8.4 Tensor Upper Index Notation Role
The upper index in tensor notation identifies the contravariant component, crucial for coordinate transformations and vector operations in multilinear algebra.
Tensor Upper Index Notation Role is the function that a superscript index plays within tensor index notation: marking a slot as contravariant, dictating that slot's transformation law under a change of coordinates, and identifying which of a tensor's several argument positions accepts a covector rather than a vector. It is the collection of duties assigned to the upper position in the notation, considered as a single coherent role rather than a list of disconnected facts.
The Transformation Duty
Carrying the Inverse Jacobian
The primary duty of an upper index is to signal that its slot transforms using the inverse of the Jacobian matrix relating one coordinate system to another; this is what makes a quantity like a vector's components v^{i} contravariant, meaning they vary inversely, in a specific technical sense, to how the coordinate basis itself varies.
Compensating for Basis Change
This transformation duty exists precisely so that the underlying vector, as a basis-independent object, remains the same when its coordinate-dependent components are recomputed in a new basis; the upper index's transformation role is the mechanism that keeps components and object consistent across a change of coordinates.
The Argument-Slot Duty
Accepting a Covector
In the interpretation of a tensor as a multilinear map, a slot marked with an upper index in the tensor's type designation (p, q) is one of the p slots that accepts a covector (an element of the dual space) as its argument; this is the structural, argument-accepting role that upper indices play once a tensor is understood in its most basis-independent formulation.
Distinguishing Slots by Their Accepted Input
A (2, 1) tensor accepts two covectors and one vector, in that order, and it is precisely the upper indices in its symbolic representation T^{ij}_{k} that mark which two of its three total slots are the covector-accepting ones, leaving the lower-indexed slot as the vector-accepting one.
Diagram of the Upper Index Role
The Upper Index Role in Common Objects
Vectors as the Archetype
A vector's components v^{i} are the clearest illustration of the upper index role: they transform contravariantly, and the vector itself, viewed as a multilinear map, accepts a single covector as input to produce a scalar, matching exactly the single upper index it carries.
Inverse Metric and Index Raising
The inverse metric tensor g^{ij}, carrying two upper indices, plays this role twice over: each of its two slots transforms contravariantly, and it is this tensor, contracted against a covector, that performs the operation of raising an index, converting a covariant slot into a contravariant one on some other tensor.
Consequences of the Upper Index Role
Determines Compatibility in Contraction
Because contraction pairs one upper index with one lower index, the upper index role directly determines which pairings are legitimate: a slot marked upper can only be validly contracted against a slot marked lower, never against another upper slot, since only an upper-lower pairing produces the Jacobian cancellation that makes the resulting contraction basis-independent.
Shapes How a Formula Is Read
Recognizing the upper index role immediately tells a reader that the corresponding quantity behaves like a vector rather than a covector under a change of coordinates, which in turn informs how that quantity is expected to combine with other tensors in a larger expression, well before any explicit computation with the formula is carried out.
Relationship to the Lower Index's Complementary Role
Mirror-Image Duties
The lower index plays an exactly complementary pair of duties — covariant transformation and vector-accepting argument slots — so that the upper and lower index roles together account for the complete range of behaviors a tensor's slots can exhibit; understanding either role in isolation is aided by seeing it as one half of this matched, mirror-image pair.
A Foundational Piece of the Overall Notation System
The upper index role, together with its lower counterpart, forms the foundational distinction on which the entire apparatus of tensor index notation — summation, contraction, raising and lowering, symmetry — is built, since none of these operations can be correctly specified without first knowing which slots are upper and which are lower.