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How do I verify a vector identity using Mathematica?
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I am trying to verify well known Frenet–Serret formulas in general setting using Mathematica.
I need to consider a general space curve $r(s)=(x(s),y(s),z(s))$ in $mathbbR^3$ and define its unit tangent $T$ and unit normal $N$ vectors by $$T=dfracdrds$$ and $$dfracdTds=kappa N.$$ Then I define the unit binomial vector $B$ in such a way that $$dfracdNds=-kappa T+tau B$$ and now I need to find an expression for $B$ only interms of derivative of $r(s)$ using this formula as the definition. The difficulty to do this computation by hand arise as formulas for $kappa, tau$ are too long.
However I do not know how I can do this symbolic computations through Mathematica. I tried to find a reference or any computation of this kind that I can use as an example, but couldn't succeed. Any help that you can do is highly appreciated.
symbolic computational-geometry vector-calculus geometric-computation
edited 3 mins ago
chris
12k440107
asked 4 hours ago
Bumblebee
20217
add a comment |Â
up vote
2
down vote
favorite
I am trying to verify well known Frenet–Serret formulas in general setting using Mathematica.
I need to consider a general space curve $r(s)=(x(s),y(s),z(s))$ in $mathbbR^3$ and define its unit tangent $T$ and unit normal $N$ vectors by $$T=dfracdrds$$ and $$dfracdTds=kappa N.$$ Then I define the unit binomial vector $B$ in such a way that $$dfracdNds=-kappa T+tau B$$ and now I need to find an expression for $B$ only interms of derivative of $r(s)$ using this formula as the definition. The difficulty to do this computation by hand arise as formulas for $kappa, tau$ are too long.
However I do not know how I can do this symbolic computations through Mathematica. I tried to find a reference or any computation of this kind that I can use as an example, but couldn't succeed. Any help that you can do is highly appreciated.
symbolic computational-geometry vector-calculus geometric-computation
edited 3 mins ago
chris
12k440107
asked 4 hours ago
Bumblebee
20217
1
What code have you tried thus far e,g, to set up the equations?
– Daniel Lichtblau
2 hours ago
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am trying to verify well known Frenet–Serret formulas in general setting using Mathematica.
I need to consider a general space curve $r(s)=(x(s),y(s),z(s))$ in $mathbbR^3$ and define its unit tangent $T$ and unit normal $N$ vectors by $$T=dfracdrds$$ and $$dfracdTds=kappa N.$$ Then I define the unit binomial vector $B$ in such a way that $$dfracdNds=-kappa T+tau B$$ and now I need to find an expression for $B$ only interms of derivative of $r(s)$ using this formula as the definition. The difficulty to do this computation by hand arise as formulas for $kappa, tau$ are too long.
However I do not know how I can do this symbolic computations through Mathematica. I tried to find a reference or any computation of this kind that I can use as an example, but couldn't succeed. Any help that you can do is highly appreciated.
symbolic computational-geometry vector-calculus geometric-computation
edited 3 mins ago
chris
12k440107
asked 4 hours ago
Bumblebee
20217
I am trying to verify well known Frenet–Serret formulas in general setting using Mathematica.
I need to consider a general space curve $r(s)=(x(s),y(s),z(s))$ in $mathbbR^3$ and define its unit tangent $T$ and unit normal $N$ vectors by $$T=dfracdrds$$ and $$dfracdTds=kappa N.$$ Then I define the unit binomial vector $B$ in such a way that $$dfracdNds=-kappa T+tau B$$ and now I need to find an expression for $B$ only interms of derivative of $r(s)$ using this formula as the definition. The difficulty to do this computation by hand arise as formulas for $kappa, tau$ are too long.
However I do not know how I can do this symbolic computations through Mathematica. I tried to find a reference or any computation of this kind that I can use as an example, but couldn't succeed. Any help that you can do is highly appreciated.
symbolic computational-geometry vector-calculus geometric-computation
symbolic computational-geometry vector-calculus geometric-computation
edited 3 mins ago
chris
12k440107
asked 4 hours ago
Bumblebee
20217
edited 3 mins ago
chris
12k440107
asked 4 hours ago
Bumblebee
20217
edited 3 mins ago
chris
12k440107
edited 3 mins ago
chris
12k440107
edited 3 mins ago
chris
12k440107
12k440107
asked 4 hours ago
Bumblebee
20217
asked 4 hours ago
Bumblebee
20217
asked 4 hours ago
Bumblebee
20217
20217
1
What code have you tried thus far e,g, to set up the equations?
