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

This page assumes you have seen nothing. We meet every letter, arrow, subscript and superscript the parent note uses, in an order where each one is built only from things already explained.

0. Coordinates — the grid you scribble on

Notice the number sits up high: $x^1$, $x^2$. Read that as "x-one, x-two" — it is not "x squared". This raised label is an index, our first piece of notation.

1. The index — a slot that counts

• $n$ = the number of dimensions (how many coordinates), so $i$ runs $1$ to $n$.
• Upper index like $V^i$ and lower index like $W_i$ will turn out to mean different kinds of thing (Section 6). For now: it is a slot number, and its height is a flag we will learn to read.

2. Rank — how many index-slots a thing has

The count of numbers is $n^{\text{rank}}$: a scalar has $n^0=1$, a vector $n^1=n$, a rank-2 tensor $n^2$. This is the whole "hierarchy" the parent note draws.

3. Scalar, vector, and "grid with meaning"

The word component just means "one of the shadow-numbers". The arrow is the real thing; the components are what a particular choice of axes reports.

4. Changing basis — the same arrow, new numbers

Look at the figure. One fixed orange arrow. On the black axes its shadows are $(V^1,V^2)$; on the teal rotated axes the shadows are $({\tilde{V}}^1,{\tilde{V}}^2)$ — different numbers, same arrow. The tilde $\stackrel{~}{}$ over a symbol always means "measured in the new coordinates".

5. The derivative and the Jacobian — the exact rule of the change

To say precisely how new numbers relate to old, we need the tool that measures "how much does one quantity change when another nudges". That tool is the derivative.

This is the central engine of the parent note: every transformation law is "attach one Jacobian factor per index". See Jacobian and the multivariable chain rule for the machine itself.

6. Upper vs lower index — contravariant vs covariant

Now the height of the index earns its keep.

The distinction between $V^i$ and $V_i$ only becomes numerical in bent or stretched coordinates, where you convert between them using the metric — see Metric tensor and Riemannian geometry.

7. The summation convention — the disappearing $\sum$

8. Kronecker delta — the identity, in index clothes

It shows up because the Jacobian times its inverse collapses to it: $\frac{\partial {\tilde{x}}^i}{\partial x^k}\frac{\partial x^k}{\partial {\tilde{x}}^j}={\delta }_j^i$ — the statement "go to new coordinates and back, and nothing happened".

9. Symmetry — a property that survives every coordinate change

Picture two grids: one that reads the same reflected across the main diagonal, one that turns into its own negative. Because the transformation law treats both indices identically, this mirror-property is the same in every frame — a coordinate-free fact, which is why physicists trust it (stress is symmetric in any axes — Stress and strain tensors (continuum mechanics)).

10. Where these bricks are heading

You now hold every symbol the parent note uses. Two live beyond this foundation and get their own pages later — flagged here only so you know they exist:

• The covariant derivative ${\nabla }_k$ (ordinary ${\partial }_k$ is not a tensor) — Covariant derivative and Christoffel symbols.
• The grand payoff, curved spacetime as a tensor equation — General Relativity — Einstein field equations.

Equipment checklist

Test yourself — each should feel obvious before you tackle the parent note.