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I always hear that there are scalar fields and they are different from vector fields in that vectors have a direction whereas temperature has not. For instance, here is a professor saying that, http://www.youtube.com/watch?v=Z8ace3NbXAA#t=259s

But, I see that scalars have direction! As much as 1-dimentional vectors do! How can you tell that positive and negative temperature has less direction than position on a line? What a nonsense? I mean, what is the difference?

I am recalling that Gilbert Strang had only one vector, which had no direction. It was 0-vector, from 0-(dimentinoal )space. Rotation and all other linear operators applied universally to all vectors. There was no exclusion for 1-dimensional vectors. So, I do not understand if 1-component vector has a direction or not. Is scalar indirectional in the same way and why don't you want to highlight the difference? Might be the math is not precise science. This is terrible :( People repeat the words, the meaning of which they do not understand, like robots. Is it taboo to think that scalar = 1d-vector? Is it taboo to ask about the difference?

Is it like single element, e.g 0, differs from the singleton, e.g. {0}?

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Well, yes, scalars form the $1$ dimensional vector space (the line). –  Berci May 7 '13 at 21:58
But, vectors have directions whereas scalars don't! How scalars can be a special case of vectors??? How can you be right over 9000 of professors of physics? Why they just don't say that it is a single-component vector and fall into contradiction instead? –  Val May 7 '13 at 22:00
If you watch 10 more seconds into that Susskind video he explains the difference. In common usage scalars refer to quantities with a single "dimension" or "component" (e.g temperature). Vectors are quantities with multiple components (e.g wind direction). Obviously any field is a vector space over itself. That's not what is meant in this case by the word vector though. –  FiveLemon May 7 '13 at 22:05
@FiveLemon: Do you mean that he contradicts to himself or what? Why should vectors have multiple components? He speaks about mathematical abstraction and vectors in math are not defined as multicomponent objects. Look into definition: vectors are objects that can be added and scaled. Simple numbers can be added and scaled. Simple, numbers (scalers) are vectors, therefore. Why should they have multiple components. Answer. –  Val May 7 '13 at 22:15
What is the SE policy on downvotes based on rudeness in the comments again? –  Simon Markett May 7 '13 at 22:16

3 Answers 3

up vote 2 down vote accepted

Yes, a 1 component vector is a vector. It is no different than a scalar field. You are allowed to say that a scalar has a direction, because it can certainly have a direction--up to 2 directions to be exact. It may be easier for you to think of a scalar as being a magnitude. A magnitude inherently carries a direction (up to 2). A higher dimension vector has a magnitude and another direction in addition to the one carried by the magnitude. This is why a full rotation is $2\pi$ radians, because for each "direction" in 2 dimensions, there is an additional one implied by the sign of the magnitude.

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A scalar field is one that assigns a single number to each point in space. The temperature field $T(x,y,z)$ is a good example. At any point, one can put a thermometer and measure the temperature. The value of the field does not have a direction, but you can make a vector from the origin to the point of interest.

A vector field assigns a vector to each point in space. The electric field $\vec E(x,y,z)$ is a good example. It has a strength and direction at each point. This is independent of the vector from the origin to the point of measurement.

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How you answer resolves the contradiction that I expose? How does it explain the difference between single-component vector and scalar? What is independent of the vector from the origin to the point of measurement? Temperature? How T(x,y,z) can be independent of the vector (x,y,z)? –  Val May 8 '13 at 6:05
@Val: You are looking at the vector from the origin to the point of measurement. Often it is not useful to view this as a vector, just as a point. The temperature at a given point is defined without reference to the origin of your coordinate system, but using the vector requires that you pick one. Then you are confusing the vector from the origin to the point of measure with (in the case of the electric field) the vector that is the field you measure. They are very distinct. Note that the function $T$ takes three inputs $x,y,z$ and produces one output. –  Ross Millikan May 8 '13 at 16:10
@Val: The function $\vec E$ takes the same three inputs and produces three outputs: $E_x,E_y,E_z$ The temperature does not have a direction, even if it has a sign. –  Ross Millikan May 8 '13 at 16:11
ok, I was sure that you was doing that mistake. Excuse me for getting you wrong. But, I do not understand how to get you. You just drill definitions. They don't explain anything. I ask you WHY temperature does not have the definition. You cheat when use 3d vector vs. 1d scalar. I see that more than one dim is what gives you the "dir" and intentionally asked compare the 1d vec vs. scalars to avoid the cheat. Yet, you do not hear me. You keep repeating the definitions, like robot. Why do you say that temp has no dir? Do people say temp goes higher because it does not have the direction? –  Val May 8 '13 at 16:28
@Val: Temperature has an ordering, in that we can compare two temperatures and find the higher or lower. That is not the same as a spatial direction. You can imagine a graph of the temperature at various places on the ground and plot temperature on the $z$ axis, but that still does not make it into a direction in space. The concept of a dot product of vectors $(x,y,t)$ doesn't make sense, for example, as it does $(x,y,z)$ –  Ross Millikan May 8 '13 at 16:55

Here is a higher brow explanation since you are interested in abstraction. A vector field is a section of the tangent bundle of a manifold. In other words, it is the (smooth) assignment of an element from the tangent space of each point.

A scalar field is a section of the product bundle with $\mathbb{R}$.

If you are working in a space whose dimension is greater than $1$, these must be different notions. It is true that vectors in $\mathbb{R}$ have direction in the sense of positive and negative (I'm ignoring issues of orientation), but that direction is not connected to the manifold in the same way as vectors in the tangent space.

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Do you mean you can do something more with vectors than adding and scaling them? So that you can do this with a 1-component vector but not with a scalar? –  Val May 8 '13 at 6:14
What do you mean by "can do"? –  Adam Saltz May 8 '13 at 11:39
Are you asking me? To answer what do you mean? –  Val May 8 '13 at 11:50
We are in whatever space we are. Can 1-component vectors do something more than scalars in that space? What is the difference between two? –  Val May 8 '13 at 12:07
Yes, because there are $1$-component vectors that aren't elements of a product bundle. E.g. the Mobius bundle. And yes, it is not clear to me what you mean by "can do." –  Adam Saltz May 8 '13 at 14:19

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