– Daniel Lichtblau
2 hours ago
add a comment |Â
1
What code have you tried thus far e,g, to set up the equations?
– Daniel Lichtblau
2 hours ago
1
1
What code have you tried thus far e,g, to set up the equations?
– Daniel Lichtblau
2 hours ago
What code have you tried thus far e,g, to set up the equations?
– Daniel Lichtblau
2 hours ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
You can use FrenetSerretSystem:
FrenetSerretSystem[x[s], y[s], z[s], s][[-1, -1]] //TeXForm
$leftfracy'(s) z''(s)-y''(s) z'(s)sqrtleft(x'(s) y''(s)-x''(s) y'(s)right)^2+left(x''(s) z'(s)-x'(s)
z''(s)right)^2+left(y'(s) z''(s)-y''(s) z'(s)right)^2,fracx''(s) z'(s)-x'(s) z''(s)sqrtleft(x'(s)
y''(s)-x''(s) y'(s)right)^2+left(x''(s) z'(s)-x'(s) z''(s)right)^2+left(y'(s) z''(s)-y''(s)
z'(s)right)^2,fracx'(s) y''(s)-x''(s) y'(s)sqrtleft(x'(s) y''(s)-x''(s) y'(s)right)^2+left(x''(s)
z'(s)-x'(s) z''(s)right)^2+left(y'(s) z''(s)-y''(s) z'(s)right)^2right$
answered 27 mins ago
Carl Woll
63k282163
add a comment |Â
up vote
0
down vote
Since N is a built-in symbol, we will use n instead of N. The required formulas are
r = x[s], y[s], z[s];
T = D[r, s]
[Kappa] = Norm[D[T, s]]
n = D[T, s]/[Kappa]
[Tau] = Norm[([Kappa]*T + D[n, s])]
B = ([Kappa]*T + D[n, s])/[Tau]
answered 35 mins ago
Alex Trounev
3,2701312
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
You can use FrenetSerretSystem:
FrenetSerretSystem[x[s], y[s], z[s], s][[-1, -1]] //TeXForm
$leftfracy'(s) z''(s)-y''(s) z'(s)sqrtleft(x'(s) y''(s)-x''(s) y'(s)right)^2+left(x''(s) z'(s)-x'(s)
z''(s)right)^2+left(y'(s) z''(s)-y''(s) z'(s)right)^2,fracx''(s) z'(s)-x'(s) z''(s)sqrtleft(x'(s)
y''(s)-x''(s) y'(s)right)^2+left(x''(s) z'(s)-x'(s) z''(s)right)^2+left(y'(s) z''(s)-y''(s)
z'(s)right)^2,fracx'(s) y''(s)-x''(s) y'(s)sqrtleft(x'(s) y''(s)-x''(s) y'(s)right)^2+left(x''(s)
z'(s)-x'(s) z''(s)right)^2+left(y'(s) z''(s)-y''(s) z'(s)right)^2right$
answered 27 mins ago
Carl Woll
63k282163
add a comment |Â
up vote
2
down vote
You can use FrenetSerretSystem:
FrenetSerretSystem[x[s], y[s], z[s], s][[-1, -1]] //TeXForm
$leftfracy'(s) z''(s)-y''(s) z'(s)sqrtleft(x'(s) y''(s)-x''(s) y'(s)right)^2+left(x''(s) z'(s)-x'(s)
z''(s)right)^2+left(y'(s) z''(s)-y''(s) z'(s)right)^2,fracx''(s) z'(s)-x'(s) z''(s)sqrtleft(x'(s)
y''(s)-x''(s) y'(s)right)^2+left(x''(s) z'(s)-x'(s) z''(s)right)^2+left(y'(s) z''(s)-y''(s)
z'(s)right)^2,fracx'(s) y''(s)-x''(s) y'(s)sqrtleft(x'(s) y''(s)-x''(s) y'(s)right)^2+left(x''(s)
z'(s)-x'(s) z''(s)right)^2+left(y'(s) z''(s)-y''(s) z'(s)right)^2right$
answered 27 mins ago
Carl Woll
63k282163
add a comment |Â
up vote
2
down vote
up vote
2
down vote
You can use FrenetSerretSystem:
FrenetSerretSystem[x[s], y[s], z[s], s][[-1, -1]] //TeXForm
