1. Visualizing 3-D mathematics
Level curves
Curves (1-D) in 3-space
Surfaces (2-D) in 3-space
Volumes (3-D) in 3-space
2. Abstracting 1-D and 2-D mathematics
a. From Number to Vector and Matrix
From Absolute
Value to Length of vector & Determinant of Matrix
From arithmetic
with numbers to arithmetic with vectors
addition (graphical)
multiplication: two types of product of vectors
b. From number-valued functions of
single variables to possibly vector-valued functions of possibly multiple
variables
c. Derivatives
Partial derivatives
Gradient
Matrix form
of derivative
From chain
rule (product of numbers) to chain rule (product of matrices)
d. Integrals
Double and
Triple Sums
Iteration
of single integrals
Substitution
using Jacobian
e. From The Fundamental Theorem of
Calculus to Green's & Stokes Theorems