Introduction to the mathematics of general relativity: Difference between revisions

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[[Image:Components stress tensor cartesian.svg|300px|right|thumb|Stress, a second-order tensor. Stress is here shown as a series of vectors on each side of the box]]

A tensor extends the concept of a vector to additional dimensionsdirections. A [[scalar (mathematics)|scalar]], that is, a simple number without a direction, would be shown on a graph as a point, a zero-dimensional object. A vector, which has a magnitude and direction, would appear on a graph as a line, which is a one-dimensional object. A tensorvector extends this concept to additional dimensions. A two-dimensional tensor would be calledis a secondfirst-order tensor. This can be viewed as a set of related vectors, movingsince init multipleholds directionsone on a planedirection.

A second-order tensor has two magnitudes and two directions, and would appear on a graph as two lines similar to the hands of a clock. The "order" of a tensor is the number of directions contained within, which is separate from the dimensions of the individual directions. A second-order tensor in two dimensions might be represented mathematically by a 2-by-2 matrix, and in three dimensions by a 3-by-3 matrix, but in both cases the matrix is "square" for a second-order tensor. A third-order tensor has three magnitudes and directions, and would be represented by a cube of numbers, 3-by-3-by-3 for directions in three dimensions, and so on.

=== Applications ===