Interactive Physics Module
Rank-2 Tensors
& Passive Rotation
Cartesian coordinates · Principal axes · Elliptical orbits
ApniPhysics Interactive Demo- Sliders a, b, c — diagonal and off-diagonal entries of the rank-2 tensor T
- Angle t=θ — rotation angle for passive coordinate transformation
- Ellipse — shape governed by tensor; semi-axes scale with eigenvalues
- Moving particle — traces the ellipse to show anisotropic response
- Principal axes — eigenvectors shown as coloured unit vectors
What Is a Rank-2 Tensor?
A rank-2 tensor is a mathematical object that encodes a linear relationship between two vectors. In 2-D Cartesian space it is represented as a 2×2 matrix; in 3-D it becomes a 3×3 matrix with nine components.
Think of it as a generalisation of a scalar and a vector. A scalar (rank-0) gives one number. A vector (rank-1) gives magnitude and direction. A rank-2 tensor gives magnitude and two directions, it maps one direction onto another.
Thus, a Rank-2 tensor represents a physical property independent of coordinates, while its matrix representation depends on the chosen basis.
General 2-D Tensor
Diagonal (Principal Axes)
After Rotation by t=θ
A Rank-2 tensor in 2D can be written as:
Under passive rotation of axes by angle \(\theta\):
2. Ellipse Representation
The quadratic form associated with the tensor is:
This equation describes an ellipse. Its principal axes correspond to eigenvectors of \(\mathbf{T}\).
Eigenvalues are obtained from:
3. Passive Rotation Explained
In passive rotation, we rotate the coordinate axes while the physical object remains fixed. The tensor itself is invariant, but its components depend on the observer’s frame.
The physical content of a tensor does not change when you rotate your coordinate axes. Only the numbers (components) change. This is what passive rotation demonstrates.
The Car Analogy
🚗 Real-World Analogy
Your Car's Inertia Tensor
Imagine you want to push a parked car. If you push from behind (along its length), it rolls easily little resistance. If you push from the side, it resists strongly because you must overcome the car's full lateral inertia.
This direction-dependent resistance cannot be captured by a single number (scalar). You need a rank-2 tensor: the moment of inertia tensor. Its two principal values are the easy-roll direction (small eigenvalue λ₁) and the hard-push direction (large eigenvalue λ₂).
In the demo, the semi-axes of the ellipse are exactly 1/√λ₁ and 1/√λ₂. A thin, elongated ellipse means strongly anisotropic like pushing a bus sideways. A circle means isotropic equal resistance in every direction.
When you rotate the angle slider t=θ, you are not physically rotating the car. You are rotating your coordinate axes like stepping sideways and re-describing the same car from a different viewpoint. The ellipse shape (physics) stays the same; the tensor components (numbers) change.
Worked Mathematical Example
Let the tensor in its principal frame (diagonal) be defined by sliders a = 3, b = 1:
T = | 3 0 |
| 0 1 |
The ellipse semi-axes are 1/√3 ≈ 0.577 (x-dir) and 1/√1 = 1 (y-dir). Now rotate the coordinate frame by θ = 45°:
| 0.707 0.707 |
// Passive rotation formula:
T' = RT· T · R
T'11 = T'22 = 2.0
T'12 = T'21 = 1.0
T' = | 2.0 1.0 |
| 1.0 2.0 |
Connecting Demo to the Formula
The trace and determinant stay constant under any rotation. In car terms: the car's total rotational inertia doesn't depend on how you stand when you measure it.
Why an Ellipse? The Unit Circle Map
A rank-2 tensor T transforms the unit circle into an ellipse. If point x lies on the unit circle (|x| = 1), then y = T·x traces the ellipse. The semi-axes equal the square roots of the eigenvalues of T (for symmetric T):
Semi-axis along ê₂ = 1/√b
// a > b → squashed along x, elongated along y
// a = b → circle (isotropic, T = λI)
The moving particle in the demo traces this ellipse parametrically. The closer it passes to an axis crossing, the stronger the tensor's resistance in that direction.
Rank-2 Tensors in the Real World
Stress Tensor
In structural engineering, the stress tensor maps internal forces per area across surfaces of different orientations exactly like rotating the car's coordinate frame.
Conductivity Tensor
In anisotropic crystals, electrical current does not flow in the direction of the applied electric field. The conductivity tensor maps E → J.
Inertia Tensor
A car's rotational resistance depends on the spin axis. The moment of inertia tensor encodes all three principal resistances plus cross-coupling terms.
Strain Tensor
Geophysicists use the strain tensor to quantify rock deformation under tectonic pressure strain maps are rank-2 tensor fields over the crust.
How to Use This Demo
Follow the steps below to explore how a rank-2 tensor changes under passive coordinate rotation. Start with the Theory tab's car analogy if this is your first encounter with tensors.
Start Diagonal
Set a and b to different values (e.g. a=3, b=1) and keep c=0. This gives a tensor in its principal frame.
Observe the Ellipse
Notice how the ellipse semi-axes align with the coordinate axes. The longer axis corresponds to the smaller eigenvalue (weaker resistance).
Rotate the Frame
Slowly drag the t=θ slider. Watch the principal axes rotate while the ellipse shape stays perfectly identical.
Enable Off-Diagonal
Drag slider c away from zero. This tilts the principal axes, creating a full symmetric tensor with off-diagonal entries.
Check Invariants
As you rotate t=θ, confirm that a + b (trace) and ab − c² (det) remain constant these are the physical invariants.
Isotropic Limit
Set a = b and c = 0. The ellipse becomes a circle. Rotating t=θ now has zero effect on the components.
Parameter Reference
| Parameter | Description | Effect on Demo |
|---|---|---|
| a | T₁₁ — principal value along ê₁ | Semi-axis along x = 1/√a |
| b | T₂₂ — principal value along ê₂ | Semi-axis along y = 1/√b |
| c | T₁₂ = T₂₁ — off-diagonal coupling | Tilts principal axes in lab frame |
| t=θ (theta) | Passive rotation angle | Rotates axes; components change, ellipse fixed |
| Ellipse | Image of unit circle under T | Visualises anisotropy |
| Particle | Point on ellipse | Shows directional tensor response |
Passive rotation changes how you describe the tensor, not the tensor itself. Moving t=θ does not rotate the physical ellipse it rotates your ruler. The invariance of the ellipse shape is the entire point of the demo.