Consider a two-dimensional spatial coordinate system S' whose coordinates (u,v) are defined by

x = u + v

y = u - v

in terms of the coordinates of a Cartesian coordinate system S. Suppose you are given a vector in S whose contravariant components are Am = (2,8). Determine the contravariant components of this vector in S'.

You'll find different notations for tensors in the literature. The so-called kernel-index notation makes it particularly easy to remember the transformation rules. In this notation you denote the components of a tensor in a transformed coordinate system (S') by putting a prime on the index, instead of using a different name for the tensor itself. You also do this for the coordinates.

The ...

Solution Summary

A detailed solution is given. A two-dimensional spatial coordinate systems is defined. The contravariant components of this vector is determined.

... Finding Kernel and Range of a transformation, checking whether 3 vectors are linear ... space of AX = 0, where A is am × n matrix, is a vector space. ...

... The problems contain operations over linear transformation on finite dimension vector spaces and also finding a basis and dimension of a vector subspace. ...

... a 1-form (or covariant vector ﬁeld) expressed in terms of the coor- dinate system x = (x1 , · · · , xn ). Determine the (covariant) transformation law for ...

... Consider the transformation N: V->V. Let g be a vector such that . First show that the vectors are linearly independent, and then (assuming V has dimension n ...

... Find the vectors, so that all points on the vector are stretched along the same vector line. This problem is asking for the eigenvectors of the transformation...

... a matrix, Finding orthogonal basis and how to find orthogonal projection onto a vector space. ... We apply the transformation to each of the basis vectors. ...

... Now suppose that ξ is a contravariant vector ﬁeld ... the contraction in the hint is scalar means that it is invariant under coordinate transformations, that is. ...