4 Vector Analysis
This section studies functions whose values are vectors rather than numbers, and the integrals that go with them.
A vector field assigns a vector to each point — a velocity to each point of a fluid, a force to each point of space. Two derivatives of such a field matter: the divergence, a scalar measuring the extent to which the field spreads out from a point, and the curl, a vector measuring its rotation. Both are expressed through the operator \(\nabla \), which is the organising notation of the whole section.
The integrals come next. A line integral adds a quantity along a curve, a surface integral over a surface. The central question throughout is when a line integral depends only on its endpoints rather than on the path taken; the answer identifies the conservative fields, which are exactly the gradients.
The section culminates in three theorems — Green’s, Stokes’ and the divergence theorem — which are one theorem in three dresses. Each says that integrating a derivative over a region equals integrating the original quantity over the boundary of that region, and each is the fundamental theorem of calculus raised a dimension.
4.2 Gradients
4.3 The Operator \(\nabla \)
4.4 divergence and curl of a Vector Field
4.5 Line Integrals
4.6 Line Integrals Of Vector Fields
4.7 The Fundamental Theorem of Line Integrals
4.8 Transformation Of Line Integrals Into Double Integrals
4.9 Parametric Surfaces and their Areas
4.10 Surface Integrals
4.11 Surface Integrals of Vector Fields
4.12 Oriented Surfaces
4.13 Stokes’ Theorem
4.14 The divergence Theorem
4.15 Practice Problems
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