what is a criterion(s) to define some quantity as a vector or scalar?

[PG1995]

Hi

As far as my limited knowledge goes, I don't think there is any definitive criterion to define some quantity as a vector or scalar. For instance, area and length are most of the time are defined to be vectors and even intuitively they seem so but still when the need comes and context requires we define them to be vectors such as in the laws related to electromagnets. I wouldn't be surprised to see mass being defined to be a vector some place where it makes sense in the given context. Please guide me with this. Thank you.

Regards
PG

 

[Ratchit]

PG,

I wouldn't be surprised to see mass being defined to be a vector some place where it makes sense in the given context.

I would be very suprised to see mass defined as a vector since it has no spacial direction like velocity does.

Ratch

 

[PG1995]

Thank you for the reply.

That's exactly what I'm driving at. Declaring velocity a vector does make sense and under 'normal' circumstances defining length and area to be scalar also makes equal sense but still when the context requires we treat them vectors. Then, why would you be surprised to see mass defined a vector under some special context? Please let me know. Thanks.

Regards
PG

 

[Ratchit]

PG,

Then, why would you be surprised to see mass defined a vector under some special context?

I never said "under special context".

Context has nothing to do with vectors. If a quantity has magnitude and spacial direction, then it is a vector. Mass has only magnitude, so it is a scaler.

Ratch

 

[PG1995]

confused between negative numbers and vector quanities

Hi

Electric current, i, is a scalar quantity but still it is assigned positive and negative directions in circuits. Likewise, there are numerous other quantities which are scalar but still are assigned negative and positive values such as temperature. In plain words, if quantities like electric current and temperature can be assigned positive and negative values then why can't they called vector quantities? Actually, I'm confused by the fact that I relate negative and positive signs with directions. Perhaps, negative sign with quantities such as current and temperature only indicates that they have a value lower than some set standard reference value such as '0'. Please help me with it. Thank you.

I think as far as electron current is concerned (which arguably is different from 'simple' electric current), it could be considered as a vector quantity under special circumstances. For instance, if we take the motion of electrons (i.e. electron current) toward the right of screen as positive and toward the left as negative. In this case electron current has both magnitude and direction.

Regards
PG

 

[steveB]

PG, this question might be answered in different ways depending on the situation and on the sophistication of the mathematical tools you are using.

I don't think there is any definitive criterion to define some quantity as a vector or scalar.

There are definitely criteria to apply, but the key question is what definitions and physical problems are you using. You know that vectors fundamentally have magnitude and direction. Additionally, we sometimes require a defined norm and transformation properties in most physical contexts. I'll assume we are talking about the latter situation. Here, the key thing about a vector is that it must transform as a vector. If you change the coordinate system (rotate the axes, for example), the components will change, but the vector magnitude and direction will not change. A scalar also has transformation properties, but here the requirement is that the scalar is invariant. So, a coordinate transformation does not change the value of the scalar, but obviously the coordinates to find that value have changed. One can proceed to higher order objects like 2nd order tensors, which can be represented as a matrix, since now we use unit dyads instead of unit vectors to identify the separate components. The key thing about all tensors, whether they be 0-order tensors (scalars), 1st order tensors (vectors or 1-forms), or higher order tensors, is that they must have components that obey transformation properties which make the tensor itself (but not the components themselves) invariant (or the same) in any coordinate transformation.

I wouldn't be surprised to see mass being defined to be a vector some place where it makes sense in the given context.

More correctly, mass can be defined in terms of mass density since it is an integral of a mass density. Although mass is rightly a scalar quantity, mass density is a higher order tensor which transforms as a 3-form in 3D space.

Electric current, i, is a scalar quantity but still it is assigned positive and negative directions in circuits. Likewise, there are numerous other quantities which are scalar but still are assigned negative and positive values such as temperature. In plain words, if quantities like electric current and temperature can be assigned positive and negative values then why can't they called vector quantities? Actually, I'm confused by the fact that I relate negative and positive signs with directions. Perhaps, negative sign with quantities such as current and temperature only indicates that they have a value lower than some set standard reference value such as '0'. Please help me with it. Thank you.

