The second differential

Posted on 30 January, 2015 by jesperg

Let M be a two-dimensional smooth manifold with local coordinates x,y. For a real-valued function f(x,y) on this manifold we have the first differential

df = f_xdx+f_ydy.

The second differential d^2f is computed by applying d to df, and using Leibniz’s rule we get

d^2f=f_{xx}dx^2+2f_{xy}dxdy+f_{yy}dy^2+f_xd^2x+f_yd^2y

This is a form on the second order tangent bundle T^2M=TTM, so while dx and dy are evaluated on some tangent vector a\partial_x+b\partial_y, the forms d^2x and d^2y are evaluated on a second order tangent vector u\partial_{dx}+v\partial_{dy}.

Equivalently we may think of dx and dy as coordinates on the tangent plane at (x,y) and d^2x and d^2y as coordinates on the second order tangent plane at (x,y,dx,dy). To be able to evaluate d^2f at some point means that we must specify not only a velocity (dx,dy) but also an acceleration (d^2x,d^2y).

One of the ways to define tangent vectors at a point is to define them as equivalence classes of curves passing through this point and agreeing up to first order. That is, the curves that share the same the same velocity at the point. The second order tangent space is then the refinement of the equivalence classes into classes of curves agreeing up to second order, these curves also have the same acceleration.

The second differential d^2f is invariant under coordinate changes, the above formula is valid in any coordinate system and this is why the extra acceleration terms are needed. But it would be nice if we could be able to specify the second differential as a pure quadratic form on the tangent bundle. This is possible in three cases.

  • When the function is critical at the point (x,y), then the first differential vanishes and the second differential becomes a quadratic form at this point.

d^2f=f_{xx}dx^2+2f_{xy}dxdy+f_{yy}dy^2

  • If the manifold is the affine plane, we have a set of preferred coordinate systems: the affine coordinates. We can then set d^2x=0 and d^2y=0, which is invariant in all affine coordinate systems and thus only keeping the quadratic form-part of d^2f. In a physics context the affine coordinate systems are exactly the inertial systems.
  • Finally, if we have a general (symmetric) connection, which is a way to specify how the accelerations are dependent on the velocites

\begin{aligned}d^2x&=-(\Gamma_{11}^1dx^2+2\Gamma_{12}^1dxdy+\Gamma_{22}^1dy^2)\\d^2y&=-(\Gamma_{11}^2dx^2+2\Gamma_{12}^2dxdy+\Gamma_{22}^2dy^2)\end{aligned}

then the second differential becomes

\begin{aligned}d^2f&=(f_{xx}-\Gamma_{11}^1f_x-\Gamma_{11}^2f_y)dx^2\\&+2(f_{xy}-\Gamma_{12}^1f_x-\Gamma_{12}^2f_y)dxdy\\&+(f_{yy}-\Gamma_{22}^1f_x-\Gamma_{22}^2f_y)dy^2\end{aligned}

This entry was posted in differential geometry and tagged . Bookmark the permalink.

Leave a comment