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Math Notes

University Mathematics

ConceptComplete

Vector Analysis and Tensors - Core Definitions

Vector analysis provides the mathematical foundation for describing physical quantities with both magnitude and direction, while tensor analysis extends these concepts to higher-dimensional objects essential in modern physics.

Vector Fields and Operations

DefinitionVector Field

A vector field on a domain D⊆RnD \subseteq \mathbb{R}^n is a mapping F:D→Rn\mathbf{F}: D \to \mathbb{R}^n that assigns a vector to each point in the domain. In Cartesian coordinates:

F(r)=Fx(x,y,z) i^+Fy(x,y,z) j^+Fz(x,y,z) k^\mathbf{F}(\mathbf{r}) = F_x(x,y,z)\,\hat{\mathbf{i}} + F_y(x,y,z)\,\hat{\mathbf{j}} + F_z(x,y,z)\,\hat{\mathbf{k}}

Common physical examples include electric fields E(r)\mathbf{E}(\mathbf{r}), magnetic fields B(r)\mathbf{B}(\mathbf{r}), and velocity fields v(r,t)\mathbf{v}(\mathbf{r},t) in fluid dynamics.

The three fundamental differential operators in vector calculus each reveal different physical properties. The gradient ∇f\nabla f of a scalar field ff points in the direction of steepest increase and is perpendicular to level surfaces. The divergence ∇⋅F\nabla \cdot \mathbf{F} measures the net outward flux density from an infinitesimal volume, appearing in conservation laws. The curl ∇×F\nabla \times \mathbf{F} measures the infinitesimal rotation or circulation density of a vector field.

Curvilinear Coordinates

Physical problems often exhibit symmetries that make Cartesian coordinates inconvenient. Curvilinear coordinate systems adapt to the geometry of the problem.

DefinitionOrthogonal Curvilinear Coordinates

A coordinate system (u1,u2,u3)(u_1, u_2, u_3) is orthogonal if the tangent vectors ei=∂r∂ui\mathbf{e}_i = \frac{\partial \mathbf{r}}{\partial u_i} are mutually perpendicular at every point. The scale factors are defined as:

hi=∣∂r∂ui∣h_i = \left|\frac{\partial \mathbf{r}}{\partial u_i}\right|

The gradient, divergence, and curl take specific forms in terms of these scale factors. For example, the gradient is:

∇f=1h1∂f∂u1e^1+1h2∂f∂u2e^2+1h3∂f∂u3e^3\nabla f = \frac{1}{h_1}\frac{\partial f}{\partial u_1}\hat{\mathbf{e}}_1 + \frac{1}{h_2}\frac{\partial f}{\partial u_2}\hat{\mathbf{e}}_2 + \frac{1}{h_3}\frac{\partial f}{\partial u_3}\hat{\mathbf{e}}_3

ExampleSpherical Coordinates

In spherical coordinates (r,θ,ϕ)(r, \theta, \phi) where θ\theta is the polar angle and ϕ\phi is the azimuthal angle, the scale factors are:

hr=1,hθ=r,hϕ=rsin⁡θh_r = 1, \quad h_\theta = r, \quad h_\phi = r\sin\theta

The Laplacian becomes:

∇2f=1r2∂∂r(r2∂f∂r)+1r2sin⁡θ∂∂θ(sin⁡θ∂f∂θ)+1r2sin⁡2θ∂2f∂ϕ2\nabla^2 f = \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial f}{\partial r}\right) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta}\left(\sin\theta\frac{\partial f}{\partial \theta}\right) + \frac{1}{r^2\sin^2\theta}\frac{\partial^2 f}{\partial \phi^2}

This form is essential for solving the hydrogen atom in quantum mechanics and gravitational problems with spherical symmetry.

Tensor Fundamentals

Tensors generalize vectors to multilinear objects that transform covariantly under coordinate changes, making them indispensable in relativity and continuum mechanics.

DefinitionTensor of Rank (p,q)

A tensor of type (p,q)(p,q) is a multilinear map that takes pp covectors and qq vectors and produces a scalar. In component notation with Einstein summation:

T=Tj1⋯jqi1⋯ip ei1⊗⋯⊗eip⊗ej1⊗⋯⊗ejqT = T^{i_1\cdots i_p}_{j_1\cdots j_q}\,\mathbf{e}_{i_1}\otimes\cdots\otimes\mathbf{e}_{i_p}\otimes\mathbf{e}^{j_1}\otimes\cdots\otimes\mathbf{e}^{j_q}

Under a coordinate transformation xi→x′ix^i \to x'^i, the components transform as:

Tj1⋯jq′i1⋯ip=∂x′i1∂xk1⋯∂x′ip∂xkp∂xℓ1∂x′j1⋯∂xℓq∂x′jqTℓ1⋯ℓqk1⋯kpT'^{i_1\cdots i_p}_{j_1\cdots j_q} = \frac{\partial x'^{i_1}}{\partial x^{k_1}}\cdots\frac{\partial x'^{i_p}}{\partial x^{k_p}}\frac{\partial x^{\ell_1}}{\partial x'^{j_1}}\cdots\frac{\partial x^{\ell_q}}{\partial x'^{j_q}}T^{k_1\cdots k_p}_{\ell_1\cdots\ell_q}

RemarkPhysical Tensor Examples
  • Stress tensor σij\sigma_{ij}: Describes force per unit area in elastic media
  • Moment of inertia tensor IijI_{ij}: Relates angular velocity to angular momentum
  • Electromagnetic field tensor FμνF_{\mu\nu}: Unifies electric and magnetic fields in special relativity
  • Metric tensor gμνg_{\mu\nu}: Encodes spacetime geometry in general relativity

The power of tensor notation lies in its coordinate-free formulation of physical laws. An equation expressed in tensor form remains valid in any coordinate system, reflecting the underlying physical principle that nature's laws are independent of our choice of coordinates.