Tensor analysis — scalars, vectors, rank-2 tensors: Question bank

Quick vocabulary refresher before you start (so no symbol here is unearned):

A tensor = a box of numbers plus a rule saying how those numbers change when you re-draw your coordinate axes.
Upper index $V^{i}$ = contravariant (transforms like a displacement arrow $d x^{i}$ ).
Lower index $W_{i}$ = covariant (transforms like a gradient $\partial ϕ / \partial x^{i}$ ).
Metric tensor $g_{ij}$ = the covariant rank-2 object that measures squared distance, $d s^{2} = g_{ij} d x^{i} d x^{j}$ . It also raises/lowers indices ( $V_{i} = g_{ij} V^{j}$ ). In flat Cartesian axes $g_{ij} = δ_{ij}$ (the identity), but in curved or non-orthonormal axes (e.g. polar, $g_{ij} = diag (1, r^{2})$ ) it is not the identity — see Metric tensor and Riemannian geometry.
Contraction = pairing one upper with one lower index and summing (see Einstein summation convention) — this drops the rank by 2.
Rotation matrix $R$ = the coordinate-change map when axes just rotate; for it, "undo" ( $R^{- 1}$ ) equals "flip across the diagonal" ( $R^{⊤}$ ).

The two pictures below fix the index bookkeeping this whole page rides on — glance at them before the traps.

Figure — Tensor analysis — scalars, vectors, rank-2 tensors

True or false — justify

TF1. "A

3 \times 3

table of my grocery prices per shop is a rank-2 tensor because it has two indices."

False — two indices is not enough; it must transform by the tensor law under a coordinate change, and grocery prices have no such rule. It's just a table.

TF2. "A scalar is any single number."

False — a scalar is a number that stays the same value at a point under any coordinate change (

\tilde{ϕ} = ϕ

). "5 apples" is a number but "the 1st component of velocity" is not a scalar because it changes with axes.

TF3. "In Cartesian coordinates

V^{i}

and

V_{i}

hold identical numbers, so upper/lower is pointless."

False in general — they match only because the metric is

g_{ij} = δ_{ij}

there. In polar or any non-orthonormal frame

V_{i} = g_{ij} V^{j}

differs from

V^{i}

, so the distinction is real.

TF4. "Eigenvalues of a stress tensor change when I rotate my axes."

False —

\tilde{T} = R T R^{⊤}

is a similarity transform, and similarity preserves eigenvalues, trace and determinant. Those are the frame-independent physics (see Linear algebra — change of basis & similarity transforms).

TF5. "The trace

T^{i}_{i}

depends on the frame like the individual entries do."

False — trace is a contraction, so the Jacobian and its inverse cancel to

δ

, leaving a genuine scalar; e.g.

T = (3113)

has trace

3 + 3 = 6

, and after any rotation the two diagonal entries still sum to

6

TF6. "The Kronecker delta

δ_{j}^{i}

is a tensor with different components in some frames."

False — it is isotropic:

\frac{\partial x ~ ^{i}}{\partial x ^{k}} \frac{\partial x ^{k}}{\partial x ~ ^{j}} = δ_{j}^{i}

, so it is literally

δ

in every frame.

TF7. "Adding a covariant tensor

A_{ij}

to a contravariant tensor

B^{ij}

gives a tensor."

False — you may only add same-type tensors; the two carry opposite Jacobian factors, so their sum has no consistent transformation rule.

TF8. "Whether a tensor is symmetric (

T_{ij} = T_{j i}

) can flip if I change coordinates."

False — symmetry is coordinate-independent for the covariant

T_{ij}

type: the same Jacobian factor sits on both slots, so swapping

i \leftrightarrow j

commutes with the transformation.

TF9. "The gradient

\partial ϕ / \partial x^{i}

is a contravariant vector."

False — it is covariant (lower index): the chain rule gives

\partial / \partial \tilde{x}^{i} = (\partial x^{k} / \partial \tilde{x}^{i}) \partial / \partial x^{k}

, the inverse Jacobian, which defines covariant behaviour.

TF10. "The ordinary partial derivative

\partial_{k} V^{i}

of a vector is a rank-2 tensor."

False — differentiating hits the position-dependent Jacobian and spits out an extra non-tensorial term. You need the covariant derivative

\nabla_{k} V^{i}

to restore tensor character.

TF11. "The Levi-Civita symbol

ε_{ij k}

transforms like an ordinary tensor."

False — it is a pseudotensor: under an orientation-reversing change (a reflection,

det J < 0

) it picks up an extra minus sign the true tensor rule would not produce. That is why cross products flip under mirror reflection.

Why questions

WHY1. Why must contravariant and covariant components use inverse Jacobians of each other?

So that a contraction

W_{i} V^{i}

has its two Jacobian factors cancel into

δ

, leaving a true scalar — the invariance we demand.

