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I'm not sure I fully understand the difference between these two terms when used in differential geometry/general relativity.

If I were to describe covariant differentiation to someone, I would say something like this:

The reason I ask is because I did have a conversation with a fellow student. I was explaining what a Christoffel symbol was and why it was needed.

I remember saying

I also tried to explain the act of "getting a number" in a way that related to QM.

I said that in QM, you "get a number" by operating on something in the bra-ket notation.

I noticed myself using "tangent space" and "vector space" interchangeably in that conversation as well.

So......how exactly are they different?

Perhaps my "visual image" of the tangent plane on a curved manifold is what has me using them as equal?

If I were to describe covariant differentiation to someone, I would say something like this:

*"On a curved manifold (imagine a basketball), you could assume a tangent space at the point "P." At that point, the tangent space would be like placing a flat piece of paper on the ball at that point P.*

If you move from point P to a new point "Q," you're no longer in that same

If you move from point P to a new point "Q," you're no longer in that same

**tangent space**and therefore need a "connection" to allow the differentiation to work for non-flat manifolds...yada yada..."The reason I ask is because I did have a conversation with a fellow student. I was explaining what a Christoffel symbol was and why it was needed.

I remember saying

*"If you take the derivative, you're no longer in the same*in the spot where I used tangent space above.**vector space**"I also tried to explain the act of "getting a number" in a way that related to QM.

I said that in QM, you "get a number" by operating on something in the bra-ket notation.

*"If you want a number for the energy, you can perform the operation*

[tex]

\langle \psi | H | \psi \rangle

[/tex]

In GR, it's not much different. If you want to "get a number," you combine vector and a dual vector. The contraction of the vector [tex]\upsilon^{\alpha}[/tex] with the dual vector [tex]f_{, \alpha}[/tex] in the form

[tex]

\frac{df}{d\lambda}=f_{, \alpha}\upsilon^{\alpha}

[/tex]

gives you a scalar, i.e. "number."[tex]

\langle \psi | H | \psi \rangle

[/tex]

In GR, it's not much different. If you want to "get a number," you combine vector and a dual vector. The contraction of the vector [tex]\upsilon^{\alpha}[/tex] with the dual vector [tex]f_{, \alpha}[/tex] in the form

[tex]

\frac{df}{d\lambda}=f_{, \alpha}\upsilon^{\alpha}

[/tex]

gives you a scalar, i.e. "number."

I noticed myself using "tangent space" and "vector space" interchangeably in that conversation as well.

So......how exactly are they different?

Perhaps my "visual image" of the tangent plane on a curved manifold is what has me using them as equal?

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