AdvancedTopic 102 of 104Engineering Mathematics
Tensors
Multi-index arrays that generalise vectors and matrices; used for stress, strain and deep learning.
In depth
A tensor generalises scalars (rank 0), vectors (rank 1) and matrices (rank 2) to arrays with several indices that transform predictably under coordinate changes. The stress and strain tensors describe forces in materials; deep learning uses multi-dimensional arrays called tensors.
Key points
- Rank = number of indices
- Einstein summation convention
- Invariants like principal stresses
ExampleThe 3×3 Cauchy stress tensor gives traction on any plane.
Tensors generalise scalars, vectors and matrices, so they describe stress, strain and anisotropic materials in a coordinate-independent way.
What is Tensors?
Multi-index arrays that generalise vectors and matrices; used for stress, strain and deep learning.
Why is it important to understand tensors?
Tensors generalise scalars, vectors and matrices, so they describe stress, strain and anisotropic materials in a coordinate-independent way.
What is the rank of a tensor?
The number of indices it has. A scalar has rank zero, a vector rank one and a stress tensor rank two.
Why is stress a tensor rather than a vector?
The force on a surface depends on the surface orientation as well as the force direction, so two indices are needed to capture all of it.
How is Tensors used in practice?
Typical applications include describing the stress state at a point in a loaded solid, writing the stiffness of a composite laminate with directional properties and computing the gradients of a deep neural network in machine learning.
What should I learn before Tensors?
Study "Z-transform and difference equations" first (topic 101 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Tensors?
Rank = number of indices. Einstein summation convention. Invariants like principal stresses.
Can you give an example of Tensors?
The 3×3 Cauchy stress tensor gives traction on any plane.
What comes after Tensors?
The next topic is "Calculus of variations", which builds directly on this one.
How difficult is Tensors?
It is classed as Advanced in Engineering Mathematics (topic 102 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.