Einstein (Index) Notation — Full Notes
Dot & double-dot, gradient operators, product rules, conversions, reading tricks, and intuition. Flat headings • wide layout • print-ready.
5. Dot & Double-Dot Products
Contractions in Einstein form. Repeated indices imply summation; never repeat an index more than twice in a term.
| Operation | Index Form | Result | Example / Notes |
|---|---|---|---|
| Vector · Vector | $a_i b_i$ | Scalar | $a\!\cdot\!b$ |
| Vector · Tensor | $a_i T_{ij}$ | Vector | $(a\!\cdot\!T)_j$ |
| Tensor · Tensor (single contraction) | $A_{ij}B_{jk}$ | Tensor | Matrix multiplication $(AB)_{ik}$ |
| Tensor :: Tensor (double dot) | $A_{ij}B_{ij}$ | Scalar | $A\!:\!B=\!\sum_{i,j}A_{ij}B_{ij}=\mathrm{tr}(A^{\mathsf T}B)$ |
$$\delta_{ij}=\begin{cases}1,&i=j\\0,&i\ne j\end{cases},\qquad \delta_{ij}A_{jk}=A_{ik}.$$
$$A_{ii}=\mathrm{tr}(A),\qquad A\!:\!B=\mathrm{tr}(A^{\mathsf T}B).$$
6. Gradient Operators in Index Form
With $\partial_i\equiv\dfrac{\partial}{\partial x_i}$. For a vector $u$, $(\nabla u)_{ij}=\partial_j u_i$.
| Operator | Definition (Index) | Result |
|---|---|---|
| Gradient | $(\nabla f)_i=\partial_i f$ | Vector |
| Divergence | $\nabla\!\cdot u=\partial_i u_i$ | Scalar |
| Curl | $(\nabla\times u)_i=\epsilon_{ijk}\partial_j u_k$ | Vector |
| Laplacian (scalar) | $\nabla^2 f=\partial_i\partial_i f$ | Scalar |
| Laplacian (vector) | $(\nabla^2 u)_i=\partial_j\partial_j u_i$ | Vector |
Index Notation Reminders
- Repeated index ⇒ summation (contraction). Never repeat an index more than twice.
- Dot product → single contraction: $a_i b_i=a\cdot b$.
- Double dot → full contraction: $A_{ij}B_{ij}=A\!:\!B=\mathrm{tr}(A^{\mathsf T}B)$.
- Gradient and divergence: $\nabla a$ → tensor; $\nabla\!\cdot a$ → scalar.
Step 1. Start from Definitions
| Concept | Index form | Vector form |
|---|---|---|
| Gradient of scalar $f$ | $(\nabla f)_i=\partial_i f$ | $\nabla f$ |
| Divergence of vector $a$ | $\nabla\!\cdot a=\partial_i a_i$ | $\nabla\!\cdot a$ |
| Laplacian of scalar $f$ | $\nabla^2 f=\partial_i\partial_i f$ | $\nabla^2 f$ |
| Laplacian of vector $u$ | $(\nabla^2 u)_i=\partial_j\partial_j u_i$ | $\nabla^2 u$ |
Step 2. Laplacian as “divergence of a gradient”
$$\nabla^2=\nabla\!\cdot\nabla.$$
Scalar $f$: $$\nabla^2 f=\nabla\!\cdot(\nabla f)=\partial_i(\partial_i f)=\partial_i\partial_i f.$$
Vector $u=(u_1,u_2,u_3)$: $$(\nabla^2 u)_i=\partial_j\partial_j u_i,\qquad \nabla^2 u=\begin{bmatrix}\nabla^2 u_1\\ \nabla^2 u_2\\ \nabla^2 u_3\end{bmatrix}.$$
Step 3. Product Rule in Index Notation
For outer product $(ab)_{ij}=a_i b_j$: $$\big(\nabla\!\cdot(ab)\big)_j=\partial_i(a_i b_j) =(\partial_i a_i)\,b_j + a_i(\partial_i b_j).$$
Step 4. Go Back to Vector Form
$\partial_i b_j=(\nabla b)_{ij}$
Substitute: $$\partial_i(a_i b_j)=(\nabla\!\cdot a)\,b_j + (a\cdot\nabla)\,b_j,$$ $$\boxed{\nabla\!\cdot(ab)=(\nabla\!\cdot a)\,b + (a\cdot\nabla)\,b.}$$
Step 5. Switch Between Forms
| You see this (vector) | Index form |
|---|---|
| $\nabla\!\cdot a$ | $\partial_i a_i$ |
| $\nabla f$ | $\partial_i f$ |
| $\nabla\!\cdot(ab)$ | $\partial_i(a_i b_j)$ |
| $(a\cdot\nabla)b$ | $a_i\,\partial_i b_j$ |
| $\nabla^2 u$ | $\partial_j\partial_j u_i$ |
How to Go Between Vector & Einstein (Index) Notation
Key principle: Twice = sum; once = free output component.
