V63.0123-1 Calculus III : Multivariable calculus
Instructor: Alex Barnett
barnett at cims.nyu.edu
tel: 212-998-3296, rm 1122 Warren Weaver Hall (WWH)
Office hours (note change):
5-6pm Mondays and 11am-noon Tuesdays.
Find the volume of cool and unusual geometric shapes.
Electric fields are vector fields.
Charges are responsible for divergence
(sources of the field).
Fluid flow velocity is also a vector field.
Vortices are localized regions of high curl.
In this course we generalize
ideas of differential and integral calculus to two and three dimensions.
As well as giving us tools to model the real, three-dimensional world
through physics, chemistry, engineering,
etc (as in the pictures above),
this is also essential to understanding functions of many variables
in more abstract situations, such as arise in
computer science, economics, and statistics and probability.
The emphasis will be on intuition, visualization, and problem-solving,
rather than theorem-lemma-proof.
I am a physicist by training and I am sure this
in class W Feb 19, 9am-10am. No calculators or cheat sheets.
You will be given the harder-to-remember formulae.
Please practise with the Sample Midterm1,
eventually checking against answers (corrected).
- Midterm1 solutions: I made a mistake in 3. Method A. You should
replace it with d = |r_o|sin theta = |r_o cross v| divided by |v|.
Best is to get notes from
lecture 10 (W Feb 26).
in class M Apr 7, 9am-10am. Calculators allowed.
You may bring a single-page formula sheet of your own (few standard
formulae will be provided). Practise with
Sample Midterm2 (including lists of Stewart practise problems),
This sample exam may be a little longer than the real one.
- Final, in usual
classroom, Wed May 7,
10am-11:50. Calculators and double-sided formula sheet allowed.
Answers (not full solutions) are here.
Sample Final (including lists of Stewart practise problems),
Handouts, corrections, and interesting links:
- Mini Quiz Solutions (W Jan 22).
Make sure you understand these before anything else!
Trigonometric identities, part of an excellent, interactive
You should instantly know double/half angle formulae, sum formulae, and
be able to sketch graphs of sin, cos, tan, cot, sec, cosec.
- Minor corrections to lecture, M Jan 27.
Math study guide including reviews of single-variable calculus,
and summary materials for our course.
Noncrossing lines in 3d.
- See how vectors are used in computer graphics:
Cross product gives unit normal vector,
giving shading (lighting) in applets like
local prof Ken Perlin's
(you need java).
- Derivative rule for scalar-vector products,
from lecture W Feb 5. You should become
familiar with this type of proof by manipulation.
Introduction to hyperbolic trig functions sinh and cosh.
- Solutions to HW4: 15.3.48, I wrote sec^2 x rather than sec^2 2x
throughout. Please correct your copy.
- Lecture Mon 3 Mar: `enjoyment of dinner' function E(y,z) =
5(30-y-z) - (y-10)^2 + z^3 plotted for you as
contours. Our search for a maximum
actually found the critical point y = 15/2, z = sqrt(5/3) shown
as an "x" on the diagram. You can see it is a saddle-point (you can test this).
The true maximum is at the top corner,
where the contours become so densely packed because of the z^3 term.
- Little page on why Lagrange multipliers work, and practise problems
with hints (note that the 3 or 4 eqns are derived in a different way
- Slightly more theoretical tutorial notes on double, triple integrals.
(Part of complete online course similar to ours, bit more theoretical).
- Errors in solutions to HW6: 15.8.4 the max and min are 26 and -26,
not 22 as I had.
16.1.4 I flipped the limits on x and y: the technique is the same, and the
true answer -11.
- Errors in solutions to HW7: 16.4.2: inner integral limits are y=0
to y=2-x. Of course if you do it as Type II that's equally good.
Link and another link
on cylindrical and spherical
- HW9 will be postponed by 1 week, until April 16. Therefore there will
only be 11 HWs. The lowest 2 will still be dropped.
You will be pleased to know my research in physics
and medical imaging relies heavily
on partial derivatives (in the form of partial differential equations)
and almost every other aspect of this course. I am not special - almost
any practicing scientist would say the same. So... keep studying - it's useful!