← Back to Math Roadmap

Vector Calculus

Line integrals, surface integrals, divergence, and curl — multivariable calculus for physics and engineering.

Vector Fields

▼
A vector field assigns a vector to every point in space. Think of wind velocity at each point in the atmosphere, or the force a magnet exerts at each point around it. The three fundamental differential operators in vector calculus are the gradient (which produces a vector field from a scalar function, pointing uphill), the divergence (which measures how much a vector field spreads out from a point — positive divergence means a source, negative means a sink), and the curl (which measures the rotation or circulation of a vector field around a point). These are linked by fundamental identities: the curl of a gradient is always zero, and the divergence of a curl is always zero.
Key identities. The gradient field has zero curl (it is irrotational). The curl field has zero divergence (it is solenoidal).
Divergence. The dot product of del (nabla) with the vector field. Measures the net outward flux per unit volume at a point.

The Integral Theorems

▼
Vector calculus is unified by three great integral theorems that relate integrals over regions to integrals over their boundaries. Green's theorem (2D) relates a double integral over a region to a line integral around its boundary. Stokes' theorem (3D) relates a surface integral of curl to a line integral around the boundary of the surface. The Divergence theorem (Gauss's theorem, 3D) relates a triple integral of divergence to a surface integral over the boundary. These are all special cases of the Generalized Stokes' Theorem from differential geometry, which states that the integral of a differential form over the boundary of a manifold equals the integral of its exterior derivative over the manifold.
Stokes' theorem. The circulation around the boundary equals the flux of curl through the surface.

Maxwell's Equations

▼
Maxwell's equations of electromagnetism are the crown jewel application of vector calculus. They are four vector calculus equations that describe all classical electromagnetic phenomena. Gauss's law (divergence of the electric field equals charge density) and Gauss's law for magnetism (divergence of the magnetic field is zero — no magnetic monopoles) use divergence. Faraday's law (curl of the electric field equals minus the time derivative of the magnetic field) and Ampere's law (curl of the magnetic field equals current plus the time derivative of the electric field) use curl. Together, they predict that light is an electromagnetic wave.

Worked Example

▼
Worked Example
Find the flux of F = (x, y, z) through the sphere x² + y² + z² = a².
Use Gauss's theorem: the divergence of F is 1+1+1 = 3.
Flux = triple integral of 3 over the sphere = 3 × (volume of sphere) = 3 × (4πa³/3) = 4πa³.

Vector Field Visualizer

▼
Explore how vector fields flow through space. See divergence (sources/sinks) and curl (rotation) in action.