## Invariants of Quadratic Differential Forms |

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2Fdudv A. N. Whitehead algebraic forms algebraic invariants angle application arbitrary functions calculated Christoffel coefficients congruence consider const contravariant system coordinates corresponding covariant and contravariant covariant derivatives covariant differentiation covariant system curves f deduce denote determined differential equations differential invariants differential parameters Edu2 equivalent symbols Euclidean space example expressed in terms finite follows form F form Gt functions f fundamental form Gaussian invariants geodesic curvature geometrical interpretation given gkhi Hence higher derivatives increments independent variables infinitesimal transformation invariant theory invariantive constituent invariants of order Jacobian linear transformation manifold Maschke Math method necessary and sufficient orthogonal ennuple orthogonal trajectories parametric curves particular plane quadratic differential form quadrilinear form quantities rank zero relations Riemann symbols satisfied second derivatives second order set of equations set of variables Similarly solution suppose system of order theorem third derivatives trans values variant

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Page 18 - I,Il,...ofthe algebraic forms F, Gt,... are a complete system of relative differential invariants for the quadratic differential form F, and if under any transformation such an invariant I becomes kl, then k is some power of the Jacobian of the transformation. If we take account of differential invariants which involve the magnitudes dx themselves...

Page 58 - Gdv2 = 0, that is to say d/ = 0. The curves are hence such that the distance between any two points on one of them, measured along that curve, is zero. Consider the more general case of A<£ =/(<£). Let the curves ^ be chosen at right angles to the curves <£, then A (<£, <//) = 0, and therefore or if du = d<f>/\/f(4>), ds?

Page 80 - ... Its position, and therefore the coordinates of all its points, are known when we know the coordinates of its middle point and the angle it makes with the x axis. The position of a sphere in space is determined by the coordinates of its centre, the angle that a line fixed in it makes with the z axis, the angle that a vertical plane through the line makes with a fixed plane, and the angle that a second line, fixed in the sphere at right angles to the first, makes with the line that is in the vertical...

Page 53 - ... an example of this has already been given in the case of the rolls for the year 19 Edward I.

Page 14 - F' is possible. We thus have the important result : The necessary and sufficient conditions in order that it shall be possible to transform a quadratic form F into another quadratic form dot F...

Page 78 - It is identically zero if [ra] is geodesic. (2) Its projection on the plane tangent to the lines i and n is equal to the curvature of the projection of the line n on the same plane. (3) It is normal to the line n. For these reasons y is called the geodesic curvature of the line n, and the congruence p.

Page 67 - ... thus associated a direction with each point in the manifold, and, if we start from a given point and proceed always in the direction associated with the point reached, we finally obtain a curve in the manifold. Hence the equations given above define a congruence of curves in the manifold, such that there is one and only one curve through each point of the manifold. Let /* define another such congruence, then the cosine of the angle between two intersecting curves, one belonging to the congruence...

Page 33 - ... transformations is then extended so as to include the transformation equations for the new variables introduced and their derivatives, and our problem is to determine all the invariants of this extended group. To do this it is necessary to obtain the infinitesimal transformations of the group, and from these we obtain a complete system of linear differential equations the solutions of which are the invariants. 34. The case of two independent variables. As a first case we consider a quadratic...

Page 37 - ... the square of the element of length of an w-fold manifold in ordinary (Euclidean) space of n + r dimensions. 38. Differential parameters for forms of rank zero. Let the form be of rank zero, then we have nn 2 arsdxrdxs = 2...

Page 80 - ... to certain dynamical restrictions in a particular problem, some changes in the coordinates do not give possible displacements. For example, if a sphere is moving on a fixed plane with pure rolling motion, all infinitesimal displacements are excluded which do not make the displacement of the point of contact of the sphere with the plane zero (to first order of small quantities).