Items tagged with differential-geometry

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Unfortunately, the Differential Geometry package is too difficult for non-mathematicians. Is there a package in the Maple for classical differential geometry?

Is there geometric or statistical meaning for ln(dy/dx) = 0?

is there any feature in vector field plot when ln(dy/dx) = 0?

How to manually switch frames in the DifferentialGeometry package? (Example: Help_Differential geometry_lesson2_exercise 4). Thank you

How to prove or disprove the flatness of the surface x = (u-v)^2, y =  u^2-3*v^2, z = (1/2)*v*(u-2*v), where u and v are real-valued parameters? Here is my try:

 

plot3d([(u-v)^2, u^2-3*v^2, (1/2)*v*(u-2*v)], u = -1 .. 1, v = -1 .. 1, axes = frame);plot3d([(u-v)^2, u^2-3*v^2, (1/2)*v*(u-2*v)], u = -1 .. 1, v = -1 .. 1, axes = frame)

 

eliminate([x = (u-v)^2, y = u^2-3*v^2, z = (1/2)*v*(u-2*v)], [u, v])

[{u = -2*(-x+2*z+(x^2-8*x*z)^(1/2))/(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = (1/2)*(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {-(x^2-8*x*z)^(1/2)*x+y*(x^2-8*x*z)^(1/2)-4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}], [{u = 2*(-x+2*z+(x^2-8*x*z)^(1/2))/(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = -(1/2)*(-2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {-(x^2-8*x*z)^(1/2)*x+y*(x^2-8*x*z)^(1/2)-4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}], [{u = -2*(x-2*z+(x^2-8*x*z)^(1/2))/(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = -(1/2)*(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {(x^2-8*x*z)^(1/2)*x-y*(x^2-8*x*z)^(1/2)+4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}], [{u = 2*(x-2*z+(x^2-8*x*z)^(1/2))/(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2), v = (1/2)*(2*(x^2-8*x*z)^(1/2)+2*x-8*z)^(1/2)}, {(x^2-8*x*z)^(1/2)*x-y*(x^2-8*x*z)^(1/2)+4*(x^2-8*x*z)^(1/2)*z+x^2-y*x+4*y*z-16*z^2}]

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NULL

I think Gaussian curvature should be used to this end. Dr. Robert J. Lopez is my hope.

Download flat.mw

I would like work in Riemann Normal Coordinates, and derive expansion in number of derivatives of the metric.

For example, starting with the expansion of the metric in Riemann Normal Coordinates (assuming this needs not be derived)

\displaystyle  \begin{array}{rcl}  g_{ij}(x)&=& \delta_{ij} -\frac 1 3 R_{iklj}x^kx^l -\frac 1 6 R_{iklj;m} x^kx^lx^m\\ &&+ (\frac2{45} R_{ilmk}R_{jpqk}- \frac 1 {20} R_{ilmj;pq})x^lx^m x^p x^q\\ &&+(-\frac 1{90} R_{iklj;mpq}+\frac 2{45} R_{iklr;m}R_{jpqr})x^kx^lx^mx^px^q\\ && +(-\frac 1 {504}R_{iklj;mpqr}+ \frac{17}{1260}R_{ikls;pq}R_{jmps}+ \frac{11}{1008}R_{ikls;q}R_{jmps;r}\\ && +\frac 1{315}R_{ilms}R_{jqrt}R_{kspt})x^kx^lx^mx^px^qx^r +O(|x|^7). \end{array}

I would like to express the lapliacian, or square-root of the determinant ... etc., in terms of this metric and derive an expansion for them.

However, I cannot even define the metric as such in Maple-Physics, because the coefficients of xk depend themselfs upon the metric to be defined.

Is there a way to do such calculations in Maple ?

Dear All

Using Lie algebra package in Maple we can easily find nilradical for given abstract algebra, but how we can find all the ideal in lower central series by taking new basis as nilradical itself?

