In chapter 8 on page 183 in figure 8.3 and in many many other figures, there is a little arrow labeled "x". This is very hard to interpret correctly. It doesn't mean what it says, and it doesn't say what it means. However, there is a simple and elegant way to clean up the situation:
I suggest carefully distinguishing between a _coordinate_ such as x and the _vector basis_ that is _induced_ by the coordinates.
For starters, position is not a vector. A position is a zero-sized point, with no magnitude or direction. The displacement from one position to another is a vector, but that's the answer to a different question. If you choose an arbitrary origin, you can construct the radius vector, i.e. the displacement from the origin, and thereby create a one-to-one correspondence between positions and vectors, but even so I insist that position, per se, is not a vector.
There is such a thing as the _direction of increasing x_. For reasons to be discussed in a moment, I strongly recommend calling this the dx direction. (There is no such thing as the "x" direction.)
For example, consider a point on the one-dimensional number line. Suppose the point x is in negative territory, which we represent to the left of the origin. Then the radius vector is a leftward-pointing vector. However, the _direction of increasing x_ is still a rightward vector. Calling this the dx direction has all the right mathematical properties. Do not think of dx as an infinitesimal, but rather as a tangent vector. It is the derivative of position with respect to the x-coordinate.
To repeat, the idea of an "x direction" is broken and cannot be fixed. However, the idea of a dx direction, i.e. the _direction of increasing x_ is well defined and well behaved.
To be consistent with the style of the book, dx should be written with a little arrow over it. (Personally I would be happy to see all such arrows go away, but I'm not going to argue the point. In the short term, consistency is more important.)
Note: We can get a lot of mileage out of idea that dx is a vector, and represents the direction of increasing x, In particular, it is priceless in the context of thermodynamics. It allows us to make sense of expressions such as dE = T dS - P dV. For hundreds of years, people tried to make sense of such expressions in terms of infinitesimals ... tried and failed. However, when you interpret it in terms of tangent vectors in a high-dimensional space, it's like turning on the lights in a huge heretofore-dark room.
The students don't need to know where the "dx" terminology comes from. If you start talking about exterior derivatives and differential topology in front of first-year students, they will run out of the room screaming, but if you just say that "dx is the direction of increasing x" most of them will swallow it without even noticing.
In chapter 8 on page 183 in figure 8.3 and in many many other figures, there is a little arrow labeled "x". This is very hard to interpret correctly. It doesn't mean what it says, and it doesn't say what it means. However, there is a simple and elegant way to clean up the situation:
To repeat, the idea of an "x direction" is broken and cannot be fixed. However, the idea of a dx direction, i.e. the _direction of increasing x_ is well defined and well behaved.
To be consistent with the style of the book, dx should be written with a little arrow over it. (Personally I would be happy to see all such arrows go away, but I'm not going to argue the point. In the short term, consistency is more important.)
The students don't need to know where the "dx" terminology comes from. If you start talking about exterior derivatives and differential topology in front of first-year students, they will run out of the room screaming, but if you just say that "dx is the direction of increasing x" most of them will swallow it without even noticing.
See #90 for a catalog of related issues.