$leftfracy'(s) z''(s)-y''(s) z'(s)sqrtleft(x'(s) y''(s)-x''(s) y'(s)right)^2+left(x''(s) z'(s)-x'(s)
z''(s)right)^2+left(y'(s) z''(s)-y''(s) z'(s)right)^2,fracx''(s) z'(s)-x'(s) z''(s)sqrtleft(x'(s)
y''(s)-x''(s) y'(s)right)^2+left(x''(s) z'(s)-x'(s) z''(s)right)^2+left(y'(s) z''(s)-y''(s)
z'(s)right)^2,fracx'(s) y''(s)-x''(s) y'(s)sqrtleft(x'(s) y''(s)-x''(s) y'(s)right)^2+left(x''(s)
z'(s)-x'(s) z''(s)right)^2+left(y'(s) z''(s)-y''(s) z'(s)right)^2right$
answered 27 mins ago
Carl Woll
63k282163
You can use FrenetSerretSystem:
FrenetSerretSystem[x[s], y[s], z[s], s][[-1, -1]] //TeXForm
$leftfracy'(s) z''(s)-y''(s) z'(s)sqrtleft(x'(s) y''(s)-x''(s) y'(s)right)^2+left(x''(s) z'(s)-x'(s)
z''(s)right)^2+left(y'(s) z''(s)-y''(s) z'(s)right)^2,fracx''(s) z'(s)-x'(s) z''(s)sqrtleft(x'(s)
y''(s)-x''(s) y'(s)right)^2+left(x''(s) z'(s)-x'(s) z''(s)right)^2+left(y'(s) z''(s)-y''(s)
z'(s)right)^2,fracx'(s) y''(s)-x''(s) y'(s)sqrtleft(x'(s) y''(s)-x''(s) y'(s)right)^2+left(x''(s)
z'(s)-x'(s) z''(s)right)^2+left(y'(s) z''(s)-y''(s) z'(s)right)^2right$
answered 27 mins ago
Carl Woll
63k282163
answered 27 mins ago
Carl Woll
63k282163
answered 27 mins ago
Carl Woll
63k282163
answered 27 mins ago
Carl Woll
63k282163
63k282163
add a comment |Â
add a comment |Â
up vote
0
down vote
Since N is a built-in symbol, we will use n instead of N. The required formulas are
r = x[s], y[s], z[s];
T = D[r, s]
[Kappa] = Norm[D[T, s]]
n = D[T, s]/[Kappa]
[Tau] = Norm[([Kappa]*T + D[n, s])]
B = ([Kappa]*T + D[n, s])/[Tau]
answered 35 mins ago
Alex Trounev
3,2701312
add a comment |Â
up vote
0
down vote
Since N is a built-in symbol, we will use n instead of N. The required formulas are
r = x[s], y[s], z[s];
T = D[r, s]
[Kappa] = Norm[D[T, s]]
n = D[T, s]/[Kappa]
[Tau] = Norm[([Kappa]*T + D[n, s])]
B = ([Kappa]*T + D[n, s])/[Tau]
answered 35 mins ago
Alex Trounev
3,2701312
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Since N is a built-in symbol, we will use n instead of N. The required formulas are
r = x[s], y[s], z[s];
T = D[r, s]
[Kappa] = Norm[D[T, s]]
n = D[T, s]/[Kappa]
[Tau] = Norm[([Kappa]*T + D[n, s])]
B = ([Kappa]*T + D[n, s])/[Tau]
answered 35 mins ago
Alex Trounev
3,2701312
Since N is a built-in symbol, we will use n instead of N. The required formulas are
r = x[s], y[s], z[s];
T = D[r, s]
[Kappa] = Norm[D[T, s]]
n = D[T, s]/[Kappa]
[Tau] = Norm[([Kappa]*T + D[n, s])]
B = ([Kappa]*T + D[n, s])/[Tau]
answered 35 mins ago
Alex Trounev
3,2701312
answered 35 mins ago
Alex Trounev
3,2701312
answered 35 mins ago
Alex Trounev
3,2701312
answered 35 mins ago
Alex Trounev
3,2701312
3,2701312
add a comment |Â
add a comment |Â
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1
What code have you tried thus far e,g, to set up the equations?
– Daniel Lichtblau
2 hours ago