# The Bridge Book

The online book The Geometry of Vector Calculus (aka the Bridge Book) by Tevian Dray and Corinne Manogue is a very good supplemental “text” for our course. While some of the notation is different from that used in Stewart’s Calculus, the explanations in the Bridge Book are very well done.

The correspondence between the sections in Stewart’s Calculus and the Bridge Book:

 Stewart’s Calculus The Bridge Book VECTORS AND VECTOR OPERATIONS section 12.2: Vectors Vectors section 12.3: The Dot Product The Dot Product, The Law of Cosines section 12.4: The Cross Product The Cross Product, The Triple Product CALCULUS ALONG CURVES section 13.2: Derivatives & Integrals of Vector Functions & section 13.4: Motion in Space Motion in Space DERIVATIVES OF MULTIVARIABLE FUNCTIONS section 14.1: Functions of Several Variables Surfaces, Level Sets section 14.4: Tangent Planes & Linear Approximations The Multivariable Differential section 14.5: Chain Rule Chain Rule, Chain Rule via Tree Diagrams section 14.6: Directional Derivatives & the Gradient Gradient, Properties of the Gradient, Product Rules, Directional Derivatives section 14.7: Maximum & Minimum Values Optimization, Curvature and the Second Derivative Test section 14.8: Lagrange Multipliers Lagrange Multipliers INTEGRATING MULTIVARIABLE FUNCTIONS sections 15.1 & 15.2: Double Integrals Review of Single Variable Integration, Double Integrals section 15.4: Applications of Double Integrals Mass and Center of Mass section 15.6: Triple Integrals Triple Integrals section 15.7: Triple Integrals in Cylindrical Coordinates & 15.8: Triple Integrals in Spherical Coordinates Curvilinear Coordinates, Triple Integrals in Cylindrical and Spherical Coordinates VECTOR CALCULUS section 16.1: Vector Fields Divergence, Curl, Visualizing Divergence and Curl section 16.2: Line Integrals Scalar Line Integrals, Arc Length, Vector Line Integrals, Use What You Know section 16.3: Fundamental Theorem for Line Integrals (Conservative Vector Fields) Independence of Path, Conservative Vector Fields, Finding Potential Functions, Visualizing Conservative Vector Fields section 16.4: Green’s Theorem & section 16.8: Stokes’ Theorem Stokes’ Theorem section 16.5: Curl & Divergence Curl, Divergence section 16.6: Parametric Surfaces & section 16.7: Surface Integrals of Vector Fields Surfaces, General Surface Elements, Activity: Surface Elements on Planes, Cylinders and Spheres, Wrap-Up: Surface Elements on Planes, Cylinders, and Spheres, Graphs of Functions, Flux, Parametric Surfaces, Scalar Surface Integrals section 16.9: Divergence Theorem The Divergence Theorem

#### Other useful sections in The Bridge Book

• The Vector Differential: An important section. We will be using the vector differential $d\boldsymbol{r}$ extensively in this class. The Bridge Book covers this extremely useful object in much greater depth than does Stewart’s Calculus.

• Vector calculus using basis vectors associated to curvilinear coordinates. You will not be responsible for this material in this course, but it may be useful to you in future courses.