Any square matrix can be written as a sum(1)where(2)is a symmetric matrix known as the symmetric part of and(3)is an antisymmetric matrix known as the antisymmetric part of . Here, is the transpose.The symmetric part of a tensor is denoted using parenthesesas(4)(5)Symbols for the symmetric and antisymmetric partsof tensors can be combined, for example(6)(Wald 1984, p. 26).
Any square matrix can be written as a sum(1)where(2)is a symmetric matrix known as the symmetric part of and(3)is an antisymmetric matrix known as the antisymmetric part of . Here, is the transpose.Any rank-2 tensor can be written as a sum of symmetricand antisymmetric parts as(4)The antisymmetric part of a tensor is sometimes denoted using the special notation(5)For a general rank- tensor,(6)where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example(7)(Wald 1984, p. 26).
The geodesic on an oblate spheroid can be computed analytically, although the resulting expression is much more unwieldy than for a simple sphere. A spheroid with equatorial radius and polar radius can be specified parametrically by(1)(2)(3)where . Using the second partial derivatives(4)(5)(6)(7)(8)(9)gives the geodesics functions as(10)(11)(12)(13)(14)(15)where(16)is the ellipticity.Since and and are explicit functions of only, we can use the special form of the geodesic equation(17)(18)(19)where is a constant depending on the starting and ending points. Integrating gives(20)where(21)(22) is an elliptic integral of the first kind with parameter , and is an elliptic integral of the third kind.Geodesics other than meridians of an oblate spheroid undulate between two parallels with latitudes equidistant from the equator. Using the Weierstrass sigma function and Weierstrass zeta function, the geodesic on the oblate spheroid..
A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called small circles. A great circle becomes a straight line in a gnomonic projection (Steinhaus 1999, pp. 220-221).The shortest path between two points on a sphere, also known as an orthodrome, is a segment of a great circle. To find the great circle (geodesic) distance between two points located at latitude and longitude of and on a sphere of radius , convert spherical coordinates to Cartesian coordinates using(1)(Note that the latitude is related to the colatitude of spherical coordinates by , so the conversion to Cartesian coordinates replaces and by and , respectively.) Now find the angle between and using the dot product,(2)(3)(4)The great circle distance is then(5)For the Earth, the equatorial radius is km, or 3963 (statute) miles. Unfortunately, the flattening..
A developable surface is a ruled surface having Gaussian curvature everywhere. Developable surfaces therefore include the cone, cylinder, elliptic cone, hyperbolic cylinder, and plane.A developable surface has the property that it can be made out of sheet metal, since such a surface must be obtainable by transformation from a plane (which has Gaussian curvature 0) and every point on such a surface lies on at least one straight line.
A ruled surface is a surface that can be swept out by moving a line in space. It therefore has a parameterization of the form(1)where is called the ruled surface directrix (also called the base curve) and is the director curve. The straight lines themselves are called rulings. The rulings of a ruled surface are asymptotic curves. Furthermore, the Gaussian curvature on a ruled regular surface is everywhere nonpositive.Examples of ruled surfaces include the elliptic hyperboloidof one sheet (a doubly ruled surface)(2)the hyperbolic paraboloid (a doublyruled surface)(3)Plücker's conoid(4)and the Möbius strip(5)(Gray 1993).The only ruled minimal surfaces are the planeand helicoid (Catalan 1842, do Carmo 1986).
A surface in 3-space can be parameterized by two variables (or coordinates) and such that(1)(2)(3)If a surface is parameterized as above, then the tangentvectors(4)(5)are useful in computing the surface area and surfaceintegral.
The center of any sphere which has a contact of (at least) first-order with a curve at a point lies in the normal plane to at . The center of any sphere which has a contact of (at least) second-order with at point , where the curvature , lies on the polar axis of corresponding to . All these spheres intersect the osculating plane of at along a circle of curvature at . The osculating sphere has centerwhere is the unit normal vector, is the unit binormal vector, is the radius of curvature, and is the torsion, and radiusand has contact of (at least) third order with .
The elastica formed by bent rods and considered in physics can be generalized to curves in a Riemannian manifold which are a critical point forwhere is the normal curvature of , is a real number, and is closed or satisfies some specified boundary condition. The curvature of an elastica must satisfywhere is the signed curvature of , is the Gaussian curvature of the oriented Riemannian surface along , is the second derivative of with respect to , and is a constant.
The curl of a vector field, denoted or (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of is the limiting value of circulation per unit area. Written explicitly,(1)where the right side is a line integral around an infinitesimal region of area that is allowed to shrink to zero via a limiting process and is the unit normal vector to this region. If , then the field is said to be an irrotational field. The symbol is variously known as "nabla" or "del."The physical significance of the curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space. It arises in fluid mechanics and elasticity theory. It is also fundamental in the theory of electromagnetism, where it arises in..