In the context of electromagnetics, current is a scalar and current density is a vector (or more accurately a 2-form that can be represented as a vector). In circuit theory, current does have a vector-like aspect in that there is a magnitude and a direction. But, here we are dealing with an abstract "circuit-topology" space, which is essentially a 1D domain. It seems that in 1D, what we call current, is really current density, but since there is no space transverse to the direction of charge flow, current density has magnitude equal to the current.

It is hard to imagine a context where temperature might be a vector, but I leave it to you to find a concrete example if you think it is possible.

I think as far as electron current is concerned (which arguably is different from 'simple' electric current), it could be considered as a vector quantity under special circumstances. For instance, if we take the motion of electrons (i.e. electron current) toward the right of screen as positive and toward the left as negative. In this case electron current has both magnitude and direction.

I would say that is correct, and this is very close to the concept of current density. Current density is an idea related to continuous charge distributions, but the reality is that current is always discrete. Hence, an electron could be assigned a current directly with it's velocity vector (i.e. I=ev, where I is current vector, e is electron scalar charge and v is the velocity vector of the electron).

 

[Ratchit]

PG,

Electric current, i, is a scalar quantity but still it is assigned positive and negative directions in circuits. Likewise, there are numerous other quantities which are scalar but still are assigned negative and positive values such as temperature. In plain words, if quantities like electric current and temperature can be assigned positive and negative values then why can't they called vector quantities? Actually, I'm confused by the fact that I relate negative and positive signs with directions. Perhaps, negative sign with quantities such as current and temperature only indicates that they have a value lower than some set standard reference value such as '0'. Please help me with it. Thank you.

Any current, whether electrical or not, is a spacial vector quantity. Current has magnitude and direction. Temperature is a scalar quantity, pure and simple. Whether their magnitudes are positive or negative has nothing to do with their classification.

Ratch

 

[steveB]

PG,



Any current, whether electrical or not, is a spacial vector quantity. Current has magnitude and direction. .

Ratch

I disagree that any current is a vector. Certainly current density is a vector, and the electron current specified by PG is too, and circuit theory currents are also. However, in EM theory we can define current to be the surface integral of current density, and in that case it becomes a scalar quantity defined relative to the established perimeter and orientation of the surface.

 

[Ratchit]

steveB,

I disagree that any current is a vector.

Disagreement noted. But current is particles in motion at a magnitude and direction. That makes it a vector in my way of thinking.

However, in EM theory we can define current to be the surface integral of current density, and in that case it becomes a scalar quantity defined relative to the established perimeter and orientation of the surface.

Why does adding up all the current differentials by integration to get a total current change it into a scalar? The total current still has a direction and magnitude.

Ratch

 

[MrAl]

Hello,


Natural current is a vector, but we often can simplify it into a non vector because the vector properties are not required to solve the application problem.
Velocity is a vector too, but we often simplify this into speed, which is non directional and even looses it's sign.

For the velocity example, a car driving along a long winding highway. We dont care what the direction is so we simplify it to speed.
On the other hand, if we want to know when two cars nearing each other will meet on roads that are straight but at angles to each other, we need the direction too.

For the current in a wire in order to calculate the power in a resistor we dont need to know if the wire is curved or straight, so we dont need the direction.
For the current to determine the magnetic field at a point in space from a curved wire we need to know the direction of the current along the wire.

So when the changing quantity is not important we can simplify the problem into one dimension, but when it's important to the Law involved we have to observe that carefully.

Another way to look at it is when you can look at the current along a single dimensional path even if it is an intrinsic path and nothing in the solution changes from the N dimensional solution, then it doesnt have to be handled as a vector.

 

[Ratchit]

Yes, oftentimes we are just interested in the magnitude of a vector quantity.

Ratch

 

[steveB]

Why does adding up all the current differentials by integration to get a total current change it into a scalar? The total current still has a direction and magnitude.

Ratch

There are two two ways I can answer that: one based on the math and the other based on the physical.

In the mathematical formulation of Amperes law we integrate both current density and displacement current over any surface bounded by a particular oriented boundary. In vector notation this amounts to integrating the dot product of the current densities with the normal to the surface element. Since dot product of two vectors makes a scalar, we are integrating over a scalar function. Hence, the math makes it clear that current, defined in this way is a scalar.

