Math 114 Notes and Home Work

Notes on vector calculus, especially the signed integrals and the fundamental theorem of calculs for the three dimensional space. Please read section 2 of the notes, especially subsection 2.4 if you don't rea other parts: section 2.4 discusses how to produce an orientation of the boundary of an oriented surface with boundary.

Warning--you may find nonsense in a textbook. In the textbook by Thomas, the induced orientation on the boundary is "defined" to be "counterclockwise with respect to the surface's normal vector", without adequate explanation as to what "counterclockwise" actually means. The pictures don't explain the meaning of "counterclockwise" either. Imagine you have a very large surface in the three dimensional space which makes all sorts of twists and turns all over the place, so that the give unit normal vector field performs all kinds of acrobatic somersaults. What is "counterclockwise" supposed to mean for such a surface? And, how do you translate such a description in concrete examples?

We will try to post homework assignments early. But please keep in mind that they will be continuously updated. Please let me know if you find errors or ambiguities. Sometimes part of assignment n may be moved to assignment n+1 or n+2; you will be notified if that happens.

Assignments

• Week of August 31, Homework 1.
• Week of September 7, Homework 2.
• Week of September 14, Homework 3.
• Week of September 21, Homework 4, due Monday, September 28.
• Week of September 28, Homework 5
• Weeks of October 5th and 12th, Homework 6
• Week of October 19, Homework 7
• Week of October 26, Homework 8
• Week of November 2, Homework 9, due Monday Nov 9.
• Week of November 9, Homework 10
• Week of November 16, Homework 11, due Monday November 23.
• Week of November 23, Homework 12, due Monday November 30.
• Week of November 30, Homework 13, due Friday December 4. A few more problems for practice are attached to assignment 13.

Notes

• Notes on 13.5; judge for yourself whether the formula for the torsion is so complicated that need to consult an "advanced text".
  
• Notes for Kepler's first and third laws. Kepler's second law (conservation of angular momentum for planetary motion), the easiest of the three, is derived in the textbook, but Kepler's first law is not. In this note you will find a derivation of Kepler's first law (2/3 of a page), plus a derivation of the third law (a bit different from the route indicated in the text). Here are some hints if you want to do it yourself without looking at the notes.
  • The trick to get the first law is to use angle as the independent variable, and rewrite Newton's law as a second order ODE for the reciprocal of r. (Note that the relation between the angle and the time is given by Kepler's second law.) The result is second order linear ODE with constant coefficents for the reciprocal of r, with respect to the angle.
  • When you integrate the latter ODE, you get an explicit formula for the orbit of the planet, which is an ellipse. The major axis and minor axis (plus the mass of the sun and the gravitational constant), determine the angular momentum (a constant for the plnet).
  • The period of planet is obtained from Kepler's second law. Using the expression of the angular momentum in terms of the major/minor axis, one sees that the square of period is equal to the cube of the major axis, times a constant depending only on the mass of the sun and the gravitational constant.

Access to MyMathLab comes with the textbook package, if you bought a new copy. We encourage you to take advantage of this facilty, which is helpful for getting the basics.

• The assignments with MyMathLab will be treated as extra credits.
• The course ID is chai78999 (Course Name: Math 114, Calculus II).

MyMathLab assignments, extra credit

• The first MyMathLab assignment for Math 114-001 was posted on August 29, 2015. Your feedback will be appreciated.
  
• The second MyMathLab assignment was posted on September 15, 2015.

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Syllabus and Core Problems