Changing Dot Product to Simple Multiplication

[Halaaku]

How does one change the dot product such that there is no dot product in between, just plain multiplication? For example, in the following:
e_b.\partial_ce^a=-\Gamma^a _bc

How do I get just an expression for \partial_ce^a?

 

[Halaaku]

Here \Gamma^a _bc = e^a.∂_ce_b

 

[mathman]

I have no knowledge of the particular symbols. However if you have the dot product of two vectors equal to a scalar, you cannot get one of the vectors from the scalar without further information. It is not enough just to know the other vector.

 

[Chestermiller]

Here \Gamma^a _bc = e^a.∂_ce_b

The partial derivative of the coordinate basis vector e_b with respect to the spatial coordinate x^c is a vector, which can be expressed at a given point as a linear combination of the coordinate basis vectors:

\frac{\partial e_b}{\partial x^c}=\Gamma^j_{bc}e_j

The \Gamma 's are the components of the vector. If we dot this equation with the duel basis vector e^a, we get:
e^a\centerdot\frac{\partial e_b}{\partial x^c}=\Gamma^a_{bc}

The trick is to figure out how to represent the \Gamma 's in terms of the partial spatial derivatives of the components of the metric tensor.