Salas & Hille: Calculus several variables, 7th ed. Wiley 1995.
4 hours of classes a week for 14 weeks = 56 hours total = 28 lectures total. Note these plans (48 total hours) leave 8 extra leftover hours for reviews, tests, screwups, extra content, etc.:
Math 127: topic estim. # lecture hours cum. total
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ch 12. VECTORS. sec 1-6, 7* 7 7
length, distance, schwartz inequality, cross product, fancier
distances. quaternions*.
ch 13. VECTOR-VALUED FNS OF 1 VARIABLE
(PARAMETRICALLY DESCRIBED CURVES). sec 1-4, 5-7* 4 11
deriv. arc length. planetary motion*.
ch 14. REAL-VALUED FNS OF SEVERAL VARIABLES 4 15
sec 1,2*,3,4,6
level sets of quadratic fns ("quadric surfaces").
partial derivatives. equality of mixed partials.
ch 15. GRADIENT, OPTIMIZING, & IMPORTANT DIFFERENTIALS
sec 1,2,4-8,9*,10 12 27
grad. various chain rules. intuitive meaning of grad.
kinds of critical points; discriminants.
constrained maxima. Lagrange's "multiplier" trick.
ch 16. DOUBLE & TRIPLE INTEGRALS sec 2-5, 7-11. 12 39
double integrals as double sums and over rectangles.
triple integrals as triple sums and over 3D boxes.
polar, cylindrical & polar coords.
differential element of volume.
jacobians, changes of variables, and another chain rule.
Newton's rootfinding method generalized to multidimensions*.
ch 17. LINE & SURFACE INTEGRALS sec 1,2,5-10.
curve integrals. potential energy fns.
surface integrals. surface area. Flux.
divergence. curl.
Green thm, divergence thm, Stokes thm.
Maxwell equations* 9 48
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* denotes "optional" coverage of that section or topic.
sec 15.10 will be postponed until after curl & line integrals in ch 17.
Homework: 33.3% Final: 33.3 Quizzes: 33.3
d-Dimensional Vectors are just d numbers. For example (2,6,8) is a 3-dimensional vector. It can be viewed as the coordinates of a point in 3-space, or it can be viewed as 3 numbers. It often happens in math we want to do the same things with all the numbers. So why should we repeat ourselves? Just say "do this" and have it understood you mean "to all of the numbers." So by inventing notations to specify VECTOR operations we can say more with less writing. Vectors are the assembly lines of mathematics.
We live in a 3-dimensional world. We draw on 2-dimensional surfaces. We are interested in surfaces: the surface of the world. The surface of your body. Etc. We are interested in curves drawn on surfaces. We want to know how these things interact. Water flows into a surface, filling the volume enclosed by that surface. How much water? What is the "flow rate"? How can we describe these all concepts with math? What are surface areas? What are volumes? How can surfaces distort? What are different coordinate systems and how are they related? This course will tell you all that.
Another important thing: we often have functions of many variables. (Your car's gas mileage as a function of the wheel size, the piston size, the IQ of the driver, the radio-speaker wattage, and the octane of the fuel.) How do you find the magic values of these 5 variables that optimize the function? And what if those variables have to obey some constraints? (E.g., to make it more realistic, IQ of driver plus speaker-wattage equals constant. Now optimize gas mileage.)
And another: physics is all about vectors in 3-dimensional space. Planets move. Forces exert. Fluids flow. Energies shift. Electric and magnetic fields do certain things and have certain properties, underlying the operation and understanding of electric motors, generators, etc. How do we describe all that with math? Vector calculus is the tool, the language, to do all that.
Finally, if we want to talk about vectors interacting with other vectors, then there are d-squared possible kinds of interactions between the d numbers in one vector and the d numbers in the other. So to describe that sort of thing, we naturally find ourselves (for this and other reasons) dealing with dXd matrices. That leads to the importance of developing notations for, and ways of dealing with, and understanding of, matrices.