MAC 2313 - Section 04 - Fall 2011
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Remainder of Chapter 16 - Vector Calculus
Parameterization of Curves and Surfaces
parameterize lines, circles, curves in general, planes, surfaces
parameterizations using spherical and cylindrical coordinates
surface area of a parameterized surface
Vector Fields
compute gradient vector fields
find potential function of a gradient vector field
Line Integrals
definition
properties
line integrals of vector fields
interpretation of line integrals over vector fields: work, circulation
compute line intergral over a parameterized curve
terminology: path independent = conservative = gradient vector field
determine whether a vector field is path-independent (items 1-3 below)
- 1) path-independent fields: curl test - if curl = 0, field is path-independent
- 2) path-independent fields: gradient fields are path-independent
- 3) path-independent fields: if a curve is closed and line integral is zero,
vector field is path-independent and so is a gradient field
find potential function of a path independent / conservative / gradient vector field
path-independent fields: us Fundamental Theorem of Calculus for Line Integrals (i.e. if curl = 0, find f and apply F.T.L.I.)
path-dependent fields: Green's Theorem: use if curve closed
Surface/Flux Integrals
definition
properties
interpretation (rate fluid flows through a surface)
surface integrals of scalar functions
surface integrals of vector fields
calculate flux through a surface given by z=f(x,y), through a cylindrical
surface, through a spherical surface, through a parameterized surface
Curl
definition
interpretation (rotation)
Stokes' Theorem - be able to use this theorem to calculate a line; use if
curve is boundary of a surface
curl test
Divergence
definition
interpretation (net outflow)
Divergence Theorem: be able to use this theorem to calculate flux
integral
Review Homework for Chapter 16, pg 1107: #1b, 15-17 odds, 25-35 odds
(previously assigned #1-13 odds)
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