The spherical distance between two points and on a sphere is the distance of the shortest path along the surface of the sphere (paths that cut through the interior of the sphere are not allowed) from to , which always lies along a great circle.For points and on the unit sphere, the spherical distance is given bywhere denotes a dot product.
A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles (like the equator). The geodesics in a space depend on the Riemannian metric, which affects the notions of distance and acceleration.Geodesics preserve a direction on a surface (Tietze 1965, pp. 26-27) and have many other interesting properties. The normal vector to any point of a geodesic arc lies along the normal to a surface at that point (Weinstock 1974, p. 65).Furthermore, no matter how badly a sphere is distorted, there exist an infinite number of closed geodesics on it. This general result, demonstrated in the early 1990s, extended earlier work by Birkhoff, who proved in 1917 that there exists at least one closed geodesic on a distorted sphere, and Lyusternik and Schnirelmann, who proved in 1923 that..
A quaternion Kähler manifold is a Riemannian manifold of dimension , , whose holonomy is, up to conjugacy, a subgroup ofbut is not a subgroup of . These manifolds are sometimes called quaternionic Kähler and are sometimes written hyphenated as quaternion-Kähler, quaternionic-Kähler, etc.Despite their name, quaternion-Kähler manifolds need not be Kähler due to the fact that all Kähler manifolds have holonomy groups which are subgroups of , whereas . Depending on the literature, such manifolds are sometimes assumed to be connected and/or orientable. In the above definition, the case for is usually excluded due to the fact that which, under Berger's classification of holonomy, implies merely that the manifold is Riemannian. The above classification can be extended to the case where by requiring that the manifold be both an Einstein manifold and self-dual.Some authors exclude this last criterion,..
A Kähler metric is a Riemannian metric on a complex manifold which gives a Kähler structure, i.e., it is a Kähler manifold with a Kähler form. However, the term "Kähler metric" can also refer to the corresponding Hermitian metric , where is the Kähler form, defined by . Here, the operator is the almost complex structure, a linear map on tangent vectors satisfying , induced by multiplication by . In coordinates , the operator satisfies and .The operator depends on the complex structure, and on a Kähler manifold, it must preserve the Kähler metric. For a metric to be Kähler, one additional condition must also be satisfied, namely that it can be expressed in terms of the metric and the complex structure. Near any point , there exists holomorphic coordinates such that the metric has the formwhere denotes the vector space tensor product; that is, it vanishes up to order two at . Hence, any geometric..
Regge calculus is a finite element method utilized in numerical relativity in attempts of describing spacetimes with few or no symmetries by way of producing numerical solutions to the Einstein field equations (Khavari 2009). It was developed initially by Italian mathematician Tullio Regge in the 1960s (Regge 1961).Modern forays into Regge's method center on the triangulation of manifolds, particularly on the discrete approximation of 4-dimensional Riemannian and Lorentzian manifolds by way of cellular complexes whose 4-dimensional triangular simplices share their boundary tetrahedra (i.e., 3-dimensional simplices) to enclose a flat piece of spacetime (Marinelli 2013). Worth noting is that Regge himself devised the framework in more generality, though noted that no such generality is lost by assuming a triangular approximation (Regge 1961).The benefit of this technique is that the structures involved are rigid and hence are..
The osculating circle of a curve at a given point is the circle that has the same tangent as at point as well as the same curvature. Just as the tangent line is the line best approximating a curve at a point , the osculating circle is the best circle that approximates the curve at (Gray 1997, p. 111).Ignoring degenerate curves such as straight lines, the osculating circle of a given curve at a given point is unique.Given a plane curve with parametric equations and parameterized by a variable , the radius of the osculating circle is simply the radius of curvature(1)where is the curvature, and the center is just the point on the evolute corresponding to ,(2)(3)Here, derivatives are taken with respect to the parameter .In addition, let denote the circle passing through three points on a curve with . Then the osculating circle is given by(4)(Gray 1997)...
Special affine curvature, also called as the equi-affine or affine curvature, is a type of curvature for a plane curve that remains unchanged under a special affine transformation.For a plane curve parametrized by , the special affine curvature is given by(1)(2)(Blaschke 1923, Guggenheimer 1977), where the prime indicates differentiation with respect to t. This reduces for a curve to(3)(4)(Blaschke 1923, Shirokov 1988), where the prime here indicated differentiation with respect to .The following table summarizes the special affine curvatures for a number of curves.curveparametrizationcatenarycircleellipsehyperbolaparabola0
A quantity which gives the inclination of a curve or line with respect to another curve or line. For a line in the -plane making an angle with the x-axis, the slope is a constant given by(1)where and are changes in the two coordinates over some distance.For a plane curve specified as , the slope is(2)for a curve specified parametrically as , the slope is(3)where and , for a curve specified as , the slope is(4)and for a curve given in polar coordinates as , the slope is(5)(Lawrence 1972, pp. 8-9).It is meaningless to talk about the slope of a curve in three-dimensional space unlessthe slope with respect to what is specified.J. Miller has undertaken a detailed study of the origin of the symbol to denote slope. The consensus seems to be that it is not known why the letter was chosen. One high school algebra textbook says the reason for is unknown, but remarks that it is interesting that the French word for "to climb" is "monter."..