| Vector notation | Einstein notation | Result type |
|---|---|---|
| $a$ component | $a_i$ | Scalar (component) |
| $\nabla f$ | $(\nabla f)_i=\partial_i f$ | Vector |
| $\nabla\!\cdot a$ | $\partial_i a_i$ | Scalar |
| $(a\cdot\nabla)f$ | $a_i\,\partial_i f$ | Scalar |
| $(a\cdot\nabla)b$ | $a_i\,\partial_i b_j$ | Vector |
| $\nabla\times a$ | $(\nabla\times a)_i=\epsilon_{ijk}\partial_j a_k$ | Vector |
| $\nabla^2 f$ | $\partial_i\partial_i f$ | Scalar |
| $\nabla^2 a$ | $(\nabla^2 a)_i=\partial_j\partial_j a_i$ | Vector |
| $\nabla(a\cdot b)$ | $\partial_i(a_j b_j)$ | Vector |
| $\nabla\!\cdot(ab)$ | $\partial_i(a_i b_j)$ | Vector |
Practice: Recognize “Repeated vs Free”
| Expression | What happens | Vector meaning |
|---|---|---|
| $a_i b_i$ | $i$ repeated → sum → scalar | $a\cdot b$ |
| $a_i b_j$ | both free → tensor | $ab$ (outer product) |
| $a_i \partial_i b_j$ | $i$ summed; $j$ free → vector | $(a\cdot\nabla)b$ |
| $\partial_i a_i$ | $i$ summed → scalar | $\nabla\!\cdot a$ |
| $\epsilon_{ijk}\partial_j a_k$ | antisymmetric | $(\nabla\times a)_i$ |
Reading Trick
- Repeated index → contraction (dot) → reduces rank.
- 0 free indices → scalar; 1 free → vector; 2 free → 2nd-order tensor; 3 free → 3rd-order tensor; etc.
- Count free indices to know the result’s type instantly.
Intuition & Correct Wording
In 3D, $a=[a_1,a_2,a_3]=(a_x,a_y,a_z)$. The symbol $a_i$ means “the $i$-th component of $a$.”
Dot product by components: $a\cdot b=a_x b_x+a_y b_y+a_z b_z \equiv a_i b_i$.
$(\nabla f)_i=\partial_i f$ is a vector; $i$ labels which component. Likewise $(\nabla^2 u)_i=\partial_j\partial_j u_i$.
$(\nabla u)_{ij}=\partial_j u_i$ has two free indices → a 2nd-order tensor (matrix).
Shortcut Map (Cheat Sheet)
| Operation | Einstein form | Result |
|---|---|---|
| Gradient of scalar | $\partial_i f$ | Vector |
| Divergence of vector | $\partial_i a_i$ | Scalar |
| Curl of vector | $\epsilon_{ijk}\partial_j a_k$ | Vector |
| Laplacian of scalar | $\partial_i\partial_i f$ | Scalar |
| Laplacian of vector | $\partial_j\partial_j a_i$ | Vector |
| Directional derivative | $a_i\,\partial_i b_j$ | Vector |
| Divergence of tensor $A$ | $\partial_i A_{ij}$ | Vector |