Vector Calculus

Prerequisites: Calculus, Linear Algebra.

• Intuitive
• Concrete
• Abstract
• Why is it interesting?
• History

Intuitive

A weather map that shows an arrow for the wind at every location is a vector field: at each point there is a direction and a strength. Temperature on the same map is a scalar field, one number per point. Vector calculus is the calculus of such fields. It asks how they change from place to place and how what happens inside a region is related to what happens on its edge.

Three derivatives do most of the work.

• The gradient of a scalar field is an arrow that points in the direction in which the field increases fastest, with a length equal to the rate of increase. On a hiking map it points straight uphill.
• The divergence of a vector field measures how much the arrows spread out from a point. A tap in a bathtub is a source (positive divergence), a drain is a sink (negative divergence).
• The curl measures how much the field swirls around a point. Put a tiny paddle wheel into flowing water: if it starts to spin, the flow has curl there.

The integral theorems connect these local quantities to the boundary. The total amount of fluid produced by all sources inside a region must flow out through its surface. The total swirl inside a loop shows up as circulation along the loop. These statements are the reason the laws of electromagnetism and fluid flow can be written either as differential equations at each point or as integral statements about regions and their boundaries.

Concrete

• A Survival Guide to Vector Calculus by Aylmer Johnson

Recommended Textbooks:

• Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John H. Hubbard and Barbara Burke Hubbard - Extremely student friendly, lots of margin notes that talk about the "soft" stuff that's so crucial to the actual practice of math. Reading just the margins jumps your mathematical maturity by 2 years.
• http://mathinsight.org/thread/vector_algebra
• A nice introduction can be found in Section 3 of Vol. 2 of Feynman's Lectures on Physics, which are available here
• A Student's Guide to Vectors and Tensors by Daniel A. Fleisch
• DIV, Grad, Curl, and All That: An Informal Text on Vector Calculus by H. M Schey
• Mathematical Tools for Physics by James Nearing

Geometric Calculus:

• Geometric Calculus R & D website

Geometric Calculus videos:

• Geometric Calculus 0 Alan Macdonald
• Tutorial on Geometric Calculus David Hestenes
• No-Nonsense Electrodynamics by Jakob Schwichtenberg, appendix "Vector Calculus"

Abstract

Gradient, curl and divergence are the exterior derivative in disguise. On $\mathbb{R}^3$ with the Euclidean metric and the standard orientation, the metric identifies vector fields with 1-forms ($V \mapsto V^\flat = V_i\, dx^i$, inverse $\sharp$), and the Hodge star identifies 1-forms with 2-forms and functions with 3-forms. Under these identifications

$$\operatorname{grad} f = (df)^\sharp , \qquad \operatorname{curl} V = \left( \star\, d V^\flat \right)^\sharp , \qquad \operatorname{div} V = \star\, d \star V^\flat .$$

So the sequence

$$\Omega^0 \xrightarrow{\ d\ } \Omega^1 \xrightarrow{\ d\ } \Omega^2 \xrightarrow{\ d\ } \Omega^3$$

of differential forms becomes functions $\xrightarrow{\text{grad}}$ vector fields $\xrightarrow{\text{curl}}$ vector fields $\xrightarrow{\text{div}}$ functions. The identity $d \circ d = 0$ contains both $\operatorname{curl}\operatorname{grad} f = 0$ and $\operatorname{div}\operatorname{curl} V = 0$.

Integral theorems

The generalized Stokes theorem $\int_M d\omega = \int_{\partial M} \omega$, for a compact oriented manifold $M$ with boundary, contains the three classical theorems: the gradient theorem for a curve ($\omega$ a function), the Kelvin-Stokes theorem for a surface ($\omega$ a 1-form) and the divergence theorem for a volume ($\omega$ a 2-form).

Topology

The converse statements ("a curl-free field is a gradient", "a divergence-free field is a curl") hold on contractible regions by the Poincaré lemma, but fail in general. On $\mathbb{R}^3$ minus the $z$-axis, the field $\frac{1}{x^2 + y^2}(-y, x, 0)$ is curl-free but not a gradient, since its line integral around the axis is $2\pi$. The obstruction is measured by the de Rham cohomology of the region. This is the mathematics behind the Aharonov-Bohm effect and magnetic monopoles.

What is special about three dimensions

The formulas above use the identification of 2-forms with vectors, which works because $\binom{3}{2} = 3$. In $n$ dimensions the gradient and the divergence generalize directly, but the curl of a vector field is a 2-form with $n(n-1)/2$ components. In Minkowski space the electric and magnetic fields combine into one 2-form $F$, and the Maxwell equations become $dF = 0$ and $d\star F = \star J$ (up to sign and unit conventions).

Why is it interesting?

Vector calculus is an important tool, whenever we want to understand systems where directions play a role. A vector is an arrow that points in some direction. Thus, a vector is a tool to denote a direction.

This is needed, for example, to describe in which spatial direction a ball moves or how a fluid flows.

A vector is the mathematical representation of a physical entity that may be characterized by size (or “magnitude”) and direction. In keeping with this definition, speed (how fast an object is going) is not represented by a vector, but velocity (how fast and in which direction an object is going) does qualify as a vector quantity. Another example of a vector quantity is force, which describes how strongly and in what direction something is being pushed or pulled. But temperature, which has magnitude but no direction, is not a vector quantityA Student's Guide to Vectors and Tensors by Daniel A. Fleisch

History

https://en.m.wikipedia.org/wiki/A_History_of_Vector_Analysis

Leads to: Continuity Equation, Coulomb's law, Cross Product, Differential Forms, Divergence, Maxwell Equations, Newtonian Mechanics, Tensor Calculus, The Navier-Stokes Equations.