# Newbie-changing the dot product to simple multiplication

1. Oct 24, 2013

### 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:
eb.$\partial$cea=-$\Gamma$a bc

How do I get just an expression for $\partial$cea?

2. Oct 24, 2013

### Halaaku

Here $\Gamma$a bc = ea.∂ceb

3. Oct 24, 2013

### 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.

4. Oct 24, 2013

### Staff: Mentor

The partial derivative of the coordinate basis vector eb with respect to the spatial coordinate xc 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 ea, 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.