Why is this true? General relativity

[Replusz]

Homework Statement

I dont understand why this is true generally. I can see it in Euclidean space, but this is for a general manifold.

Relevant Equations

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"their scalar product is..." why?

 

[Chestermiller]

Why would you think it is not true for a general manifold?

 

[Replusz]

Why would you think it is not true for a general manifold?

At the moment I don't think anything, and its Maths, not what people think. This is a lecture notes so I assume its correct. But I don't see where it comes from, I.e. Why its true
Even for euclidean space i see it intuitively not mathematically

 

[George Jones]

Homework Statement: I don't understand why this is true generally. I can see it in Euclidean space, but this is for a general manifold.

Write $\mathbf{u}$ and $\mathbf{v}$ in terms of the basis vectors $\left\{ \partial / \partial x^\mu \right\}$, and substitute these expressions into $g \left( \mathbf{u} , \mathbf{v} \right)$.

 

[dextercioby]

A little nitpicking about terminology. There is no "general manifold" statement really accurate.
The sensible way to make the scalar product of $\mathbb{R}^n$ "more general" is to consider manifolds with metric, aka (pseudo-)Riemann-ian manifolds. A general manifold needn't have a metric.