Vector & Tensor Calculus: Fields and How They Change

Fields and the three derivatives

A field assigns a value to every point of space. A scalar field like temperature assigns a number; a vector field like the electric field or a fluid velocity assigns an arrow. Vector calculus studies how such fields change from point to point, built from a single operator, the del operator \nabla=(\partial_x,\partial_y,\partial_z).

Three combinations matter. The gradient \nabla\phi of a scalar points in the direction of steepest increase and measures its slope. The divergence \nabla\cdot\vec{F} of a vector field measures how much it spreads out of a point — its source or sink strength. The curl \nabla\times\vec{F} measures its local rotation or circulation.

In index notation these are compact: (\nabla\phi)_i=\partial_i\phi, \nabla\cdot\vec{F}=\partial_i F_i, and (\nabla\times\vec{F})_i=\varepsilon_{ijk}\partial_j F_k. Two identities follow in one line each from the Levi-Civita algebra: the curl of a gradient vanishes, and the divergence of a curl vanishes.

\nabla\times(\nabla\phi)=0,\qquad \nabla\cdot(\nabla\times\vec{F})=0

These two identities are why a curl-free field has a scalar potential and a divergence-free field has a vector potential — the backbone of electromagnetism.

The integral theorems

Local rates of change connect to global fluxes through two great theorems. The divergence theorem says the divergence summed over a volume equals the net flux out through its boundary surface. This is exactly Gauss's law in disguise.

\int_V (\nabla\cdot\vec{F})\,dV=\oint_{\partial V}\vec{F}\cdot d\vec{A}

The divergence (Gauss) theorem: what is produced inside equals what flows out.

The curl theorem (Stokes' theorem) says the curl integrated over a surface equals the circulation around its bounding loop — the content of Ampère's and Faraday's laws.

\int_S (\nabla\times\vec{F})\cdot d\vec{A}=\oint_{\partial S}\vec{F}\cdot d\vec{\ell}

Stokes' theorem: the swirl over a surface equals the flow around its edge.

Curvilinear coordinates

Cartesian coordinates are rarely the natural choice. A problem with spherical symmetry — an atom, a planet, a point charge — cries out for spherical coordinates; a wire or a pipe wants cylindrical. This is the realm of curvilinear coordinates, and switching is not free: distances stretch, and the operators grow scale factors.

\nabla^2\phi=\frac{1}{r^2}\frac{\partial}{\partial r}\!\left(r^2\frac{\partial\phi}{\partial r}\right)+\frac{1}{r^2\sin\theta}\frac{\partial}{\partial\theta}\!\left(\sin\theta\frac{\partial\phi}{\partial\theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2\phi}{\partial\varphi^2}

The Laplacian in spherical coordinates. Its messiness is the price of matching the symmetry — and separating it is where the Legendre polynomials and spherical harmonics are born.

Tensors: beyond vectors

A scalar is one number; a vector is three, one per direction. A tensor of rank two is nine — one for each pair of directions. It is best pictured as a machine that eats a vector and returns a vector, generally rotated and rescaled. The inertia tensor turns an angular velocity into an angular momentum; the stress tensor turns a surface orientation into a force per area.

What defines a tensor is not the grid of numbers but how those numbers transform when you rotate the axes. Each index brings one factor of the rotation matrix. This is the whole point: an equation written as tensor = tensor holds in every coordinate system at once, so it can express a law of physics that cannot depend on your choice of axes.

T'_{ij}=R_{ik}\,R_{jl}\,T_{kl}

The transformation law for a rank-2 Cartesian tensor: one rotation factor R per index. This rule, not the array itself, is the definition.