# Math 2110 Learning Activities (Spring 2022)

### A Few Useful Links:

Paul’s Online Math Notes, Calculus III

GeoGebra 3D Calculator

3D Function Grapher (also does Parametric Surfaces!)

2D Vector Field Generator

### GeoGebra Resources:

Graphing with GeoGebra: Intro to GeoGebra

Surfaces: Have a look at the equations and graphs of common surfaces

https://www.geogebra.org/3d/s52k8uur: Visualizing Traces and Partial Derivatives

Tangent Plane Demo: Visualize Tangent Planes and see their Connection to Partial Derivatives

Triple Integrals: See how to build up the domain of a triple integral and visualize if it is Type X, Y or Z.

Coordinate Systems: See how Cartesian, cylindrical and spherical coordinates work

https://www.geogebra.org/3d/unfhkaps:Line Integrals of Vector and Scalar Fields

https://www.geogebra.org/m/jb7z9wnb:Fundamental Theorem of Line Integrals

Collection of Demos: Student Selected Collection of Various Demos

### Learning Activities, By Section:

 Section Topic Learning Activities 12.1 Three-Dimensional Coordinate Systems Visualizing the Octants Corners of a Box Planes Circles vs Cylinder/Line vs Plane 12.6 Cylinders and Quadric Surfaces Quadric Surfaces Notes on Sketching and Identifying Surfaces Surfaces: Have a look at the equations and graphs of common surfaces in GeoGebra 12.2 Vectors Visualizing Vectors in 3D Vector Components Unit Vectors and Normalization 12.3 The Dot Product Dot Products in 3D Direction Angles (*useful application) Projections (*useful application) 12.4 The Cross Product Cross Products Volume of a Parallelepiped (*useful application) 12.5 Equations of Lines & Planes Line in 3D (static) vs Line in 3D (variable) An Example Relating Lines and Planes 14.1 Functions of Several Variables A Gallery of Examples Level Curves of an (Elliptic) Paraboloid Level Curves of a Gaussian Surface (Bell) 14.2 Limits and Continuity (*interesting topic, neither covered in class nor on exams) Limits Handout 14.3 Partial Derivatives Cross Sections https://www.geogebra.org/3d/s52k8uur: Visualizing Traces and Partial Derivatives in GeoGebra 14.4 Tangent Planes and Linear Approximation Linear Approximation Tangent Plane Demo: Visualize Tangent Planes and see their Connection to Partial Derivatives in GeoGebra 14.5 The Chain Rule Chain Rule Practice Exercises (with Solutions) 14.6 Directional Derivatives and the Gradient Vector Directional Derivatives Video Directional Derivatives 14.7 Maximum and Minimum Values Max and Min Video Types of Extrema 14.8 Lagrange Multipliers Absolute Max/Min Video Lagrange Multipliers Video Lagrange Multipliers Practice Exercises (with Solutions) 15.1 Double Integrals over Rectangles Double Integrals over Rectangles Video Riemann Sum in 2D Riemann Sum in 3D 15.2 Double Integrals over General Regions Double Integrals over General Regions Video 15.3 Double Integrals in Polar Coordinates 15.6 Triple Integrals Modeling Solids Handout (Play-Doh!) Surface Area Video (*helpful for Chapter 16) Triple Integrals/Double Integrals in Polar Video Triple Integrals: See how to build up the domain of a triple integral and visualize if it is Type X, Y or Z in GeoGebra 15.9 Change of Variables in Multiple Integrals Change of Variables Video Change of Variables Practice Exercises (with Solutions) 15.7 Triple Integrals in Cylindrical Coordinates Visualizing Cylindrical Coordinates Triple Integrals in Cylindrical Video 15.8 Triple Integrals in Spherical Coordinates Visualizing Spherical Coordinates Basic Shapes in Spherical Coordinates Spherical Coordinates on GeoGebra (use slider bars at top left to change variables) Summary of Triple Integrals (All Coordinate Systems) Coordinate Systems: See how Cartesian, cylindrical and spherical coordinates work using GeoGebra 13.1 Vector Functions A Space Curve Another Space Curve (Helix) Line Segment A Gallery of Examples 13.2 Derivatives and Integrals of Vector Functions 13.3 Arc Length Linear Approximation of Arc Length Practice Problems with Visualizations 16.2 Line Integrals (Scalar Functions) 16.1 Vector Fields Simple 2D Vector Field Interesting 2D Vector Field Simple 3D Vector Field Interesting 3D Vector Field 16.2 Line Integrals (Vector Fields) https://www.geogebra.org/3d/unfhkaps:Line Integrals of Vector and Scalar Fields in GeoGebra 16.3 The Fundamental Theorem for Line Integrals Fundamental Theorem for Line Integrals Video https://www.geogebra.org/m/jb7z9wnb:Fundamental Theorem of Line Integrals in GeoGebra 16.4 Green’s Theorem Green’s Theorem Video Summary of Line Integrals (methods and deciding what to use) 16.5 Curl and Divergence The Idea Behind Curl Subtleties about Curl 16.6 Parametric Surfaces and Their Areas A Gallery of Examples Parametric Surfaces (from URI) 16.7 Surface Integrals (Scalar Functions) Practice Problems with Visualizations 16.7 Surface Integrals (Vector Fields) 16.9 Divergence Theorem 16.8 Stokes’ Theorem Summary of Surface Integrals (decision-making and methods)