MATH 205 Multivariable Calculus, fall '20
Multivariable Calculus is a continuation of Calculus II, but is mostly concerned with generalizing the main concepts from Calculus I, such as differentiation and integration, to higher dimensions. Topics will cover include vector algebra and geometry; parametric curves in 3-dimensional space; differentiation and integration of scalar functions of several variables with applications to maximum-minimum problems and finding areas and volumes; differentiation and integration of vector fields; and Green’s, Stokes’, and Divergence Theorems (see below for a more detailed list of topics). At the end of the course, you should have a good understanding of how calculus of functions of two or more variables works and be able to apply this knowledge to a variety of problems arising in math and physics, such as motion in space, approximation of functions, optimization, density, mass, etc.
The more general objective of this course is to continue to provide you with a deeper understanding and working knowledge of calculus, while in the process strengthening your analytical skills, increasing your ability to communicate mathematics symbolically and orally, making you comfortable with reading and understanding mathematics on your own, and developing an appreciation for calculus as one of the greatest intellectual developments in history.
Here is a more detailed list of topics we will cover:
Vector algebra and geometry
vector algebra and interpretation in space
dot product, cross product, orthogonality, direction angles, projection
Quadric surfaces
equations and visualizations of cylinders, paraboloids, hyperboloids, etc.
Calculus of one-parameter vector functions in space
parametric equations of lines and curves
vector equations of lines and curves
limits and continuity
derivatives of vector functions
tangent vectors and lines
interaction of the derivative with dot and cross products
integrals of vector functions
arc length and curvature
velocity and acceleration
Calculus of functions of two or more variables
surfaces and level curves
limits and continuity
differentiation
partial derivatives
Clairaut’s Theorem
tangent planes
linear approximation (time permitting)
differentials
directional derivatives, the gradient, and interpretations
optimization
Lagrange multipliers (time permitting)
integration
double integrals over rectangles and their properties
integral as volume
average value
iterated integrals and Fubini’s Theorem
double integrals over general regions
double integrals in polar coordinates
density, mass, center of mass
surface area
Vector calculus
vector fields in the plane and the three-space
gradient fields
line integrals
line integral of vector fields and work along a curve
The Fundamental Theorem for Line Integrals
conservative vector fields and independence of path
Green’s Theorem
curl and divergence of a vector field and interpretations
parametric equations of surfaces
grid curves
tangent planes
surface area (time permitting)
surface integrals
surface integrals (flux) of vector fields
Stokes’ Theorem
Divergence Theorem (time permitting)