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Math 202: Advanced Calculus
Text: Vector Calculus by Marsden & Tromba
Chapter 1: The Geometry of Euclidean Space 5
Vectors in three-dimensional space
The inner product
The cross product
Cylindrical and spherical coordinates
n-dimensional Euclidean space
Review exercises for chapter 1
Chapter 2: Differentiation 8
The geometry of real-valued functions
Limits and continuity
Differentiation
Properties of the derivative
Gradients and directional derivatives
Iterated partial derivatives
Review exercises for chapter 2
Chapter 3: Vector-Valued Functions 8
Paths and velocity
Arc length
Vector fields
Divergence and curl of a vector field
Vector differential calculus
Review exercises for chapter 3
Chapter 4: Higher-order Derivatives; Maxima and Minima 8
Taylor theorem
Extrema of real-valued functions
Constrained extrema nd lagrange multipliers
The implicity function theorem (optional)
Some applications
Review exercises for chapter 4
Chapter 5: Integration 8
Introduction
The double integral over a rectangle
The double integral over more general regions
Changing the order of integration
Improper integrals (optional)
The triple integral
The geometry of maps from R2 to R2
The change of variables theorem
Review exercises for chapter 5
Chapter 6: Integrals over Paths and Surfaces 6
The path integral
Line integrals
Parametrized surfaces
Area of a surface
Integrals of scalar functions over surfaces
Surface integrals of vector functions
Chapter 7: Vector Analysis 8
Greens theorem
Stokes theorem
Conservative fields
Gauss theorem
Applications to physics and differential equations (optional)
Differential forms (optional)
Review exercises for chapter 7
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