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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

This preview shows half of the first page. to view the full 2 pages of the document. 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
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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