)=a\) and \(\gamma(t)=\phi_t(a)\). Since the flow is stationary we have that
\begin{equation}
\phi_{s+t}(a)=\phi_s(\phi_t(a))=\phi_t(\phi_s(a)).
\end{equation}
Also,
\begin{equation}
\phi_{-t}(\phi_t(a))=a,
\end{equation}
where we understand \(\phi_{-t}(a)\) to mean the location of a particle a time \(t\) before it was at \(a\). So, understanding \(\phi_0\) to be the identity map and \(\phi_{-t}=\phi_t^{-1}\) we have a 1-paramter group of maps, which, assuming they are smooth, are each diffeomorphisms, \(\phi_t:\RR^3\mapto\RR^3\), collectively called a flow. Given such a flow we obtain a velocity (vector) field as
\begin{equation}
\mathbf{v}_a=\left.\frac{d\phi_t(a)}{dt}\right|_0.
\end{equation}
An individual curve \(\gamma\) in the flow is then called an integral curve through \(a\) of the vector field \(\mathbf{v}\). All this generalises to \(n\)-dimensions. A flow \(\phi_t:\RR^n\mapto\RR^n\) gives rise to a (velocity) vector field on \(\RR^n\). Conversely, suppose we have some vector field, \(\mathbf{v}\), then we can wonder about the existence of integral curves through the points of our space having the vectors of the vector field as tangents. Such an integral curve would have to satisfy,
\begin{equation}
\mathbf{v}_{\gamma(0)}(f)=\left.\frac{df(\gamma(t))}{dt}\right|_0
\end{equation}
for any function, \(f\), so that considering in turn the coordinate functions, \(x^i\), we have a system of differential equations,
\begin{equation}
\frac{dx^i(t)}{dt}=v^i(x^1(t),\dots,x^n(t))
\end{equation}
where the \(v^i\) are the components of the vector field and \(x^i(t)=x^i(\gamma(t))\). The theorem on the existence and uniqueness of the solution of this system of equations and hence of the corresponding integral curves and flow is the following.
Theorem If \(\mathbf{v}\) is a smooth vector field defined on \(\RR^n\) then for each point \(a\in\RR^n\) there is a curve \(\gamma:I\mapto\RR^n\) (\(I\) an open interval in \(\RR\) containing 0) such that \(\gamma(0)=a\) and
\begin{equation}
\frac{d\gamma(t)}{dt}=\mathbf{v}_{\gamma(t)}
\end{equation}
for all \(t\in I\) and any two such curves are equal on the intersection of their domains. Furthermore, there is a neighbourhood \(U_a\) of \(a\) and an interval \(I_\epsilon=(-\epsilon,\epsilon)\) such that for all \(t\in I_\epsilon\) and \(b\in U_a\) there is a curve \(t\mapsto\phi_t(b)\) satisfying
\begin{equation}
\frac{d\phi_t(b)}{dt}=\mathbf{v}_{\phi_t(b)}
\end{equation}
which is a flow on \(U_a\) — a local flow.
Linear vector fields on \(\RR^n\)
Suppose we have a linear transformation \(X\) of the vector space \(\RR^n\). Then to any point \(a\in\RR^n\) we can associate an element \(Xa\) which we can understand as a vector in \(T_a(\RR^n)\). The previous theorem tells us that at any point \(a\) we can find a solution to the system of differential equations,
\begin{equation}
\frac{d\gamma(t)}{dt}=X(\gamma(t)),
\end{equation}
valid in some open interval around 0 with \(\gamma(0)=a\). Let’s construct this solution explicitly. We seek a power series solution
\begin{equation}
\gamma(t)=\sum_{k=0}^\infty t^ka_k
\end{equation}
such that \(a_0=a\) and where we understand \(\gamma(t)=(x^1(\gamma(t)),\dots,x^n(\gamma(t)))\) and \(a_k=(x^1(a_k),\dots,x^n(a_k))\). Plugging the power series into the differential equation we obtain,
\begin{equation}
\sum_{k=1}^\infty kt^{k-1}a_k=\sum_{k=0}^\infty t^kXa_k,
\end{equation}
from which we obtain the recurrence relation,
\begin{equation}
a_{k+1}=\frac{1}{k+1}Xa_k,
\end{equation}
which itself leads to,
\begin{equation}
a_k=\frac{1}{k!}X^k(a_0)=\frac{1}{k!}X^ka,
\end{equation}
so that,
\begin{equation}
\gamma(t)=\sum_{k=0}^\infty\frac{t^kX^k}{k!}a=\exp(tX)a,
\end{equation}
where we’ve introduced the matrix exponential which, as we’ve already mentioned, converges for any matrix \(X\). It’s not difficult to show that this solution is unique. In this case the flow defined by \(\phi_t=\exp(tX)\) is defined on the whole of \(\RR^n\) for all times \(t\).