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0 rank tensor vs 1D vector
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$begingroup$
What is the difference between zero-rank tensor $x$ (scalar) and 1D vector $[x]$?
As far as I understand tensor is anything which can be measured and different measures can be transformed into eachother. That is, there are different basises for looking at one object.
Is lengh a scalar (zero rank tensor)?
I think it is not.
ex.:
- physical parameter: writing pen's length
- tensor: $l$
- length in inches: $[5.511811023622]$
- length in centimeters: $[14]$
- transformation law: 1cm = 2.54inch
so $l$ is a scalar, but on the other hand it's a tensor of rank 1 since "physical parameter of length is invariant, only it's measures (in different units) are".
The same example can be made with classical example of temperature (which is used as a primer of zero rank tensor most in any book) in C and K units. I'm confused.
tensor-calculus
edited 2 hours ago
Szabolcs Berecz
1031
asked 2 hours ago
coobitcoobit
350110
$endgroup$
add a comment |
$begingroup$
What is the difference between zero-rank tensor $x$ (scalar) and 1D vector $[x]$?
As far as I understand tensor is anything which can be measured and different measures can be transformed into eachother. That is, there are different basises for looking at one object.
Is lengh a scalar (zero rank tensor)?
I think it is not.
ex.:
- physical parameter: writing pen's length
- tensor: $l$
- length in inches: $[5.511811023622]$
- length in centimeters: $[14]$
- transformation law: 1cm = 2.54inch
so $l$ is a scalar, but on the other hand it's a tensor of rank 1 since "physical parameter of length is invariant, only it's measures (in different units) are".
The same example can be made with classical example of temperature (which is used as a primer of zero rank tensor most in any book) in C and K units. I'm confused.
tensor-calculus
edited 2 hours ago
Szabolcs Berecz
1031
asked 2 hours ago
coobitcoobit
350110
$endgroup$
add a comment |
$begingroup$
What is the difference between zero-rank tensor $x$ (scalar) and 1D vector $[x]$?
As far as I understand tensor is anything which can be measured and different measures can be transformed into eachother. That is, there are different basises for looking at one object.
Is lengh a scalar (zero rank tensor)?
I think it is not.
ex.:
- physical parameter: writing pen's length
- tensor: $l$
- length in inches: $[5.511811023622]$
- length in centimeters: $[14]$
- transformation law: 1cm = 2.54inch
so $l$ is a scalar, but on the other hand it's a tensor of rank 1 since "physical parameter of length is invariant, only it's measures (in different units) are".
The same example can be made with classical example of temperature (which is used as a primer of zero rank tensor most in any book) in C and K units. I'm confused.
tensor-calculus
edited 2 hours ago
Szabolcs Berecz
1031
asked 2 hours ago
coobitcoobit
350110
$endgroup$
What is the difference between zero-rank tensor $x$ (scalar) and 1D vector $[x]$?
As far as I understand tensor is anything which can be measured and different measures can be transformed into eachother. That is, there are different basises for looking at one object.
Is lengh a scalar (zero rank tensor)?
I think it is not.
ex.:
- physical parameter: writing pen's length
- tensor: $l$
- length in inches: $[5.511811023622]$
- length in centimeters: $[14]$
- transformation law: 1cm = 2.54inch
so $l$ is a scalar, but on the other hand it's a tensor of rank 1 since "physical parameter of length is invariant, only it's measures (in different units) are".
The same example can be made with classical example of temperature (which is used as a primer of zero rank tensor most in any book) in C and K units. I'm confused.
tensor-calculus
tensor-calculus
edited 2 hours ago
Szabolcs Berecz
1031
asked 2 hours ago
coobitcoobit
350110
edited 2 hours ago
Szabolcs Berecz
1031
asked 2 hours ago
coobitcoobit
350110
edited 2 hours ago
Szabolcs Berecz
1031
edited 2 hours ago
Szabolcs Berecz
1031
edited 2 hours ago
Szabolcs Berecz
1031
1031
asked 2 hours ago
coobitcoobit
350110
asked 2 hours ago
coobitcoobit
350110
asked 2 hours ago
coobitcoobit
350110
350110
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
“Scalar”, “vector”, and “tensor” have no meaning without specifying the group of transformations. In physics we focus on groups such as rotations, Galilean transformations, Lorentz transformations, Poincaire transformations, and gauge transformations because these are symmetries of various physical theories, built in to reflect symmetries of the natural world.
