Total Derivative
Back
to Physics World
Back to Geometrical Math for GR
Our task
is to find the derivative of a vector in terms of its components on a particular
basis. The result will contain the total derivative along the curve. Consider
the 4-vector x.
Expand x in terms
of a coordinate basis vectors
Take the
derivative of x
with respect to proper time to obtain
Use the
chain rule for the derivative of the basis vectors
Write ¶bem as
a linear combination of the basis vectors. i.e.
where
are
referred to as Christoffel symbols of the second kind. Substitute
Eq. (4) into Eq. (3) to obtain
Substitute
Eq. (5) into Eq. (2) to get
Relabel
indices in the second term and factor out common basis vectors
Express
the terms in brackets on the right side as
This
expression is known as the total derivative along the curve. Substituting
into Eq. (6) we obtain our final result
Finding
the second derivative is trivial since we known that the term on the left side
is a vector. Applying the total derivative a second time we’d have
This
last result will be useful when the equation of geodesic deviation is
calculated.