Introduction
Beyond vectors, mechanics employs scalars (single value), pseudovectors (angular momentum, torque), and tensors (inertia tensor, stress tensor) for complete physical descriptions. Understanding these mathematical objects is essential for advanced mechanics and continuum physics where simple vectors are insufficient.
Scalars and Their Properties
Scalars are quantities with magnitude only, no direction. They are invariant under coordinate rotations: temperature T, mass m, time t, speed v (not velocity). Scalars transform as rank-0 tensors. Scalar fields assign a scalar value to each point in space: pressure field p(x,y,z), temperature distribution T(x,y,z). Scalar multiplication of vectors changes magnitude but not direction.
\nIntroduction
Beyond vectors, mechanics employs scalars (single value), pseudovectors (angular momentum, torque), and tensors (inertia tensor, stress tensor) for complete physical descriptions. Understanding these mathematical objects is essential for advanced mechanics and continuum physics where simple vectors are insufficient.
Scalars and Their Properties
Scalars are quantities with magnitude only, no direction. They are invariant under coordinate rotations: temperature T, mass m, time t, speed v (not velocity). Scalars transform as rank-0 tensors. Scalar fields assign a scalar value to each point in space: pressure field p(x,y,z), temperature distribution T(x,y,z). Scalar multiplication of vectors changes magnitude but not direction.
\nPseudovectors (Axial Vectors)
Pseudovectors (or axial vectors) arise from cross products and transform differently under reflection. True vectors (polar vectors) reverse direction under mirror reflection; pseudovectors do not. Examples: angular momentum L = r×p, torque τ = r×F, magnetic field B. Under coordinate inversion (x→-x, y→-y, z→-z): polar vectors change sign, pseudovectors remain unchanged. This distinction matters in crystal physics and parity considerations.
Introduction to Tensors
Tensors generalize scalars (rank-0) and vectors (rank-1) to higher ranks. A rank-2 tensor in 3D has 9 components Tᵢⱼ that transform in a specific way under coordinate changes. Important tensors in mechanics: inertia tensor Iᵢⱼ relating angular momentum to angular velocity (Lᵢ = Σⱼ Iᵢⱼωⱼ), stress tensor σᵢⱼ giving force per unit area in direction j on surface with normal i, strain tensor εᵢⱼ describing deformation.
\nInertia Tensor
The inertia tensor describes mass distribution relative to rotation axes. For a rigid body, Ixx = ∫(y2+z2)dm, Ixy = -∫xy dm, etc. Principal axes are those where the inertia tensor is diagonal: I1, I2, I3 are principal moments. Angular momentum is L = I·ω (matrix multiplication). For rotation about principal axis: L = Iω. Kinetic energy of rotation: K = 1/2ω·I·ω = 1/2(I1ω₲ + I2ω22 + I3ω32).
Tensor Operations and Applications
Tensor contraction reduces rank: Tᵢᵢ (summed) is the trace, a scalar. Outer product of vectors produces tensor: AᵢBⱼ = Tᵢⱼ. Symmetric tensors have Tᵢⱼ = Tⱼᵢ; antisymmetric have Tᵢⱼ = -Tⱼᵢ. Any tensor decomposes into symmetric and antisymmetric parts. In elasticity theory, the stiffness tensor Cᵢⱼₖₗ (rank-4) relates stress to strain: σᵢⱼ = Σₖₗ Cᵢⱼₖₗεₖₗ. Tensor analysis is fundamental to general relativity and continuum mechanics.
\n