The physical reason is that current is just a measure of rate of charges that flow through a defined orifice. Whether the charges are directed perpendicular to the orifice or obliquely, is not relevant. Only whether the net is in or out has meaning, which relates to sign and orientation of the orifice.

 

[Ratchit]

steveB,

In the mathematical formulation of Amperes law we integrate both current density and displacement current over any surface bounded by a particular oriented boundary. In vector notation this amounts to integrating the dot product of the current densities with the normal to the surface element. Since dot product of two vectors makes a scalar, we are integrating over a scalar function. Hence, the math makes it clear that current, defined in this way is a scalar.

There are at least three different Ampere's laws, and they pertain to magnetic fields. However, the dot product of the current density and the surface vector do make the scalar magnitude of the current. But, since the current has a direction of its charge flow, it is a vector by itself.

The physical reason is that current is just a measure of rate of charges that flow through a defined orifice. Whether the charges are directed perpendicular to the orifice or obliquely, is not relevant. Only whether the net is in or out has meaning, which relates to sign and orientation of the orifice.

Just because the current direction is not of interest, or is irrelevant for one's purposes, does not mean that current is not a vector.

Ratch

 

[steveB]

I'll have to provide a clearer presentation of what I'm trying to say. I'll wait to see if PG is following me first, and then write out some equations to clarify. I'm on vacation traveling in mountains and using my cell phone, so it's not easy to write equations now anyway.

 

[steveB]

I disagree that any current is a vector.

So, I'd like to make a comment here before hearing from PG and before (possibly) going into clearer descriptions. It is well known that current is a scalar quantity. Every text book says this and any online search will confirm this. Despite my absolute confidence in this known and excepted science, I was open minded enough to see and acknowledge that we often treat current as a vector entity. My contention is that we do that by simplifying current density, which is certainly representable as a vector (it's really a 2-form however from a strict geometrical perspective). However, current is not the same as current density, so really current is always a scalar, if we want to be rigorous.


However, I hate rigor and being overly zealous with definitions, which is why I'm more flexible and open. All I said is that i disagree that any current is a vector, which only requires that I find one example where current is not a vector. Since every text book says current is a scalar, I will leave it to others to find one example to prove my point. I can think of many cases where the idea of current having a clear direction is completely untenable.

 

[Ratchit]

steveB,

However, I hate rigor and being overly zealous with definitions, which is why I'm more flexible and open. All I said is that i disagree that any current is a vector, which only requires that I find one example where current is not a vector. Since every text book says current is a scalar, I will leave it to others to find one example to prove my point. I can think of many cases where the idea of current having a clear direction is completely untenable.

I think you mean to disprove your point.

OK, you are on, challenge accepted. From: Semiconductor Fundamentals, by Robert F. Pierret, Second Edition, Volume #1. Look at the middle of the page in the attachment. Think of some examples, like a whirlpool in a river, or a cyclotron, where the direction of the current is changing constantly. Now, if I had a particle gun like a machine gun firing north, and another firing east, I could vectorize the output of each gun, but I would not know how to combine or add those vectors in any way that makes sense. That does not mean, however, that the output from those guns are not vectors (magnitude of mass in a direction).

Ratch

 

[steveB]

I did mean "prove" my point, not "disprove". Anyone who finds an example of current that can't have a direction, will prove my point.

You are not following my logic. I say that "I disagree that any current is a vector". I acknowledged that some currents are formulated as vectors, but I only need to find one example of a current that can't have a direction to prove my point that not all currents are vectors. I have several examples in mind, but I haven't stated them. All you did was provide a text that considers current a vector, which does nothing to help or hurt the logic of what I said. Also, this text also says that "it is generally thought that current is a scalar", then they say "but it is obviously a vector". I may not agree with their precise wording there, but, yes in that context a vector concept does make sense. However, a vector formulation doesn't make sense in all cases, which is my point.

Also, guess why it is generally thought that current is a scalar? It's because that's how it's defined. Arguing against definitions is pointless. However, allowing flexibility to modifying definitions is useful, which is why I'm not saying the vector concept is wrong, but it is limited to special cases, with the case you showed being one of many examples.

But, in the final analysis, any current you claim is a vector is really "current density" cleverly disguised as a current by wrapping up the cross sectional dependence into the quantity itself. If you (and others and even I) want to call that vector a current by definition, then there is no point arguing with that definition. But, this does not invalidate to original and proper definition, and the original definition is the one that can cover the general case, which is the essence of my original point.

