Intro to Differential Geometry and General Relativity - S. Warner Episode 12 pps

... we chose?Answer Yes.Question But how can we interpret this strange object?Answer Just as a covariant vector field converts contravariant fields into scalars (seeSection 3) we shall see that ... t = a to t.Then s is an invertible function of t, and, using s as a parameter, ||dxi/ds||2 is constant, and equals 1 if C is space-like and -1 if it is time-like.Conversely, if t is any ... x1 sin x2 sin x3 sin x4 … cos xn-1yn = r sin x1 sin x2 sin x3 sin x4 … sin xn-1 cos xnyn+1 = r sin x1 sin x2 sin x3 sin x4 … sin xn-1 sin xn.(d) The torus...

... identify densities with n−forms and n−form with densities. Thus we may integrate n−forms. The change ofvariables formula then holds for orientation preserving diffeomorphisms as doesStokes theorem.2.11 ... standard basis of Rn. So giving aframe is the same as giving an ordered basis of V and we will sometimes writef = (f1, . . . , fn).If A ∈ Gl(n) then A is an isomorphism of Rnwith itself, ... whosecolumns are the vectors Xuu, Xu and Xv. Replacing the first column by Xuvgives a corresponding expression for f, and replacing the first column by Xv vgives the expression for g. Substituting...

... these vectors to the curve.That is, does the vector V(4) remain parallel, and do the vectors {V(1), V(2), V(3), V(4)}remain orthogonal in the sense of 8.2?Answer If X and Y are vector fields, ... orthogonal tangent vectors at a general point, and sketch the resulting vectors.4. Contravariant and Covariant Vector FieldsQuestion How are the local coordinates of a given tangent vector for one chart ... Thus this vector has local and ambient coordinates equal to each other, and equal to dxidt = åi,which are the same as the original coordinates. In other words, the tangent vectors are “thesame”...

... need to specify a path every time we wanta tangent vector!Notes 3.7 (1) Under the one -to- one correspondence in the proposition, the standard basis vectors in Encorrespond to the tangent vectors ... tangent vectors at a general point, and sketch the resulting vectors. 244. Contravariant and Covariant Vector FieldsQuestion How are the local coordinates of a given tangent vector for one ... contravariant vectors “are” just tangent vectors: the contravariant vector vicorresponds to the tangent vector given byv = vi ∂∂xi ,so we shall henceforth refer to tangent vectors as contravariant...

... formula:Euclidean 3- space: d(x, y) = (y1 - x1)2+(y2 - x2)2+(y3 - x3)2 Minkowski 4- space: d(x, y) = (y1 - x1)2+(y2 - x2)2+(y3 - x3)2 - c2(y 4 - x 4 )2.Geometrically, ... rules for each of the following, and hence decide whether ornot they are tensors. Sub -and superscripted quantities (other than coordinates) areunderstood to be tensors. 36(a) dXijdt(b) ∂xi∂xj ... M into avector space. Note that we cannot expect to obtain a vector field by adding a covariant field to a contravariant field.Exercise Set 4 1. Suppose that Xj is a contravariant vector...