The length of a writing pen is a scalar under rotations and Galilean transformations. This is a significant physical fact about our world.
But the fact that you can measure its length in various units is not significant, because units are inventions of humans, not of Nature. Physicists never say that the length of a writing pen “transforms” because you can choose to measure it in different length units. Different units such as inches and centimeters for a particular physical quantity like length do not have any physical significance at all.
Going back to your original question, the difference between a scalar and a vector under rotations should now be obvious: a scalar is a single number that stays the same under a rotation, while a vector is a directed quantity that requires three numbers to describe it, and under rotations these numbers transform into linear combinations of each other, as specified by the relevant rotation matrix.
Under any other transformation group, the distinction between scalars and vectors is similar.
answered 2 hours ago
G. SmithG. Smith
10.2k11428
$endgroup$
$begingroup$
I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
$endgroup$
– coobit
55 mins ago
$begingroup$
@coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
$endgroup$
– Chiral Anomaly
44 mins ago
$begingroup$
Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
$endgroup$
– G. Smith
19 mins ago
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
“Scalar”, “vector”, and “tensor” have no meaning without specifying the group of transformations. In physics we focus on groups such as rotations, Galilean transformations, Lorentz transformations, Poincaire transformations, and gauge transformations because these are symmetries of various physical theories, built in to reflect symmetries of the natural world.
The length of a writing pen is a scalar under rotations and Galilean transformations. This is a significant physical fact about our world.
But the fact that you can measure its length in various units is not significant, because units are inventions of humans, not of Nature. Physicists never say that the length of a writing pen “transforms” because you can choose to measure it in different length units. Different units such as inches and centimeters for a particular physical quantity like length do not have any physical significance at all.
Going back to your original question, the difference between a scalar and a vector under rotations should now be obvious: a scalar is a single number that stays the same under a rotation, while a vector is a directed quantity that requires three numbers to describe it, and under rotations these numbers transform into linear combinations of each other, as specified by the relevant rotation matrix.
Under any other transformation group, the distinction between scalars and vectors is similar.
answered 2 hours ago
G. SmithG. Smith
10.2k11428
$endgroup$
$begingroup$
I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
$endgroup$
– coobit
55 mins ago
$begingroup$
@coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
$endgroup$
– Chiral Anomaly
44 mins ago
$begingroup$
Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
$endgroup$
– G. Smith
19 mins ago
add a comment |
$begingroup$
“Scalar”, “vector”, and “tensor” have no meaning without specifying the group of transformations. In physics we focus on groups such as rotations, Galilean transformations, Lorentz transformations, Poincaire transformations, and gauge transformations because these are symmetries of various physical theories, built in to reflect symmetries of the natural world.
The length of a writing pen is a scalar under rotations and Galilean transformations. This is a significant physical fact about our world.
But the fact that you can measure its length in various units is not significant, because units are inventions of humans, not of Nature. Physicists never say that the length of a writing pen “transforms” because you can choose to measure it in different length units. Different units such as inches and centimeters for a particular physical quantity like length do not have any physical significance at all.
Going back to your original question, the difference between a scalar and a vector under rotations should now be obvious: a scalar is a single number that stays the same under a rotation, while a vector is a directed quantity that requires three numbers to describe it, and under rotations these numbers transform into linear combinations of each other, as specified by the relevant rotation matrix.
