1.2.60 Tensor Vector Component Definition

Tensor Vector Component Definition is the specification of the individual numerical entries that make up the component array of a rank-one, type (1, 0) tensor, that is, a tensor with a single upper index and no lower indices, identifying each entry as the coefficient of one particular basis vector in the expansion of the vector relative to a chosen basis of the underlying vector space. A tensor vector component is therefore a special case of a general tensor component, restricted to the situation of a single contravariant index and no covariant indices, and it is the building block from which the entire coordinate representation of a vector is assembled.

Definition via Basis Expansion

The Contravariant Vector

A type (1, 0) tensor v is an element of the vector space V itself, also called a contravariant vector. Given a basis e_1, ..., e_n of V, the vector v can be written uniquely as a linear combination of the basis vectors.

The coefficients v^1, v^2, ..., v^n appearing in this expansion are the tensor vector components of v relative to the chosen basis, each one a single element of the underlying field.

Single Upper Index

Because a type (1, 0) tensor has exactly one contravariant slot and no covariant slots, its components carry exactly one upper index, v^i, and no lower index. This distinguishes vector components from covector components, which carry a single lower index instead.

Extraction of a Component

Using the Dual Basis

If e^1, ..., e^n is the dual basis associated with e_1, ..., e_n, satisfying e^i(e_j) equal to 1 when i = j and 0 otherwise, then each vector component can be recovered by applying the corresponding dual basis element to the vector.

This shows that a vector component is not an arbitrary number but a specific value obtained by pairing the vector with the appropriate element of the dual space.

Transformation of Vector Components

Contravariant Transformation Law

Under a change of basis described by a matrix A, relating a new basis to the old one, the vector components transform using the matrix A itself, which is the origin of the term "contravariant": the components transform inversely to how the basis vectors transform, so that the vector v as an abstract object remains unchanged.

Invariance of the Vector Itself

Although the numerical values v^1, ..., v^n change when the basis changes, the combination v^i e_i remains equal to the same abstract vector v in every basis, since the transformation applied to the components exactly compensates for the transformation applied to the basis vectors.

Geometric Interpretation

Components as Coordinates

Each tensor vector component can be interpreted as the coordinate of the vector along one of the chosen basis directions. In the familiar case of ordinary three-dimensional space with the standard basis, the components v^1, v^2, v^3 correspond to what are commonly labeled the x, y, and z coordinates of the vector.

Arrow Representation

A vector with components v^1, v^2 in a two-dimensional space can be drawn as an arrow from the origin to the point with those coordinates, illustrating how the components determine both the magnitude and the direction of the vector relative to the chosen basis.

In this diagram, the dashed lines show how the vector v decomposes into its component v^1 along the horizontal basis direction and its component v^2 along the vertical basis direction.

Special Values

Zero Vector

A vector all of whose components equal zero in some basis, v^i = 0 for every i, has all components equal to zero in every other basis as well, since the transformation of an all-zero array under any linear transformation remains all zero. This vector is the zero vector of V.

Standard Basis Vector Components

The i-th standard basis vector e_i itself has components that equal 1 in position i and 0 in every other position, a pattern known as the Kronecker delta pattern, v^k = δ^k_i.