# Thread: How is the derivative of a vector field Y on a curve (almost) the covariant of Y?

1. ## How is the derivative of a vector field Y on a curve (almost) the covariant of Y?

This two-part problem is from O'Neill's Elementary Differential Geometry, section 2.5. It is in the second statement of the second part where I'm unclear, beginning with "Thus."

Full problem:

Let $\displaystyle$W$$be a vector field defined on a region containing a regular curve \displaystyle a$$. Then $\displaystyle$W\circ a$$(i.e. \displaystyle W$$ composed with $\displaystyle$a$$) is a vector field on \displaystyle a$$ called the restriction of $\displaystyle$W$$to \displaystyle a$$.

1. Prove that $\displaystyle$\nabla_{a^{\prime}(t)}W = (W(a))^{\prime}(t)$$. 2. Deduce that the straight line in Definition 5.1 (below) may be replaced by any curve with initial velocity \displaystyle v$$. Thus the derivative $\displaystyle$Y^{\prime}$$of a vector field \displaystyle Y$$ on a curve $\displaystyle$a$$is (almost) \displaystyle \nabla_{a^{\prime}}Y$$.

Definition 5.1. Let $\displaystyle$W$$be a vector field on \displaystyle \mathbb{R}^3$$, and let $\displaystyle$v$$be a tangent vector to \displaystyle \mathbb{R}^3$$ at the point $\displaystyle$p$$. The the covariant derivative of \displaystyle W$$ with respect to $\displaystyle$v$$is the tangent vector \displaystyle (W(p+tv))^\prime(0)$$ at the point $\displaystyle$p$$. The following definition is useful for part (2) since it distinguishes between a vector field and a vector field on a curve. Definition 2.2. A vector field on a curve \displaystyle a$$ from $\displaystyle$I$$to \displaystyle \mathbb{R}^3$$ is a function that assigns to each number $\displaystyle$t$$in \displaystyle I$$ a tangent vector $\displaystyle$Y(t)$$to \displaystyle \mathbb{R}^3$$ at the point $\displaystyle$a(t)$$. My attempt, and where I'm stuck: Part (1) was fairly straight-forward, using the definitions of covariant derivative and what it means to differentiate a composition. Part (2) has two parts. My approach to the first part is the following, and I believe it to be correct. The idea is to define a function on curves \displaystyle a(t)$$ and show that it agrees with the covariant derivative with respect to a vector $\displaystyle$v$$at a point \displaystyle p$$ for all curves $\displaystyle$a(t)$$such that \displaystyle a(0)=p$$ and $\displaystyle$a^\prime(0)=v$$. Part (1) can be used to show the function is well-defined and that it indeed equals the covariant. The second part, starting at "Thus" is where I'm having trouble. It's with the use of the word "almost." To me, if \displaystyle Y$$ is a vector field on a curve $\displaystyle$a$$, then using the definition of covariant, straight-line or otherwise, makes no sense because \displaystyle Y\circ a$$ is not defined (i.e., $\displaystyle$Y$$is not defined on \displaystyle \mathbb{R}^3$$, only $\displaystyle$\mathbb{R}$$). So I thought this might be the almost part. However, what's 'almost' about it? I was thinking that perhaps given \displaystyle Y$$ on $\displaystyle$a$$that a vector field \displaystyle \overline{Y}$$ could be defined such that $\displaystyle$\overline{Y}\circ a = Y$$, and that then \displaystyle Y^\prime$$ would equal the covariant of $\displaystyle$\overline{Y}$$instead of \displaystyle Y$$. However, I don't think I can 'always' define such a $\displaystyle$\overline{Y}.