|
|
Math 252 (Calculus III) -- "Why must I take this course?"
Related Courses and Articulation Possibilities
The following is not meant to be exhaustive, or even official; rather, it is topic-oriented in relation to the ideas and techniques briefly introduced in Math 252, which has as a prerequisite Math 151, with a grade of C or better.
Following the order of the chapters in Multivariable Calculus by James Stewart, 5th edition (2003):
 Chapter 13 - Vectors
- vectors in 2- and 3-space
- dot and cross products
- lines and planes
- quadric surfaces
- cylindrical and spherical coordinates
This chapter is mainly analytic geometry in Euclidean 3-space.
From these ideas one could progress to:
- Math 511, Projective Geometry (prerequisite Math 254) or to
- Math 512, Non-Euclidean Geometry (prereq. Math 122 or 151)
Chapter 14 - Vector functions
- vectors as parametrized curves in space
- derivatives and integrals of vector functions
- arc length and curvature
- velocity and acceleration
- tangent and normal vectors to a curve
- Kepler's Laws
This chapter deals with calculus of curves not necessarily restricted to 2- or even 3-space.
There are many Physics and Astronomy applications; also, following curvature on to torsion, to the T, N, B basis vectors and to the Frenet-Serret equations, one is knocking on the doors of Differential Geometry, given at UCSD.
The introduction to this is Math 533, Vector Calculus, last given here by Prof. Villone in Spring 2003 (prereq. Math 254 or 342A).
Chapter 15 - Partial derivatives
- limits
- continuity and differentiability of ƒ(x,y) or ƒ(x,y,z)
- tangent planes
- linear approximations
- several generalizations of the chain rule
- directional derivatives and gradients
- extrema of functions of several variables, including the method of Lagrange multipliers
This chapter deals with several ways to generalize the notion of rate of change to functions of several variables.
A more sophisticated and in some respects simpler treatment of these functions is through linear algebra, using derivative matrices, and this is done in Math 342A, Methods of Applied Math I (prereq. Math 252), which also is an introduction to ordinary differential equations. Related closely also are:
- Math 337, Elementary Differential Equations (prereq. Math 151)
- Math 531, Partial Differential Equations (prereq. Math 252 and 337)
Chapter 16 - Multiple Integrals
- double Integrals over regions
- iterated Integrals
- polar double Integrals
- applications of them such as mass-density-moments of inertia and probability
- surface area
- triple Integrals in rectangular, cylindrical, and spherical coordinates
- change of variable
- Jacobians
This chapter deals with volume and integration of functions of several variables over areas and volumes.
Of course there are many applications to physics and engineering here; besides being required for majors in Aeronautical, Electrical; Civil, and Mechanical Engineering, Math 252 is an explicit prerequisite to
- Physics 197, Wave Motion, Optics, etc.
- Chemistry 410, Physical Chemistry
- Astronomy 340, Spherical Astronomy
- Engineering 280, Methods of Analysis
- Math 562, Mathematical Methods of Operations Research (which has Math 254 as a prereq. also)
Chapter 17 - Vector Calculus
- vector and force fields
- conservative fields
- line integrals and their fundamental theorem
- path independence
- Green's theorem
- divergence and curl and the vector version of Green's theorem
- parametrized surfaces and surface areas from this standpoint [usually simpler than and hence preferred to the Ch. 16 treatment]
- surface integrals
- Stokes' theorem
- Gauss' divergence theorem
- generalized path independence
Vector calculus opens the door to many powerful principles of physics and engineering, such as:
- Maxwell's equations
- ideas of flux across a surface (e.g., electrical flux) or divergence out of a 3-dimensional region (e.g., hydrodynamic or plasma)
- heat transfer
- fluid dynamics, etc.
Chapter 17 also opens the door to even more powerful techniques and areas of study within mathematics:
- Math 534A, Advanced Calculus I (prereq. Math 342A, or Math 245 and 254)
- Math 532, Functions of a Complex Variable (prereq. Math 252)
- Math 521A, Abstract Algebra I (prereq. Math 245 and 252)
- Math 522, Number Theory (prereq. Math 245 and 252)
- Calculus on Manifolds, given at UCSD
Usually, one does not get past surface integrals in the above list of topics, or sometimes even Green's theorem, which is the last topic on the official course outline.
Math 252 as a Requirement
The above is meant to be informative only, without claiming definitiveness or completeness as to articulation or official university requirements. However, it's nice sometimes to know a little something about where a specific course fits into the general academic scheme of things. Perhaps needless to say, Math 252 is required for all six of the baccalaureate degree programs in mathematics:
- AB in Math
- AB in Single Subject Math Credential
- BS in Math with emphasis in either
- Applied Math, or
- Computational Science, or
- Science, or
- Mathematical Finance
- BS programs in Statistics
- BS in Statistics
- BS in Statistics with emphasis in Actuarial Science
|
