Vectors (incl. axial/psuedovectors)/Scalars

[bon]

Homework Statement

Basically, trying to determine whether the following are scalars of vectors: area and angle of polarization.

Homework Equations

None

The Attempt at a Solution

My guess was that area would be a scalar rather than a vector, but my notes say that it is a psuedovector - is this true? In what sense does it have direction? And why is it axial rather than a proper vector?

Angle of polarization is a scalar I'm guessing, as its just a magnitude (angle) rather than direction?

 

[berkeman]

Homework Statement

Basically, trying to determine whether the following are scalars of vectors: area and angle of polarization.

Homework Equations

None

The Attempt at a Solution

My guess was that area would be a scalar rather than a vector, but my notes say that it is a psuedovector - is this true? In what sense does it have direction? And why is it axial rather than a proper vector?

Angle of polarization is a scalar I'm guessing, as its just a magnitude (angle) rather than direction?

(I think you have a small typo in the question. You mean "scalars or vectors", right?)

What are the definitions of scalars and vectors? How are they fundamentally different?

 

[bon]

(I think you have a small typo in the question. You mean "scalars or vectors", right?)

What are the definitions of scalars and vectors? How are they fundamentally different?

yes i did make a typo, sorry.

vectors have direction, scalars don't. both have magnitude. hence my being confused of the fact that my notes say area is a psuedovector.

 

[berkeman]

yes i did make a typo, sorry.

vectors have direction, scalars don't. both have magnitude. hence my being confused of the fact that my notes say area is a psuedovector.

Sorry, what's the definition of a pseudovector?

 

[bon]

easier than me trying to explain it:

http://en.wikipedia.org/wiki/Pseudovector

 

[vela]

Do you recall doing surface integrals, like when applying Gauss's law?

 

[bon]

Haven't covered that yet, sorry :(

 

[berkeman]

Here is one application where you can see how an area could be a pseudovector:

http://www.math.ucdavis.edu/~daddel...plications/Determinant/Determinant/node4.html

And I think you're right about the angle of rotation, as long as the plane of rotation is specified and the zero angle and rotation direction are also specified (not sure about all of that though).

 

[jambaugh]

To say "area is a pseudo-vector" you need to qualify a couple of things.

First that you're working in 3-space. (otherwise they are at best bi-vectors = rank two tensors)

Second that you're talking about oriented area elements (which are negative when you reverse orientation) typically defined by two vectors as the parallelogram they define.

The thing to remember is that "vectors" and "scalars" and such are so defined for arbitrary linear transformations on the space not just for rotations. A pseudo-scalar transforms like a volume element, being unchanged by rotations but scaling by the determinant of a more general transformation matrix. Since a vector defining a line element plus an area element together define a volume element (parallelepiped = oblique box) the area element must be a pseudo-vector (in 3 dim).

The vector associated with an area element is the normal vector of length numerically equal to the area. Then dotting a vector with this "pseudo-vector" gives the "pseudo-scalar" volume.

The types and dimensions follow the combinatorics of the the binomial coefficients. In 3 dim you have: 1,3,3,1 corresponding to scalar,vector, bivector=pseudo-vector, pseudo-scalar.
In four dimensions you would have: 1, 4, 6, 4, 1 and so:
scalar, vector, bivector(=pseudo-bivector), trivector=pseudo-vector, pseudo-scalar.

In five dimensions you would have 1,5, 10, 10, 5, 1 and so ...

 

[bon]

wow okay thanks for your help. So am i right in thinking angle of polarization is a scalar?