The issue really goes beyond this simple logic and formulation of definitions that I just described. It's really a matter of fundamental laws of nature. The beauty of Ampere's law as reformulated by Maxwell ...

[latex]I=\int \vec{H} \cdot d\vec{l}=\int\int \vec{J}\cdot d\vec{s}+\frac{d}{dt}\int\int \vec{D}\cdot d\vec{s}[/latex]

... is that the current is defined relative to the contour that we evaluate the H field over. This closed contour then defines an infinite number of surfaces that attach to that contour. Well, this is the crux of the problem with establishing a direction for the current I. Aside from the fact that the math defines current to be a scalar, there is no one clear direction to assign to the current. Not only is there a distribution of directions over any one surface, but there is no one special surface to evaluate the current density J and displacement current density dD/dt. You get the same answer for any surface, but all those surfaces have different distributions of vector directions for J and D. There is also no way to assign a direction to the contour because the contour is large and of arbitrary shape in 3D space.

This is the general case we are talking about here. This is why vector currents are very dubious entities. However, current density J and displacement current density dD/dt have clear vector meanings in general.

 

[MrAl]

So, I'd like to make a comment here before hearing from PG and before (possibly) going into clearer descriptions. It is well known that current is a scalar quantity. Every text book says this and any online search will confirm this. Despite my absolute confidence in this known and excepted science, I was open minded enough to see and acknowledge that we often treat current as a vector entity. My contention is that we do that by simplifying current density, which is certainly representable as a vector (it's really a 2-form however from a strict geometrical perspective). However, current is not the same as current density, so really current is always a scalar, if we want to be rigorous.


However, I hate rigor and being overly zealous with definitions, which is why I'm more flexible and open. All I said is that i disagree that any current is a vector, which only requires that I find one example where current is not a vector. Since every text book says current is a scalar, I will leave it to others to find one example to prove my point. I can think of many cases where the idea of current having a clear direction is completely untenable.

Hello there Steve,


Yes the current often enters as a scalar, even in electromagnetics, but then we still have to consider the direction so instead of making current a vector we make the differential wire length a vector. That's interesting:
**broken link removed**

But then we have (search) "conservation of vector current" which actually states "vector current".

Although not as direct, we also have the AC view of current as a vector.

 

[steveB]

Hi MrAl,

Yes, good points there. Also, there are formulations of the Biot Savart Law based on the current density rather than the current. However, the formulation using current is often more direct and simpler when we have filamentary current paths due to wires.

https://en.wikipedia.org/wiki/Biot-Savart_law

In the above reference, as you pointed out, the Biot Savart law based on current is using scalar current I and vector path elements, but the Biot Savart law based on current density is using the vector current density J along with volume elements.

Hence, we see the scalar nature of current, and the vector nature of current density is firmly embedded in the proper mathematical formulation of electromagnetic theory.

 

[MrAl]

Hello Steve,


Yes current enters into the equation as a scalar, because we dont have to know it's direction. But the reason we dont have to know it's direction is because we indirectly track it's direction by tracking the direction of the wire. But the wire itself is not causing (or reacting to) the magnetic field, it is the moving electrons that are causing (or reacting to) the field. We've simplified it such that we've tracked the behavior via the wire turn rather than the electron as it turns through the wire, but if we could make the electron turn without the wire then what could we say was causing the behavior. You probably would say, "the path". And i agree, so without a wire we still have "the path" to consider. But isnt that just a little bit strange, that we would rather consider it to be something that doesnt even exist over something that is really the root cause of the behavior?

The truck travels down the highway. We see the truck turn. The truck turning causes a big wind to blow on a motorcycle parked on the side of the road with a kickstand to hold it upright. The motorcycle falls over because of the wind. Did the truck cause the motorcycle to fall over, or did the turn in the road? Keep in mind that if the truck was not traveling down the road the motorcycle would have never fallen because the turn in the road can not push the motorcycle over on it's own. We could also say that if the turn in the road was not there the cycle would not have fallen either. So which is it then. We pick the one that simplifies the mathematics the most and/or leads to the most generalizations.

Any thoughts on this one?