SYSTEMATIC MATHEMATICS

Vector Calculus

This course contains 41 visible chapters. Imagine that every point in moving water has an arrow showing its speed and direction. How much work is done along a path? How much water crosses a surface? Does the flow spin locally, and can a boundary measurement reveal what happens inside? Start by drawing and reading these arrow fields. Define gradient, divergence, curl, product rules, and the Laplacian through visible local changes, including the scale factors hidden by cylindrical and spherical coordinates; then build path accumulation, surface flow, and orientation from small pieces. Green’s, Stokes’, and the Divergence theorems are reached as carefully checked local-to-global statements, not as formulas to memorize. Radial forces, implicit surfaces, vorticity, fixed and moving conservation volumes, material derivatives, Green’s boundary-energy identity, harmonic uniqueness, Helmholtz decomposition, holes, singular points, reversed directions, and missing hypotheses are tested whenever they can change an answer.

Before this course: Completed Multivariable Calculus and its prerequisites. Students must be able to differentiate scalar and vector-valued functions, use gradients and Jacobians, parametrize curves, compute double and triple integrals with coordinate changes, and work with dot and cross products.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Vector fields, flow lines, and domains

Objective: Why can two distinct integral curves not cross at a regular point of a C¹ field?

A vector field assigns a direction and magnitude to every admissible point; its domain may contain holes or singularities that later control global conclusions.

Start from the definition: A vector field on D⊆Rⁿ is a map F:D→Rⁿ; an integral curve r(t) satisfies r′(t)=F(r(t)). The result to establish is: If F is continuously differentiable near a point, a unique local integral curve passes through that point.

The exact worked question is: Find the integral curves of F(x,y)=(x,−y) through (a,b). Compare it with this failure boundary: For F(x,y)=(−y,x)/(x²+y²), the origin is excluded; drawing an arrow there or treating the domain as all of R² erases the singularity.

If F is continuously differentiable near a point, a unique local integral curve passes through that point.

Rewrite r′=F(r) as a first-order autonomous differential system with the initial point specified.

Continuous differentiability makes F locally Lipschitz, so nearby velocity arrows cannot separate solutions arbitrarily fast.

The local existence-and-uniqueness theorem then supplies one trajectory through the point on a sufficiently short interval.

Find the integral curves of F(x,y)=(x,−y) through (a,b).

  1. The system is x′=x and y′=−y.
  2. Solving separately gives x=Ceᵗ and y=De⁻ᵗ.
  3. The initial point sets C=a and D=b, so xy=ab along each trajectory.

Result: The integral curve is r(t)=(aeᵗ,be⁻ᵗ), with hyperbolic trajectories except on the axes.