Under any other transformation group, the distinction between scalars and vectors is similar.
answered 2 hours ago
G. SmithG. Smith
10.2k11428
$endgroup$
$begingroup$
I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
$endgroup$
– coobit
55 mins ago
$begingroup$
@coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
$endgroup$
– Chiral Anomaly
44 mins ago
$begingroup$
Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
$endgroup$
– G. Smith
19 mins ago
add a comment |
$begingroup$
“Scalar”, “vector”, and “tensor” have no meaning without specifying the group of transformations. In physics we focus on groups such as rotations, Galilean transformations, Lorentz transformations, Poincaire transformations, and gauge transformations because these are symmetries of various physical theories, built in to reflect symmetries of the natural world.
The length of a writing pen is a scalar under rotations and Galilean transformations. This is a significant physical fact about our world.
But the fact that you can measure its length in various units is not significant, because units are inventions of humans, not of Nature. Physicists never say that the length of a writing pen “transforms” because you can choose to measure it in different length units. Different units such as inches and centimeters for a particular physical quantity like length do not have any physical significance at all.
Going back to your original question, the difference between a scalar and a vector under rotations should now be obvious: a scalar is a single number that stays the same under a rotation, while a vector is a directed quantity that requires three numbers to describe it, and under rotations these numbers transform into linear combinations of each other, as specified by the relevant rotation matrix.
Under any other transformation group, the distinction between scalars and vectors is similar.
answered 2 hours ago
G. SmithG. Smith
10.2k11428
$endgroup$
“Scalar”, “vector”, and “tensor” have no meaning without specifying the group of transformations. In physics we focus on groups such as rotations, Galilean transformations, Lorentz transformations, Poincaire transformations, and gauge transformations because these are symmetries of various physical theories, built in to reflect symmetries of the natural world.
The length of a writing pen is a scalar under rotations and Galilean transformations. This is a significant physical fact about our world.
But the fact that you can measure its length in various units is not significant, because units are inventions of humans, not of Nature. Physicists never say that the length of a writing pen “transforms” because you can choose to measure it in different length units. Different units such as inches and centimeters for a particular physical quantity like length do not have any physical significance at all.
Going back to your original question, the difference between a scalar and a vector under rotations should now be obvious: a scalar is a single number that stays the same under a rotation, while a vector is a directed quantity that requires three numbers to describe it, and under rotations these numbers transform into linear combinations of each other, as specified by the relevant rotation matrix.
Under any other transformation group, the distinction between scalars and vectors is similar.
answered 2 hours ago
G. SmithG. Smith
10.2k11428
edited 1 hour ago
answered 2 hours ago
G. SmithG. Smith
10.2k11428
answered 2 hours ago
G. SmithG. Smith
10.2k11428
answered 2 hours ago
G. SmithG. Smith
10.2k11428
10.2k11428
$begingroup$
I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
$endgroup$
– coobit
55 mins ago
$begingroup$
@coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
$endgroup$
– Chiral Anomaly
44 mins ago
$begingroup$
Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
$endgroup$
– G. Smith
19 mins ago
add a comment |
$begingroup$
I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
$endgroup$
– coobit
55 mins ago
$begingroup$
@coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
$endgroup$
– Chiral Anomaly
44 mins ago
$begingroup$
Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
$endgroup$
– G. Smith
19 mins ago
$begingroup$
I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
$endgroup$
– coobit
55 mins ago
$begingroup$
I'm sorry if it might sound dumb, but ... Is 1D vector invariant under rotation? I mean is there rotation in 1D space? If so how it's different from scalar?
$endgroup$
– coobit
55 mins ago
$begingroup$
@coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
$endgroup$
– Chiral Anomaly
44 mins ago
$begingroup$
@coobit Consider the group of reflections along that one dimension. A vector changes sign, but a scalar doesn't.
$endgroup$
– Chiral Anomaly
44 mins ago
$begingroup$
Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
$endgroup$
– G. Smith
19 mins ago
$begingroup$
Whoops, I completely overlooked the fact that you were asking about 1D. (Since you had referred to scalars as rank 0, I was thinking "rank 1" , not "1D", when you said "vector".) There are no proper rotations in 1D. As @ChiralAnomaly explains, you can consider 1D reflections, and scalar and vectors transform differently under these, even though both are only a single number.
$endgroup$
– G. Smith
19 mins ago
add a comment |
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