How to prove that the angle between two vectors on a closed plane curve is given by a line integral
$begingroup$
i have to prove that for the angle between the two position vectors at the edge of a part of a closed plane curve is given by the following integral:
$theta$ $=$ $k$ *$int_c {rover r^2} times dr $
Where c is the arc of the curve, k is the curvature vector and r is the position vector.
My thinking went along the following lines: The angle is generally given as $Delta theta $ $=$ $Delta Sover rho$ where $rho$ is the curvature (so ${1over k}$) and $Delta S$ is the length of the arc. But this leads nowhere near what I want to prove.
My next thought was to divide the arc to infinitesimal vectors $r_n$ and find the total angle through the inner product of each vector. But this leads to a summation of an $arccos$ function that isn't particularly useful.
I'd appreciate your help a lot.
vector-analysis line-integrals
asked Jan 15 at 20:31
Mark KleiverMark Kleiver
11
$endgroup$
add a comment |
$begingroup$
i have to prove that for the angle between the two position vectors at the edge of a part of a closed plane curve is given by the following integral:
$theta$ $=$ $k$ *$int_c {rover r^2} times dr $
Where c is the arc of the curve, k is the curvature vector and r is the position vector.
My thinking went along the following lines: The angle is generally given as $Delta theta $ $=$ $Delta Sover rho$ where $rho$ is the curvature (so ${1over k}$) and $Delta S$ is the length of the arc. But this leads nowhere near what I want to prove.
My next thought was to divide the arc to infinitesimal vectors $r_n$ and find the total angle through the inner product of each vector. But this leads to a summation of an $arccos$ function that isn't particularly useful.
I'd appreciate your help a lot.
vector-analysis line-integrals
asked Jan 15 at 20:31
Mark KleiverMark Kleiver
11
$endgroup$
add a comment |
$begingroup$
i have to prove that for the angle between the two position vectors at the edge of a part of a closed plane curve is given by the following integral:
$theta$ $=$ $k$ *$int_c {rover r^2} times dr $
Where c is the arc of the curve, k is the curvature vector and r is the position vector.
My thinking went along the following lines: The angle is generally given as $Delta theta $ $=$ $Delta Sover rho$ where $rho$ is the curvature (so ${1over k}$) and $Delta S$ is the length of the arc. But this leads nowhere near what I want to prove.
My next thought was to divide the arc to infinitesimal vectors $r_n$ and find the total angle through the inner product of each vector. But this leads to a summation of an $arccos$ function that isn't particularly useful.
I'd appreciate your help a lot.
vector-analysis line-integrals
asked Jan 15 at 20:31
Mark KleiverMark Kleiver
11
$endgroup$
i have to prove that for the angle between the two position vectors at the edge of a part of a closed plane curve is given by the following integral:
$theta$ $=$ $k$ *$int_c {rover r^2} times dr $
Where c is the arc of the curve, k is the curvature vector and r is the position vector.
My thinking went along the following lines: The angle is generally given as $Delta theta $ $=$ $Delta Sover rho$ where $rho$ is the curvature (so ${1over k}$) and $Delta S$ is the length of the arc. But this leads nowhere near what I want to prove.
My next thought was to divide the arc to infinitesimal vectors $r_n$ and find the total angle through the inner product of each vector. But this leads to a summation of an $arccos$ function that isn't particularly useful.
I'd appreciate your help a lot.
vector-analysis line-integrals
vector-analysis line-integrals
asked Jan 15 at 20:31
Mark KleiverMark Kleiver
11
asked Jan 15 at 20:31
Mark KleiverMark Kleiver
11
asked Jan 15 at 20:31
Mark KleiverMark Kleiver
11
asked Jan 15 at 20:31
Mark KleiverMark Kleiver
11
asked Jan 15 at 20:31
Mark KleiverMark Kleiver
11
11
add a comment |
add a comment |
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