A positive number such that a lamina or solid body with moment of inertia about an axis and mass is given byPickover (1995) defines a generalization of as a function quantifying the spatial extent of the structure of a curve and given bywhere is the length distribution function. Small compact patterns have small .
Let be a locally Euclidean coordinate system. Then(1)Now plug in(2)(3)to obtain(4)Reading off the coefficients from(5)gives(6)(7)(8)Making a change of coordinates gives(9)(10)(11)(12)(13)(14)
A topology induced by the metric defined on a metric space . The open sets are all subsets that can be realized as the unions of open ballswhere , and .The metric topology makes a T2-space. Given two distinct points and of , their distance is certainly positive, so the open balls and are disjoint neighborhoods of and , respectively.
The Euclidean metric is the function that assigns to any two vectors in Euclidean -space and the number(1)and so gives the "standard" distance between any two vectors in .The Euclidean metric in Euclidean three-space is given by(2)giving the line element(3)(4)where Einstein summation has been used.
Let be a compact -dimensional oriented Riemannian manifold without boundary, let be a group representation of by orthogonal matrices, and let be the associated vector bundle. Suppose further that the Laplacian is strictly negative on where is the linear space of differential k-forms on with values in . In this context, the analytic torsion is defined as the positive real root ofwhere the -function is defined byfor the collection of eigenvalues of , the restriction of to the collection of bundle sections of the sheaf .Intrinsic to the above computation is that is a real manifold. However, there is a collection of literature on analytic torsion for complex manifolds, the construction of which is nearly identical to the construction given above. Analytic torsion on complex manifolds is sometimes called del bar torsion...
Given a differential operator on the space of differential forms, an eigenform is a form such that(1)for some constant . For example, on the torus, the Dirac operator acts on the form(2)giving(3)i.e., .
The level set of a differentiable function corresponding to a real value is the set of pointsFor example, the level set of the function corresponding to the value is the sphere with center and radius .If , the level set is a plane curve known as a level curve. If , the level set is a surface known as a level surface.
A differential ideal is an ideal in the ring of smooth forms on a manifold . That is, it is closed under addition, scalar multiplication, and wedge product with an arbitrary form. The ideal is called integrable if, whenever , then also , where is the exterior derivative.For example, in , the ideal(1)where the are arbitrary smooth functions, is an integrable differential ideal. However, if the second term were of the form , then the ideal would not be integrable because it would not contain .Given an integral differential ideal on , a smooth map is called integrable if the pullback of every form vanishes on , i.e., . In coordinates, an integral manifold solves a system of partial differential equations. For example, using above, a map from an open set in is integral if(2)(3)(4)(5)Conversely, any system of partial differential equations can be expressed as an integrable differential ideal on a jet bundle. For instance, on corresponds to on ...
The term metric signature refers to the signature of a metric tensor on a smooth manifold , a tool which quantifies the numbers of positive, zero, and negative infinitesimal distances of tangent vectors in the tangent bundle of and which is most easily defined in terms of the signatures of a number of related structures.Most commonly, one identifies the signature of a metric tensor with the signature of the quadratic form induced by on any of the tangent spaces for points . Indeed, given an orthogonal vector basis for any tangent space , the action of on arbitrary vectors and in is given by(1)whereby the signature of is defined to be the signature of any of the forms , i.e., the ordered triple of positive, negatives, and zero values for the inner products . This value is well-defined due to the fact that the signature of remains the same for all points in . For non-degenerate quadratic forms, the value will always satisfy , whereby the signature of will be..
Lorentzian -space is the inner product space consisting of the vector space together with the -dimensional Lorentzian inner product.In the event that the metric signature is used, Lorentzian -space is denoted ; the notation is used analogously with the metric signature .The Lorentzian inner product induces a norm on Lorentzian space, whereby the squared norm of a vector has the form(1)Rewriting (where by definition), the norm in (0) can be written as(2)In particular, the norm induced by the Lorentzian inner product fails to be positive definite, whereby it makes sense to classify vectors in -dimensional Lorentzian space into types based on the sign of their squared norm, e.g., as spacelike, timelike, and lightlike. The collection of all lightlike vectors in Lorentzian -space is known as the light cone, which is further separated into lightlike vectors which are positive and negative lightlike. A similar distinction is made for positive..