Download MATHEMATICS 2030

MATHEMATICS 2030
Summary
1. Vectors and analytic geometry in R2 and R3
a. Vectors, dot product, cross product (trigonometric and analytic definitions and
arithmetic properties)
b. Equations of lines and planes (vector form, scalar forms), equations of the
intersection of two planes and of the plane containing two intersecting lines
2. Differential calculus of curves, surfaces, and vector-valued functions
a. Parameterized curves, level curves, tangent vectors, unit tangent and normal
vectors, velocity and acceleration vectors
b. Parameterized surfaces, level surfaces, partial derivatives, tangent planes, total
differential and linear approximations, Chain Rule (total derivative, total partial
derivatives), directional derivatives, gradient, higher partial derivatives and
Clairaut’s Theorem, optimization (maxima and minima)
c. Vector fields, gradient (conservative) vector fields, conditions for conservativity
over domains in R2 and R3, calculation of the primitives of a conservative vector
field
3. Integral calculus of curves, surfaces, and vector-valued functions
a. Arc length of a curve parameterized x, y, z, or some parameter t, line integrals of
functions with respect to arc length, line integrals of vector fields over curves
b. Multiple integrals in two and three variables—area and volume elements, area,
volume, Fubini’s Theorem and iterated integrals over domains in R2 and R3
(computation, determination of limits of integration)
c. Surface area and surface integrals (time permitting)
4. Polar, cylindrical, and spherical coordinates
a. Definitions and parameterization of curves and surfaces, conversions between
Cartesian, cylindrical, and spherical coordinates in R3.
b. Area element and integrals in polar coordinates
c. Volume element and integrals in cylindrical and spherical coordinates
5. The Fundamental Theorem of Calculus in two and three variables
a. Fundamental Theorem of Calculus applied to conservative vector fields
b. Closed curves and independence of path
c. Open, connected, and simply connected sets in R2 and R3, equivalent conditions
on suitable domains in R2 and R3 for a vector field to be conservative
d. Green’s Theorem