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.

 

[sardar]

To change the dot product to simple multiplication, you would need to remove the dot product symbol (·) and replace it with the multiplication symbol (×) between the two vectors or matrices. This would result in a multiplication of the individual elements instead of a scalar product.

In the given example, to get an expression for \partialcea, you would simply multiply the vectors eb and -\Gammaa bc, resulting in:

\partialcea = -eb × \Gammaa bc