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What is the difference between a path and a vector field?

From what I understand the unit vectors $\mathbf i$, $\mathbf j$, and $\mathbf k$ are actually vector fields (constant vector fields to be exact). Then if we have a path $$\mathbf r(t) = x(t)\mathbf i + y(t)\mathbf j + z(t)\mathbf k$$ it's just a linear combination of vector fields.

$1)$ So doesn't that make it a vector field as well?

$2)$ So then are paths just a specific type of vector field or are they different concepts?

me_ravi_
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user240968
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    Domain matters. Where is $\mathbf r(t)$ defined? Where is a vector field defined? –  May 16 '15 at 06:07
  • So the difference is that $\mathbf i: \Bbb R^3 \to \Bbb R^3$ and $\mathbf r: \Bbb R\to \Bbb R^3$? I guess $\mathbf r(t) = x(t)\mathbf i(x(t), y(t), z(t)) + \cdots = x(t)\mathbf i(t) + \cdots$ in the case of the path $\mathbf r$? So is a path like a restriction of a vector field? – user240968 May 16 '15 at 06:13
  • No, a function on $\mathbb{R}$ is not in any natural way a restriction of a function on $\mathbb{R}^3$. –  May 16 '15 at 06:33

2 Answers2

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A vector field assigns a vector to every point in the space, a path only assigns a vector to a subset of points consisting of a curve of some sort.

Gregory Grant
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the unit vectors $\mathbf i$, $\mathbf j$, and $\mathbf k$ are actually vector fields

Not really. These vectors are elements of $\mathbb{R}^3$. An element of $\mathbb{R}^3$ is not the same as a map into $\mathbb{R}^3$.

Now, one can consider a constant map into $\mathbb{R}^3$ which takes the value $\mathbf i$. But... there are many such constant maps, one for each set imaginable. There is

  • the constant map from $\mathbb{Z}$ to $\mathbb{R}^3$, which takes value $\mathbf i$ on every integer
  • the constant map from $\mathbb{R}$ to $\mathbb{R}^3$, which takes value $\mathbf i$ on every real number
  • the constant map from $\mathbb{R}^2$ to $\mathbb{R}^3$, from $\mathbb{R}^5$ to $\mathbb{R}^3$... from $[0,3]^3\times \mathbb{Z}\times \mathbb R$...

So, which one of these "is actually" $\mathbf i$? Probably none.

In some contexts it may convenient to use the same letter $\mathbf i$ for the constant map $\mathbb{R}^3\to \mathbb{R}^3$ that takes the value $\mathbf i$ everywhere. But it should be understood that this is using the same letter for a different, though related object. (A convenient abuse of notation.)

So: no, these are different things. A path is a map from $\mathbb{R}$ to $\mathbb{R}^3$.

A vector field is a map from $\mathbb{R}^3$ to $\mathbb{R}^3$ (actually, this is also an abuse of notation, which will become obvious later, when you study vector fields on manifolds).

These are very different concepts, and not just because the domains are different but because of what we do with them.