MAT 205:  Vector Calculus          Topics for final exam
The exam will cover all of the topics covered by the first two exams (see the corresponding summaray sheets), plus topics in chapters 12 and 13 as described below. About half of the exam will concentrate on chapters 12 and 13, with the other half divided between the other topics. The exam will be about two hours long, with some very easy questions and some more difficult ones. Expect at least one word problem, if not two or seventeen. Agent Orange may or may not make an appearance, depending on his other assignments. You should be familiar with all of the following topics:


Double Integrals: Understand the definition of a double integral as a volume, and as an interated integral. Be able to compute double integrals over both rectangular and general regions. Be able to change the order of integration in a double integral. Be able to set up and compute double integrals in polar coordinates in addition to in rectangular coordinates.


Triple integrals: Understand the definition of a triple integral, and be able to set up and compute them. Be able to exchange the order of integration. Triple integrals in spherical or cylindrical coordinates will not be explicitly required, but feel free to use them if you know how.


Surface Area: This is really just the surface integral $\int\negthinspace\negthinspace\int_{S} 1~dS$, but you can treat it separately if it makes you happier.


Vector Fields: Know what a vector field is, and how to tell if a vector field is conservative or not. If a field is conservative, be able to find a potential function for it.


Line Integrals: Know what a line integral is, and how to compute it. If a vector field is conservative, be able to use the fundamental theorem to compute the line integral in terms of the potential. Understand the equivalence between independence of path of the line integral $\int_C F\cdot dR$ and the existence of a potential f such that $F=\nabla f$.


Green's theorem: Know the statement of Green's theorem and how to apply it to compute a line integral.


Surface integrals: Know the definition of a surface integral, and how to compute them, both for parametric surfaces and surfaces which are graphs of the form z=f(x,y). Know the relationship between a surface integral and the flux of a vector field through a surface.


Curl and divergence: If F is a vector field on ${\Bbb{R}}^3$, know the definition of the curl of F ( $\nabla \times F$) and divergence of F ( $\nabla\cdot F$). Know the interpretation of curl as a measurement of the rotation of a vector field, and divergence as measuring its compressability. Know that ${\rm curl}(\nabla F)=0$, and that ${\rm div}~{\rm curl}(F)=0$.


Stokes' Theorem: Know the statement of Stokes' theorem, and how to apply it.


Gauss' Theorem: Know the statement of Gauss' theorem, and how to apply it.


 

Scott Sutherland
2000-12-12