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MATH 2162.01 Study Guide - Final Guide: Divergence Theorem, Lagrange Multiplier, Spherical Coordinate System


Department
Mathematics
Course Code
MATH 2162.01
Professor
All
Study Guide
Final

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SPRING 2013 MATH 2162.01 FINAL REVIEW
TAE EUN KIM
We’ve covered a lot of materials in this course starting from the theory of sequences and
series to Stokes’ theorem and the divergence theorem. Reviewing all of them thoroughly
can be a very daunting task and one may easily lose sight of what’s more important and
what’s less so. In addition, we don’t have enough time. So here I prepare a short handout
to help you prepare for the final exam. Good luck!
Part 1. Some Important Topics
Make your own summary sheet by filling in details of the following list.
(1) Power series: To find the radius of convergence and the interval of convergence
of a given power series;
(a) Use the ratio test or the root test.
(b) Make sure you check the convergence at the end points.
(2) Polynomial approximation: the n-th order Taylor polynomial of f(x)centered at
x=ais ...
(3) Polar coordinates:
(a) Arc length formula
(b) Area formula
(4) Geometric application of vector products:
(a) Angle between two vectors (dot product)
(b) Orthogonal projection (dot product)
(c) Area of the parallelogram formed by two vectors (cross product)
(5) Optimization of multivariable functions:
(a) The Second Derivative Test
(b) The Lagrange multiplier
(6) Finding volume of a solid using double integrals and triple integrals:
(a) Double integrals: Cartesian and polar coordinates
(b) Triple integrals: Cartesian, cylindrical, and spherical coordinates
(7) Line integrals:
(a) Scalar function vs. vector field
(b) Conservative vector field and the Fundamental Theorem of Line Integrals
(8) Surface integrals:
(a) Scalar function vs. vector field
(b) Parametrization of surface
(9) Generalized Fundamental Theorems of Calculus:
(a) Green’s theorem (circulation version) and Stokes’ theorem
(b) Green’s theorem (flux version) and the Divergence theorem
Date: April 28, 2013.
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