What Topics Are Covered in Calculus III?

[GoldPheonix]

I'm teaching myself calculus III off of various websites including wikipedia.

However, I don't know exactly where to begin. I've started by looking at the partial derivative and parts of vector calculus.

My calculus learning so far has been:

Calc I:
Limits
Basic and complicated single-variable differential calculus (chain rule, power rule, et cetera. Differentiation isn't hard.)
Basic single-variable integral calculus (power rule, et cetera)

Calc II:
Infinite limits, sequences and series. (Taylor series, power series, et cetera)
Improper integrals, work as an integral, integration by parts, all three rotated volume integration, introduction to conical sections, and introduction to polar/parametric/vector calculus.

So what would most colleges teach after this in calc III?

 

[ice109]

why don't you just buy a used textbook

 

[GoldPheonix]

Because I'm going to have to buy one in a semester... And this is just for something to do over the summer.

 

[cristo]

Why don't you find out which textbook you'll be using next semester and buy that one and study through it? It's a lot easier learning out of a textbook, since it's written in a logical order, as opposed to various webpages.

 

[lurflurf]

There is significant variation in topics, rigor, and depth of coverage

functions of several variables
limits
sequences and series
partial derivatives
multiple integration
change of variable
chain rule
implict function thm
inverse function thm
determinants
basic linear algebra
Jacobian
Hessian
optimization
lagrange multiplier
differentials
partial differential equations
conical sections parametric equations
vector algebra
curl grad div
identities for curl grad div
line,surface,volume integration
fundamental theorems of vector calculus about 30
liebniz rules for integration and differentiating products
differential forms (if lucky because 30 ftovc->1)

In summary do every thing from single variable again with several variables

 

[SRode]

everything lurflurf stated, didnt have jacobian, hessian, liebniz in mine. Heavy concentration to stokes theorm, greens theorm, fields and flux and applications of mathamatical physics. Also course will be different depending whether it is specifically for math majors or applied science majors, and whether there is a Calculus 4 class or if it ends at 3

 

[Saladsamurai]

If you have realplayer (which can be obtained free) check out this guy's lectures... http://press.princeton.edu/video/banner/8351.html They are not labeled I, II, III, but some of the topics might be covered.

Casey

 

[leakin99]

Here is an actual textbook about calculus which has been made available online. It has problems and answers included for the odd numbered problems. Just follow the format of the book.
http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm
Here is the Calc III website for the summer course that was given at my school, it has a schedule with topics. You can follow it if you want. Should be able to cover the stuff in 3 – 4 weeks: http://www.math.ubc.ca/~warcode/m200/index.html

 

[proton]

Here is an actual textbook about calculus which has been made available online. It has problems and answers included for the odd numbered problems. Just follow the format of the book.
http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm
Here is the Calc III website for the summer course that was given at my school, it has a schedule with topics. You can follow it if you want. Should be able to cover the stuff in 3 – 4 weeks: http://www.math.ubc.ca/~warcode/m200/index.html

that seems similar to what I learned in my Calc III class except mine also included Vector calculus: div, grad, curl, and green's, stoke's and divergence thm

 

[sam1]

My school used Stewart's 5th edition Calculus for calc I-III, and we separated it like this. I think this is more or less the standard breakup:

calc I, chapters 1-5:
- functions
- limits and derivatives (defn of limit, limits at infinity, continuity, etc.)
- rules for differentiation (product, chain, quotient, logs, hyperbolic functions, linear approximations)
- applications of differentiation (max/min, MVT, l'hospital, optimization, Newton's method, antiderivatives)
- intro to integrals (FTC, substitutions, areas/distances, definite/indefinite)

calc II, chapters 6-11:
- applications of integration (volumes, work as an integral, areas between curves)
- techniques of integration (parts, trig integrals, trig substitution, partial fractions, numerical methods, improper integrals)
- applications again (arc length, surface area, applications to physics, bio, econ, etc. are given in the book)
- basic differential equations (first order separable, first order linear)
- parametric/polar coordinates (intro/defns, calc with polar/parametric, arc lengths, conics)
- sequences/series (integral test, comparison tests, convergence, power series, taylor/maclaurin, etc.)

calc III, chapters 12-16:
- vectors/review of geometry/linalg (vectors, dot/cross product, eqns of lines/planes, cylindrical/spherical coords)
- vector functions (space curves, calc with vector functions, arc length/curvature)
- partial derivatives (multivariable functions, limits/continuity, linear approximations, directional derivatives, grad, max/min, lagrange multipliers)
- multiple integrals (double/triple integrals over rectangles, over general regions, in spherical/cylindrical coordinates, change of variables)
- vector calculus (vector fields, line integrals, green's theorem, curl, div, surface integrals, stokes' theorem, gauss/divergence theorem)