Lecture 34, Monday, 11/17/08:
Section 14.6: Surface integrals
Topics
- Surfaces defined by function z=f(x,y) for (x,y) in some region R
- Surfaces defined by parametrization r(u,v) for (u,v) in some
region R
- Surface area element dS (formulas (6.1) and (6.2))
- Surface integrals of scalar functions
- Surface integrals over vector fields, flux
- Mass and center of mass of surface
Read
Section 14.6. The vast majority of problems can be done by expressing
the surface in the form z=f(x,y) and using the simpler formula (6.1) for dS.
However, you should know the formula (6.2) for parametrized
surfaces, and you should be able to apply it in simple cases such
as spherical surfaces (where there is a natural parametrization given by
spherical coordinates phi and theta). You can skim through the formal
definitions (Riemann sums, etc.) and the proofs of the various formulas.
Homework
Section 14.6: 19, 21, 25, 29, 31, 37, 39, 49
(The first three problems above (19, 21, 25 in 14.6) are ordinary
surface area computations of the type that came up in 13.4 and that could
have been given in 13.4.)
Solutions.
Notes
As in the case of line integrals, there are two basic types of surface
integrals: integrals over scalar functions g(x,y,z),
and integrals over vector fields F, multiplied by a
unit normal vector n. The first type is a simple extension of
the double integral that gives a surface area (from Section 13.4), with
the integrand 1 replaced by a more general function. The second type is
analogous a line integral over F dr. Just like this line
integral, which can be interpreted as a work integral, this latter
surface integral has a physical interpretation: it denotes the net flow
("flux") of the vector field F across the surface S.
The vector version of a surface integral arises as the left side of
the Divergence Theorem, one of the fundamental theorems of vector
calculus, which can be thought of as a 3-dimensional version of Green's
theorem.
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Last modified Wed 26 Nov 2008 08:59:10 AM CST