WHY2. Why is the stress tensor rotated as

R T R^{⊤}

rather than just

R T

A rank-2 tensor has two index slots; each slot must be rotated separately, one by

R

and one by

R^{⊤}

(its inverse Jacobian). One factor would only transform one slot.

WHY3. Why does the Quotient Theorem let us declare

g_{ij}

a tensor without checking components by hand?

Because

g_{ij} d x^{i} d x^{j}

is invariant for the arbitrary vector

d x^{i}

; the only way that can hold for all

d x^{i}

is if

g_{ij}

carries the matching covariant Jacobians — the theorem hands us the transformation rule for free.

WHY4. Why can we always split any rank-2 tensor into symmetric + antisymmetric parts, and why is that useful?

The identity

T_{ij} = \frac{1}{2} (T_{ij} + T_{j i}) + \frac{1}{2} (T_{ij} - T_{j i})

is pure algebra, and each part is separately a tensor whose symmetry survives every coordinate change — so the split has physical meaning (e.g. strain vs rotation in Stress and strain tensors (continuum mechanics)).

WHY5. Why do we care that the off-diagonals of stress vanished after a

4 5^{\circ}

rotation?

Zero off-diagonals means no shear in those axes — we found the principal axes, and the diagonal entries are the coordinate-independent eigenvalues, the real load the material feels.

WHY6. Why is the ordinary derivative not enough in curved space, forcing Christoffel symbols in?

The basis vectors themselves turn/stretch from point to point;

\partial_{k}

can't see that motion, so we add correction terms (Christoffel symbols) to track the changing basis — the covariant derivative. This is the machinery behind General Relativity — Einstein field equations.

WHY7. Why does the chain rule appear everywhere in tensor transformation laws?

Every Jacobian factor

\partial \tilde{x} / \partial x

is the chain-rule link between old and new coordinates; tensors are built from objects (

d x^{i}

\partial_{i} ϕ

) whose transformation the chain rule already dictates (see Jacobian and the multivariable chain rule).

Edge cases

EC1. Rank-0 case: how many Jacobian factors multiply a scalar under a coordinate change?

Zero — with no indices there is nothing to drag a Jacobian, which is exactly why

\tilde{ϕ} = ϕ

(pure invariance).

EC2. Zero tensor: is the all-zeros array a valid tensor, and does it stay zero in every frame?

Yes — applying any Jacobian factors to zeros gives zeros, so the zero tensor is genuinely zero in all frames (this is why tensor equations "= 0" are frame-safe).

EC3. Degenerate change of coordinates: what breaks if the Jacobian is singular (determinant zero)?

The map isn't invertible, so

\partial x^{k} / \partial \tilde{x}^{i}

doesn't exist and covariant/mixed rules collapse — you cannot legally change to such coordinates at that point.

EC4. A tensor that is symmetric and antisymmetric (

T_{ij} = T_{j i}

and

T_{ij} = - T_{j i}

): what is it?

It must be the zero tensor, since the two conditions force

T_{ij} = - T_{ij}

, hence

T_{ij} = 0

everywhere.

EC5. Identity-like rotation (

R = I

, i.e. no rotation): what does

R T R^{⊤}

give and why is that a good sanity check?

It returns

T

unchanged, confirming the "do-nothing" coordinate change leaves components alone — a mandatory consistency check on any transformation rule.

EC6. Highest-symmetry frame: for a symmetric rank-2 tensor, is a frame with all off-diagonals zero always reachable?

Yes — a real symmetric tensor is diagonalisable by an orthogonal rotation (its principal axes), so such a frame always exists; the diagonal entries are the invariant eigenvalues.

EC7. One-dimensional space (

n = 1

): how many components does a rank-

r

tensor have, and what does that reveal?

Exactly

n^{r} = 1^{r} = 1

— every rank collapses to a single number, showing the whole upper/lower distinction only bites when

n \geq 2

EC8. Orientation flip (a reflection,

det J < 0

): what distinguishes a true tensor from a pseudotensor / density here?

A true tensor obeys the plain Jacobian rule; a pseudotensor (like

ε_{ij k}

) gains an extra

sign (det J)

, and a density gains a power of

det J

— both are the tell-tale corner cases where "transforms like a tensor" is subtly violated.

Recall Self-test before moving on

Cover every answer above and re-derive the reason, not the verdict. A grid of numbers is a tensor only when… ::: it obeys the tensor transformation law under coordinate change. Contraction changes rank by… ::: minus 2 (one upper + one lower index removed). Symmetric-and-antisymmetric forces the tensor to be… ::: identically zero. The Levi-Civita symbol is not a true tensor because… ::: it flips sign under orientation-reversing transforms (a pseudotensor).

Back to the parent topic.

Question bank — Tensor analysis — scalars, vectors, rank-2 tensors

True or false — justify

Spot the error

Why questions

Edge cases