Rod of length 4. One end on z-axis. One end on sphere of radius sqrt(3) centered at origin. Find Tangent...
Question: Considered Manifold M in R^4 of the set of positions of a rod of length 4 where one end lies on the positive z-axis
and the other end lies on a sphere of radius sqrt(3) centered at the origin.
Find the tangent space to M at the point (1,1,1) on the sphere.
I've approached this problem two ways and have come up with two different answers so I'm looking for some help.
[First Method] is a variation of the implicit function theorem. I tried to define M as a system of functions
F1(x1, x2, x3) are the set of points that would be on the sphere.
F2(x4) is the coordinate of the end of the rod that is on the z-axis. I used geometry to find x4 if I'm given x1, x2, x3.
Essentially, if I'm given the coordinates of one end on the sphere then implicitly I know the coordinate the other end is on the z-axis.
F3(x1, x2, x3, x4) is the distance between the ends of the rods in 3-space.
I defined M = (F1, F2, F3), took the Jacabion, then found the kernel. My result was a single vector.
[Second Method] I tried to parameterize the 3 functions F1, F2, F3.
I expressed x1, x2, x3, x4 in terms of parameters phi and theta.
I found the Jacobian of the parameterized matrix I defined.
Then used subsition to find the phi and theta values for the point (1,1,1) on the sphere.
This resulted in two vectors. First one is the partial derivative with respect to phi, second is the partial derivative with respect to theta.
The Tangent Space is the span of these two vectors.
Sorry for the long post but I'm kind of lost and not sure how to solve this problem.
Any help or insight is greatly appreciated.
multivariable-calculus manifolds
asked Nov 24 at 4:32
Michael Romero Jr.
113
add a comment |
Question: Considered Manifold M in R^4 of the set of positions of a rod of length 4 where one end lies on the positive z-axis
and the other end lies on a sphere of radius sqrt(3) centered at the origin.
Find the tangent space to M at the point (1,1,1) on the sphere.
I've approached this problem two ways and have come up with two different answers so I'm looking for some help.
[First Method] is a variation of the implicit function theorem. I tried to define M as a system of functions
F1(x1, x2, x3) are the set of points that would be on the sphere.
F2(x4) is the coordinate of the end of the rod that is on the z-axis. I used geometry to find x4 if I'm given x1, x2, x3.
Essentially, if I'm given the coordinates of one end on the sphere then implicitly I know the coordinate the other end is on the z-axis.
F3(x1, x2, x3, x4) is the distance between the ends of the rods in 3-space.
I defined M = (F1, F2, F3), took the Jacabion, then found the kernel. My result was a single vector.
[Second Method] I tried to parameterize the 3 functions F1, F2, F3.
I expressed x1, x2, x3, x4 in terms of parameters phi and theta.
I found the Jacobian of the parameterized matrix I defined.
Then used subsition to find the phi and theta values for the point (1,1,1) on the sphere.
This resulted in two vectors. First one is the partial derivative with respect to phi, second is the partial derivative with respect to theta.
The Tangent Space is the span of these two vectors.
Sorry for the long post but I'm kind of lost and not sure how to solve this problem.
Any help or insight is greatly appreciated.
multivariable-calculus manifolds
asked Nov 24 at 4:32
Michael Romero Jr.
113
add a comment |
Question: Considered Manifold M in R^4 of the set of positions of a rod of length 4 where one end lies on the positive z-axis
and the other end lies on a sphere of radius sqrt(3) centered at the origin.
Find the tangent space to M at the point (1,1,1) on the sphere.
I've approached this problem two ways and have come up with two different answers so I'm looking for some help.
[First Method] is a variation of the implicit function theorem. I tried to define M as a system of functions
F1(x1, x2, x3) are the set of points that would be on the sphere.
F2(x4) is the coordinate of the end of the rod that is on the z-axis. I used geometry to find x4 if I'm given x1, x2, x3.
Essentially, if I'm given the coordinates of one end on the sphere then implicitly I know the coordinate the other end is on the z-axis.
F3(x1, x2, x3, x4) is the distance between the ends of the rods in 3-space.
I defined M = (F1, F2, F3), took the Jacabion, then found the kernel. My result was a single vector.
[Second Method] I tried to parameterize the 3 functions F1, F2, F3.
I expressed x1, x2, x3, x4 in terms of parameters phi and theta.
I found the Jacobian of the parameterized matrix I defined.
Then used subsition to find the phi and theta values for the point (1,1,1) on the sphere.
This resulted in two vectors. First one is the partial derivative with respect to phi, second is the partial derivative with respect to theta.
The Tangent Space is the span of these two vectors.
Sorry for the long post but I'm kind of lost and not sure how to solve this problem.
Any help or insight is greatly appreciated.
multivariable-calculus manifolds
asked Nov 24 at 4:32
Michael Romero Jr.
113
Question: Considered Manifold M in R^4 of the set of positions of a rod of length 4 where one end lies on the positive z-axis
and the other end lies on a sphere of radius sqrt(3) centered at the origin.
Find the tangent space to M at the point (1,1,1) on the sphere.
I've approached this problem two ways and have come up with two different answers so I'm looking for some help.
[First Method] is a variation of the implicit function theorem. I tried to define M as a system of functions
F1(x1, x2, x3) are the set of points that would be on the sphere.
F2(x4) is the coordinate of the end of the rod that is on the z-axis. I used geometry to find x4 if I'm given x1, x2, x3.
Essentially, if I'm given the coordinates of one end on the sphere then implicitly I know the coordinate the other end is on the z-axis.
F3(x1, x2, x3, x4) is the distance between the ends of the rods in 3-space.
I defined M = (F1, F2, F3), took the Jacabion, then found the kernel. My result was a single vector.
[Second Method] I tried to parameterize the 3 functions F1, F2, F3.
I expressed x1, x2, x3, x4 in terms of parameters phi and theta.
I found the Jacobian of the parameterized matrix I defined.
Then used subsition to find the phi and theta values for the point (1,1,1) on the sphere.
This resulted in two vectors. First one is the partial derivative with respect to phi, second is the partial derivative with respect to theta.
The Tangent Space is the span of these two vectors.
Sorry for the long post but I'm kind of lost and not sure how to solve this problem.
Any help or insight is greatly appreciated.
multivariable-calculus manifolds
multivariable-calculus manifolds
asked Nov 24 at 4:32
Michael Romero Jr.
113
asked Nov 24 at 4:32
Michael Romero Jr.
113
asked Nov 24 at 4:32
Michael Romero Jr.
113
asked Nov 24 at 4:32
Michael Romero Jr.
113
asked Nov 24 at 4:32
Michael Romero Jr.
113
113
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