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Proof of curvature of vector and orthogonality of vector derivative to itself
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In the proof of curvature of a vector r(t) we take the first derivative of r(t) to be orthogonal to the vector itself. But isn't it true only for r(t) with constant magnitude?
calculus
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Srilakshmidaran
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In the proof of curvature of a vector r(t) we take the first derivative of r(t) to be orthogonal to the vector itself. But isn't it true only for r(t) with constant magnitude?
calculus
asked yesterday
Srilakshmidaran
12
New contributor
Srilakshmidaran is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Yes. In general $r'(t)$ is not orthogonal to $r(t)$ itself. It happens if and only if the norm of $r(t)$ is constant.
– Dog_69
yesterday
Then how do we apply it to a curvature where r(t) changes?
– Srilakshmidaran
yesterday
The general expression for the curvature is $$kappa(t)=frac{|r'(t)times r''(t)|}{|r'(t)|^3} .$$
– Dog_69
yesterday
add a comment |
up vote
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up vote
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down vote
favorite
In the proof of curvature of a vector r(t) we take the first derivative of r(t) to be orthogonal to the vector itself. But isn't it true only for r(t) with constant magnitude?
calculus
asked yesterday
Srilakshmidaran
12
New contributor
Srilakshmidaran is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
In the proof of curvature of a vector r(t) we take the first derivative of r(t) to be orthogonal to the vector itself. But isn't it true only for r(t) with constant magnitude?
calculus
calculus
asked yesterday
Srilakshmidaran
12
New contributor
Srilakshmidaran is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Srilakshmidaran
12
New contributor
Srilakshmidaran is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Srilakshmidaran
12
New contributor
Srilakshmidaran is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
Srilakshmidaran
12
asked yesterday
Srilakshmidaran
12
12
New contributor
Srilakshmidaran is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Srilakshmidaran is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Srilakshmidaran is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Yes. In general $r'(t)$ is not orthogonal to $r(t)$ itself. It happens if and only if the norm of $r(t)$ is constant.
– Dog_69
yesterday
Then how do we apply it to a curvature where r(t) changes?
– Srilakshmidaran
yesterday
The general expression for the curvature is $$kappa(t)=frac{|r'(t)times r''(t)|}{|r'(t)|^3} .$$
– Dog_69
yesterday
add a comment |
Yes. In general $r'(t)$ is not orthogonal to $r(t)$ itself. It happens if and only if the norm of $r(t)$ is constant.
– Dog_69
yesterday
Then how do we apply it to a curvature where r(t) changes?
– Srilakshmidaran
yesterday
The general expression for the curvature is $$kappa(t)=frac{|r'(t)times r''(t)|}{|r'(t)|^3} .$$
– Dog_69
yesterday
Yes. In general $r'(t)$ is not orthogonal to $r(t)$ itself. It happens if and only if the norm of $r(t)$ is constant.
– Dog_69
yesterday
Yes. In general $r'(t)$ is not orthogonal to $r(t)$ itself. It happens if and only if the norm of $r(t)$ is constant.
– Dog_69
yesterday
Then how do we apply it to a curvature where r(t) changes?
– Srilakshmidaran
yesterday
Then how do we apply it to a curvature where r(t) changes?
– Srilakshmidaran
yesterday
The general expression for the curvature is $$kappa(t)=frac{|r'(t)times r''(t)|}{|r'(t)|^3} .$$
– Dog_69
yesterday
The general expression for the curvature is $$kappa(t)=frac{|r'(t)times r''(t)|}{|r'(t)|^3} .$$
– Dog_69
yesterday
add a comment |
1 Answer
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(At first, you don’t prove but define curvature instead.) Curvature deals with orthogonality, sort of: it may be defined as the normal component of the acceleration, divided by the square of the velocity.
answered yesterday
Michael Hoppe
10.6k31733
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
(At first, you don’t prove but define curvature instead.) Curvature deals with orthogonality, sort of: it may be defined as the normal component of the acceleration, divided by the square of the velocity.
answered yesterday
Michael Hoppe
10.6k31733
add a comment |
up vote
0
down vote
(At first, you don’t prove but define curvature instead.) Curvature deals with orthogonality, sort of: it may be defined as the normal component of the acceleration, divided by the square of the velocity.
answered yesterday
Michael Hoppe
10.6k31733
add a comment |
up vote
0
down vote
up vote
0
down vote
(At first, you don’t prove but define curvature instead.) Curvature deals with orthogonality, sort of: it may be defined as the normal component of the acceleration, divided by the square of the velocity.
answered yesterday
Michael Hoppe
10.6k31733
(At first, you don’t prove but define curvature instead.) Curvature deals with orthogonality, sort of: it may be defined as the normal component of the acceleration, divided by the square of the velocity.
answered yesterday
Michael Hoppe
10.6k31733
answered yesterday
Michael Hoppe
10.6k31733
answered yesterday
Michael Hoppe
10.6k31733
answered yesterday
Michael Hoppe
10.6k31733
10.6k31733
add a comment |
add a comment |
Srilakshmidaran is a new contributor. Be nice, and check out our Code of Conduct.
Srilakshmidaran is a new contributor. Be nice, and check out our Code of Conduct.
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Yes. In general $r'(t)$ is not orthogonal to $r(t)$ itself. It happens if and only if the norm of $r(t)$ is constant.
– Dog_69
yesterday
Then how do we apply it to a curvature where r(t) changes?
– Srilakshmidaran
yesterday
The general expression for the curvature is $$kappa(t)=frac{|r'(t)times r''(t)|}{|r'(t)|^3} .$$
– Dog_69
yesterday