Course Handout Page
Handouts and Other Resources
Math 241   Spring 2025

Review   Helpful Review/Preparation for MML HW 00.1-00.6.   most items (many are links) will not be distributed in class

• Unit Circle
• Integration Basics  This blue sheet is distributed in class and recalls Calc. I & II integration basics.
• Solving system of linear equations   (LibreTexts link)
• Math 142 covered from Ch 11:  (look towards the end of your Math 142 notes)
  • 2 Dimensional Parametric Curves, along with corresponding tangent lines and arc length (§11.1-11.2)
  • 2 Dimentsional Polar Coordinates, along with corresponding area and arc length (§11.3, 11.5).
  Basically: Calc 2 is in 2D while Calc 3 is in 3D.
  Math 241 will generalize Math 142 concepts from 2D to 3D.
  Below is some Math 142 review material.   Remark. Math 142 no longer covers §11.4 (graphing/sketching polar equations, e.g. r= cos θ).
  • Math 142 Notes
    • §11.1-11.2. Prof. G. Math 142 Lecture Notes   Parametric Curves/Tangent lines & Arc Length
    • §11.3-11.5. Prof. G. Math 142 Lecture Notes    Polar Coords with Area & Arc Length
    • §11.1-11.2. Schaum's Outline: Calculus. Ch 37:   Parameterized curves in 2D: 1st&2nd derivative, tangent line, arc length.
  • Prof. G. Math 142 Handouts
    • in 2D: §11.1-11.2 Prof. G's Handout - Parametization: Definitions and Calculus   [soln]
    • in 2D: §11.3-11.5 Prof G's Handout - Polar Coordinates
    • in 2D: §11.2-11.5 Prof. G's Handout - Parametric Curves and Polar Coordinates Basics   [soln]
  further Math 142 review items, visit Prof G's: Math 142 Handout page (with 46 handouts) and Math 142 home page.

Chapter 12-16     (A * indicates will not be distributed in class.)

1. Finding the determinate of a 2x2 or 3x3 matrix.^* (In Appendix B. Needed in §12.4)
   • Prof. G.'s explanation, with a color-coded example  (source: textbook's appendix B.1)
   • video by "patrickJMT"  (Finding the Determinant of a 3 x 3 Matrix)
   • video by "Professor Dave Explains"  first 4.5 minutes  (Evaluating the Determinant of a Matrix)
2. 12.2 Vector Algebraic Operations   (soln)
3. 12.3+12.4 Dot Product and Cross Product: Overview  
4. 12.3 Vector Projections and Scalar Components  1 page summary
5. 12.3+12.4 Formulas for Dot and Cross products involving θ: why do they hold? 
6. 12.4 Right-Hand-Rule Picture  
7. 12.4 Hint on MML GeoGebra question 2 pages. not distributed in class
8. 12.5 Summary of Lines and Planes 2 pages
9. 12.6 Hyperbolic Paraboloid (saddle) 4 pages, first 2 pages handed out in class
10. 13.1 Curves in space and their tangents lecture outline
11. 13.2 Integral of paths lecture outline
12. 13.3 Summary of Arc Length   2 page
13. 13.4 Curvature and Normal Vectors - a practice example^* 
14. 14.1 Open and Friends 2 page summary
15. 14.3 Partial Derivatives Definitions/Theorems 2 pages
16. 14.4 Chain Rule   lecture outline
17. 14.5 Directional Derivative and Gradient Vector  lecture outline
18. 14.7 Extreme Vaules and Saddle Points   detailed notes with lots of examples
19. 14.7 Summary of Extreme Vaules and Saddle Points   2 page summary
20. 14.6 Tangent Planes  lecture outline
21. 15.1-15.3 Double (i.e. Iterated) Integals in Carestian/Rectangular Coordinates   lecture outline
22. 15.1-15.3 Solutions to lecture   lecture solutions
23. 15.4 Math 142 Handout to Review Polar Coordinates*  
24. 15.4 Double (i.e. Iterated) Integals in Polar Coordinates   lecture outline
25. 16.1 Line Integals of Scalar-valued Functions   updated lecture outline   (soln)
26. 16.2 Line Integals of Vector-valued Functions   updated lecture outline
27. 16.4 Green's Theorem   lecture outline
28. 15.5 Triple Integral in Rectangular (Cartesian) Coords   lecture outline
29. 15.5 Details on the Double Integrations from class examples   details in pink
30. Final-like Practice Problems   (soln)