Introduction
Vector calculus extends ordinary calculus to vector-valued quantities and to fields (quantities defined throughout a region of space), giving the mathematical language used constantly in mechanics, electromagnetism, and fluid flow. This chapter is largely the rigorous mathematical companion to ideas met more physically in the Physics syllabus (Mechanics, Electricity and Magnetism).
1. Vector algebra and products
Vectors are added component-wise and combined via the dot product (A·B = |A||B|cosθ, producing a scalar) and cross product (A×B, producing a vector perpendicular to both, with magnitude |A||B|sinθ). These operations obey specific algebraic rules: the dot product is commutative, while the cross product is anti-commutative (A×B = −B×A) — a distinction that matters when manipulating vector expressions or proving vector identities.
2. Differentiation of vectors
A vector function of a single variable, r(t), can be differentiated component-wise: dr/dt is itself a vector, tangent to the curve traced by r(t) (this is the basis of velocity and acceleration as derivatives of position, and of curvature calculations). Product-rule-type identities apply, but care is required with order for cross products, since A×B ≠ B×A: d/dt(A×B) = (dA/dt)×B + A×(dB/dt), with the order of each term preserved.
3. Gradient, divergence, and curl
These three operators (already introduced physically in Physics Chapter I) are studied here as formal differential operators built from the vector operator ∇ = (∂/∂x, ∂/∂y, ∂/∂z):
- Gradient (∇f): applied to a scalar field, produces a vector field pointing toward steepest increase.
- Divergence (∇·F): applied to a vector field, produces a scalar measuring net “outflow” at a point.
- Curl (∇×F): applied to a vector field, produces a vector measuring local rotation.
Useful identities include ∇×(∇f) = 0 (the curl of any gradient is always zero — a gradient field can never have “circulation”) and ∇·(∇×F) = 0 (the divergence of any curl is always zero) — both are algebraic consequences of the equality of mixed partial derivatives, and both have direct physical meaning (a conservative field, being a gradient, cannot do net work around a closed loop; magnetic fields, being curls of a vector potential, have no sources or sinks).
4. Line, surface, and volume integrals
A line integral sums a scalar or vector quantity along a curve (used to compute work done by a force along a path, ∫F·dr). A surface integral sums a quantity over a two-dimensional surface (used to compute flux, ∫∫F·dA — the total “flow” of a vector field through a surface). A volume integral sums a quantity throughout a three-dimensional region. Correctly parametrising the curve, surface, or volume (choosing suitable variables and limits) is usually the main practical challenge in evaluating these integrals.
5. Integral theorems: Green’s, Gauss’s (divergence), and Stokes’ theorems
These theorems connect integrals over a region to integrals over its boundary, and are among the most powerful results in vector calculus:
- Green’s theorem relates a line integral around a closed curve in the plane to a double integral over the enclosed region.
- Gauss’s (divergence) theorem relates the flux of a vector field through a closed surface to the volume integral of its divergence throughout the enclosed volume — this is the mathematical foundation of Gauss’s law in electromagnetism (Physics Chapter I of Paper II).
- Stokes’ theorem relates the line integral of a vector field around a closed curve to the surface integral of its curl over any surface bounded by that curve — the mathematical foundation of both Ampère’s and Faraday’s laws (Physics Chapter I of Paper II).
Each theorem trades a harder integral (over a region) for an easier one (over its boundary, or vice versa), and each carries specific conditions (continuity, orientation, closed boundaries) that must be checked before applying it.
Key Takeaways
- Vector algebra’s dot and cross products obey different symmetry rules (commutative vs. anti-commutative), which must be respected when differentiating vector products.
- Gradient, divergence, and curl are the three fundamental differential operators on fields; ∇×(∇f) = 0 and ∇·(∇×F) = 0 are essential, frequently used identities.
- Line, surface, and volume integrals generalise ordinary integration to curves, surfaces, and regions in space, with correct parametrisation being the key practical skill.
- Green’s, Gauss’s, and Stokes’ theorems each relate an integral over a region to an integral over its boundary, directly underlying the integral form of Maxwell’s equations.