Please see following;

 

with(DifferentialGeometry); with(LieAlgebras)

DGsetup([x, y, t, u, v])

`frame name: Euc`

(1)
Euc > 

VectorFields := evalDG([D_v, D_v*x+D_y*t, 2*D_t*t-2*D_u*u-D_v*v+D_y*y, t*D_v, D_v*y+D_u, D_t, D_x, D_x*t+D_u, 2*D_v*x+D_x*y, -D_t*t+2*D_u*u+2*D_v*v+D_x*x, D_y])

[_DG([["vector", "Euc", []], [[[5], 1]]]), _DG([["vector", "Euc", []], [[[2], t], [[5], x]]]), _DG([["vector", "Euc", []], [[[2], y], [[3], 2*t], [[4], -2*u], [[5], -v]]]), _DG([["vector", "Euc", []], [[[5], t]]]), _DG([["vector", "Euc", []], [[[4], 1], [[5], y]]]), _DG([["vector", "Euc", []], [[[3], 1]]]), _DG([["vector", "Euc", []], [[[1], 1]]]), _DG([["vector", "Euc", []], [[[1], t], [[4], 1]]]), _DG([["vector", "Euc", []], [[[1], y], [[5], 2*x]]]), _DG([["vector", "Euc", []], [[[1], x], [[3], -t], [[4], 2*u], [[5], 2*v]]]), _DG([["vector", "Euc", []], [[[2], 1]]])]

(2)
Euc > 

L1 := LieAlgebraData(VectorFields)

_DG([["LieAlgebra", "L1", [11]], [[[1, 3, 1], -1], [[1, 10, 1], 2], [[2, 3, 2], -1], [[2, 5, 4], 1], [[2, 6, 11], -1], [[2, 7, 1], -1], [[2, 8, 4], -1], [[2, 9, 5], -1], [[2, 9, 8], 1], [[2, 10, 2], 1], [[3, 4, 4], 3], [[3, 5, 5], 2], [[3, 6, 6], -2], [[3, 8, 8], 2], [[3, 9, 9], 1], [[3, 11, 11], -1], [[4, 6, 1], -1], [[4, 10, 4], 3], [[5, 10, 5], 2], [[5, 11, 1], -1], [[6, 8, 7], 1], [[6, 10, 6], -1], [[7, 9, 1], 2], [[7, 10, 7], 1], [[8, 9, 4], 2], [[8, 10, 8], 2], [[9, 10, 9], 1], [[9, 11, 7], -1]]])

(3)
Euc > 

DGsetup(L1)

`Lie algebra: L1`

(4)
L1 > 

MultiplicationTable("LieTable"):

L1 > 

N := Nilradical(L1)

[_DG([["vector", "L1", []], [[[1], 1]]]), _DG([["vector", "L1", []], [[[2], 1]]]), _DG([["vector", "L1", []], [[[4], 1]]]), _DG([["vector", "L1", []], [[[5], 1]]]), _DG([["vector", "L1", []], [[[6], 1]]]), _DG([["vector", "L1", []], [[[7], 1]]]), _DG([["vector", "L1", []], [[[8], 1]]]), _DG([["vector", "L1", []], [[[9], 1]]]), _DG([["vector", "L1", []], [[[11], 1]]])]

(5)
L1 > 

Query(N, "Nilpotent")

true

(6)
L1 > 

Query(N, "Solvable")

true

(7)

Taking N as new basis , how we can find all ideals in lower central series of this solvable ideal N?

 

Download [944]_Structure_of_Lie_algebra.mw

Regards

Dear All

I intend to convert Lie algebra into abstract Lie algebra with the aid of Maple packages DifferentialGeometry and Lie algebras. I want to do this for classification of Lie algebra for group inavriant solution, can anybody help me out.....?

Regards

Hi;

Could anyone help me to compute the lie derivative of the function h:R^3-->R with respect to the vector-valued function f:R^5-->R^3 below?

f(x,y,L,u,v) = [x + u;
y + v;
L]

and

h(x,y,L) = sqrt((y-x)^2 + (L)^2)
 

Thank you in advance.

 

 

 

 

I noticed exactly the same expression in Help file of Maple2015.2 and Online Help (http://www.maplesoft.com/support/help/Maple/view.aspx?path=DifferentialGeometry/DGsetup). 

Currently Online Help of DGsetup has been reverted to 'normal expression' and we cannot see the expression anymore by Online Help.  Help of Maple2015.2 is still kept unchanged or possibly awaiting update, though such expression has never appeared with Maple2015.2 runs.

Maple18 had a minor update of 18.02. Maple18.02 in hand does not produce such expression. Had you tried it by Maple18?    I am rather interested in the intension of developer to modify the original familiar expression.

Best regards, Stev Eland

I would like to pay attention to http://www.scientificcomputing.com/atlas-032408.aspx . It seems to be a powerful tool to research, to teach, and to learn.

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