微積分 III：高階課程（Flex）

In this course, students will be provided with the opportunity to learn and/or strengthen their knowledge of advanced calculus concepts. All knowledge levels are welcome.

This is an advanced course, and students are expected to have prior knowledge of Calculus. If you are concerned about how your student will perform in this class, please consider enrolling in Calculus II first, which is listed here: https://outschool.com/classes/calculus-ii-the-semester-course-flex-JrXxrrKr

Each week, students will be provided with a Nearpod lesson. Students will not need a Nearpod account to access the material, but the student's name will be requested so that the teacher knows who completes the work. Lessons will include a lecture video explaining the topic as well as problems for the student to try on their own. The teacher will provide solutions for all problems so that students are able to check their work. If there are additional questions, messages will be responded to within 24 hours.

The schedule of topics for this course is as follows:
Week 1:
-Parametrizations of Plane Curves
-Calculus with Parametric Curves
-Polar Coordinates

Week 2:
-Graphing Polar Coordinate Equations
-Areas and Lengths in Polar Coordinates
-Conic Sections

Week 3:
-Conics in Polar Coordinates
-Three-Dimensional Coordinate Systems
-Vectors

Week 4:
-The Dot Product
-The Cross Product
-Lines and Planes in Space

Week 5:
-Cylinders and Quadric Surfaces
-Curves in Space and Their Tangents
-Integrals of Vector Functions; Projectile Motion

Week 6:
-Arc Length in Space
-Curvature and Normal Vectors of a Curve
-Tangential and Normal Components of Acceleration

Week 7:
-Velocity and Acceleration in Polar Coordinates
-Functions of Several Variables
-Limits and Continuity in Higher Dimensions

Week 8:
-Partial Derivatives
-The Chain Rule
-Directional Derivatives and Gradient Vectors

Week 9:
-Tangent Planes and Differentials
-Extreme Values and Saddle Points
-Lagrange Multipliers

Week 10:
-Taylor's Formula for Two Variables
-Partial Derivatives with Constrained Variables
-Double and Iterated Integrals over Rectangles

Week 11:
-Double Integrals over General Regions
-Area by Double Integration

Week 12:
-Double Integrals in Polar Form
-Triple Integrals in Rectangular Coordinates

Week 13:
-Moments and Centers of Mass
-Triple Integrals in Cylindrical and Spherical Coordinates

Week 14:
-Substitutions in Multiple Integrals
-Line Integrals

Week 15:
-Vector Fields and Line Integrals; Work, Circulation, and Flux
-Path Independence, Conservative Fields, and Potential Functions

Week 16:
-Green's Theorem in the Plane
-Surfaces and Area

Week 17:
-Surface Integrals
-Stokes' Theorem

Week 18:
-The Divergence Theorem and Unified Theory
-Second-Order Linear Equations

Week 19:
-Non-homogeneous Linear Equations
-Applications

Week 20:
-Euler Equations
-Power Series Solutions

Please do not hesitate to reach out with any questions. I hope to see you in class soon!