16: Vector Calculus
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- Nov 10, 2020
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- 16.1: Vector Fields
- A two-dimensional vector field is a function that maps each point (x,y) in R2 to a two-dimensional vector ⟨u,v⟩, and similarly a three-dimensional vector field maps (x,y,z) to ⟨u,v,w⟩.
- 16.2: Line Integrals
- We have so far integrated "over'' intervals, areas, and volumes with single, double, and triple integrals. We now investigate integration over or "along'' a curve---"line integrals'' are really "curve integrals''.
- 16.3: The Fundamental Theorem of Line Integrals
- Fundamental Theorem of Line Integrals, like the Fundamental Theorem of Calculus, says roughly that if we integrate a "derivative-like function'' ( f′ or ∇f ) the result depends only on the values of the original function (f) at the endpoints.
- 16.4: Green's Theorem
- Green's theorem argues that to compute a certain sort of integral over a region, we may do a computation on the boundary of the region that involves one fewer integrations. This describes the Fundamental Theorem of Calculus and the Fundamental Theorem of Line Integrals: to compute a single integral over an interval, we do a computation on the boundary (the endpoints) that involves one fewer integrations, namely, no integrations at all.
- 16.5: Divergence and Curl
- Divergence and curl are two measurements of vector fields and both are most easily understood by thinking of the vector field as representing as fluid flow. The divergence measures the tendency of the fluid to collect or disperse at a point, and curl measures the tendency of the fluid to swirl around the point. Divergence is a scalar, that is, a single number, while curl is itself a vector.
- 16.6: Vector Functions for Surfaces
- We have dealt extensively with vector equations for curves, r(t)=⟨x(t),y(t),z(t)⟩ . A similar technique can be used to represent surfaces in a way that is more general than the equations for surfaces we have used so far.
- 16.8: Stokes's Theorem
- Further applications and proof of Stokes Theorem is presented.
- 16.9: The Divergence Theorem
- The third version of Green's Theorem can be coverted into another equation: the Divergence Theorem. This theorem related, under suitable conditions, the integral of a vector function in a region of three dimensional space and to an integral over is its boundary surface.
