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

by OC2554284

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**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

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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