Multivariable Calculus

Author: Don Shimamoto

I: Preliminaries

II: Vector-valued function of one variable

Chapter 2: Path and Curves

• Parametrizations
• Velocity, acceleration, speed, arclength
• Intgrals with respect to arclength
• The geometry of curves: tangent and normal vectors
• The cross product
• The geometry of space curves: Frenet vectors
• Curvature and torsion
• The Frenet-Serret formulas
• The classification of space curves
• Exercise for Chapter 2

III: Real-Valued Functions

Chapter 3: Real-valued functions: Preliminaries

• Graphs and level sets
• More surface in $\mathbb{R}^3$
• The equation of a plane in $\mathbb{R}^3$
• Open sets
• Continuity
• Some properties of continuous functions
• The Cauchy-Schwarz and triangle inequalities
• Limits
• Exercise for chapter 3

Chapter 4: Real-valued functions: Differentiation

• The first-order approximation
• Conditions for differentiability
• The mean value theorem
• The $C^1$ test
• The Little Chain Rule
• Directional derivatives
• $\nabla f$ as normal vector
• Higher-order partial derivatives
• Smooth functions
• Max/Min: Critical points
• Classifying nondegenrate critical points
• Max/Min: Lagrange Multipliers
• Exercise for chapter 4

Chapter 5: Real-valued fuctions: Integration

• Volume and iterated integrals
• The double integral
• Interpretations of the double integral
• Parametrization of surfaces
  • Polar Coordinates $(r, \theta)$ in $\mathbb{R}^2$
  • Cylindrical coordinates $(r, \theta, z)$ in $\mathbb{R}^3$
  • Spherical coordinates $(p, \phi , \theta)$ in $\mathbb{R}^3$
• Integrals with respect to surface area
• Triple integrals and beyond
• Exercise for chapter 5

IV: Vector-valued Functions

Chapter 6: Differentiability and the Chain rule

• Continuity revisited
• Differentiability revisited
• The chain rule : a conceptual approach
• The chain rule : a computational approach
• Exercise for chapter 6

Chapter 7: Change of Variables

• Change of vaiables for double integrals
• A word about substitution
• Examples : Linear changes of variables, symmetry
• Change of variables for $n-$fold integrals
• Exercise for chapter 7