Math 202: Advanced Calculus
Text: Vector Calculus by Marsden & Tromba
Chapter 1:
1.1
1.2
1.3
1.4
1.5
Chapter 2:
2.1
2.2
2.3
2.4
2.5
2.6
Chapter 3:
3.1
3.2
3.3
3.4
3.5
Chapter 4:
4.1
4.2
4.3
4.4
4.5
Chapter 5:
5.1
5.2
5.3
5.4
5.5
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
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
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
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
Integration
Introduction
The double integral over a rectangle
The double integral over more general regions
Changing the order of integration
Improper integrals (optional)
8
5.6
5.7
5.8
Chapter 6:
6.1
6.2
6.3
6.4
6.5
6.6
Chapter 7:
7.1
7.2
7.3
7.4
7.5
7.6
The triple integral
The geometry of maps from R2 to R2
The change of variables theorem
Review exercises for chapter 5
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
Vector Analysis
8
Green’s theorem
Stokes’ theorem
Conservative fields
Gauss’ theorem
Applications to physics and differential equations (optional)
Differential forms (optional)
Review exercises for chapter 7