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University Math Lab: Systems & Structures FREE
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Introduction to Tensors
Learning Objectives
- Understand tensor notation
- Perform tensor operations
- Apply to physics
Key Concepts
Tensor
Multilinear map; generalizes scalars, vectors, and matrices
Rank
Number of indices: scalar(0), vector(1), matrix(2)
Theory
**Tensors** generalize scalars, vectors, and matrices:
- Rank 0: Scalar (temperature)
- Rank 1: Vector (velocity)
- Rank 2: Matrix (stress tensor)
**Einstein summation:** Repeated index implies summation.
Tᵢⱼuⱼ means Σⱼ Tᵢⱼuⱼ
**Operations:**
- Contraction: Tᵢᵢ = T₁₁ + T₂₂ + T₃₃ (trace)
- Outer product: (Aᵢ)(Bⱼ) = Cᵢⱼ
Examples
Stress tensor σᵢⱼ at a point
Solution: 3×3 matrix where σᵢⱼ = force in j-direction on face with normal in i-direction