Friday, Jan 19, Lecture 5, Week 2

14.3 Cross product. Discussion of handedness (orientation).

Monday, Jan 22, Lecture 6, Week 3

Lecture on matrices and determinants of n × n matrices - not in text except for a minimal discussion on page 604-605. Determinants and cross products, 14.3.

Friday, Jan 26, Lecture 7-8, Week 3

Functions of several variables (15.1). Partial derivatives (15.2).

Monday, Jan 29, Lecture 9, Week 4

Limits and continuity (15.3). We began differentiability (15.4).

Wednesday, Jan 31, Lecture 10, Week 4

Differentiability (15.4). Directional derivatives and gradients (15.5). We have denoted the gradient of f by del(f) where del is an inverted majuscule delta. Another popular notation is grad(f).

Friday, Feb 2, Lecture 11, Week 4

Chain rule and implicit differentiation (15.6).

Monday, Feb 5, Lecture 12, Week 5

More on gradients. Tangent plane to level surfaces and graphs (15.7) Maxima and minima (beginning 15.8). I still need to discuss using the tangent plane for approximation (by differentials).

Wednesday, Feb 7, Lecture 13, Week 5

Approximation by differentials (tangent space approximation to graph) - 15.7. Maxima and minima for functions of several variables (15.8). Second partial derivatives test.

Friday, Feb 9, Lecture 14, Week 5

Lagrange's method for constrained extremization (15.9). This material will not be on the test on Wednesday, but you will definitely see Lagrange multipliers on the final exam.

Monday, Feb 12, Review, Week 6

Review. Don't be bashful! Bring (mathematical) questions. If you don't bring questions then I will not have much to say! The test will cover 13.2, 13.3, 14.1-14.4, 15.1-15.8 so about the first 10 "lessons" as listed on the inside front cover of the Study Guide.

Wednesday, Feb 14, Test, Week 6

In class test. Bring a calculator or two - one to drop and one to use! Bring a notesheet, but only one. Turn off phones, etc.

Friday, Feb 16, Lecture 15, Week 6

We covered 16.1 and 16.2 - double integrals and iterated integrals.

Monday, Feb 19, Lecturer 16, Week 7

16.3 - Double integrals over general regions.

Wednesday, Feb 21, Lecturer 17, Week 7

Last day to withdraw. 12.6, 16.4 - Polar coordinates. Change of coordinates in double integrals -

• general coordinates, x=x(u,v), y=y(u,v)
• polar coordinates, x=r cos(θ), y=r sin(θ)
• hyperbolic coordinates, x=u²-v², y=2uv

Many coordinate systems are important in physics and applications in other areas but our text deals only with polar coordinates (in this section). Bummer! Modern calculus texts omit a lot of the fun. (See however the text, section 16.8, p. 725, for a brief description of the general change of coordinates that we derived in class.)

Friday, Feb 23, Lecturer 18, Week 7

16.7, 16.8 - Triple integrals and iterated triple integrals. Change of coordinates in triple integrals -

• general coordinates, x=x(u,v,w), y=y(u,v,w), z=z(u,v,w)
• cylindrical coordinates, x=r cos(θ), y=r sin(θ), z=z

We will discuss spherical coordinates (16.8) on Monday.

Monday, Feb 26, Lecturer 19, Week 8

14.7, 16.8 - Triple integrals in spherical coordinates. The volume of a ball and the surface area of a ball - Archimedes' equal area theorem for a cylinder circumscribing a sphere. We obtained Archimedes' result by differentiating the volume of a sphere with respect to the radius. One can also obtain it by computing the surface area as a double integral - see Section 16.6. (Of course, Archimedes had none of these tools available, though he did have an informal limit argument that he used to discover results, but not to prove them.) By the way, we used the volume of the ball, which may be derived from another theorem of Archimedes, that the volume of the ball is 2/3 the volume of the circumscribing cylinder (simple to prove now, but not 2200+ years ago).

Wednesday, Feb 28, Lecturer 20, Week 8

(16.5, 16.6, 16.7) Applications of double and triple integrals.

Friday, Mar 2, Lecturer 21, Week 8

Applications of multiple integrals continued.

Monday, Mar 5, Lecturer 22, Week 9

(13.4, 14.4, 13.5, 14.5) Vector valued functions and parametrized curves (moving point). Velocity, speed, arc length, acceleration, unit tangent, principal unit normal, curvature, re-parametrization by arc length

Wednesday, Mar 7, Lecturer 23, Week 9

(13.5, 14.5) Parametric curves continued: principal unit normal, principal unit binormal, torsion, Serret-Frenet formulae (Frederic-Jean Frenet and Joseph Alfred Serret, mid 1800's).

Friday, Mar 9, Lecturer 24, Week 9

Example: Charged particle moving in an electric field. (Also an advertisement for our differential equations course.)

Monday, Mar 12, Lecturer 25, Week 10

Review. If I remember to bring the forms we'll do teaching evaluations today. so bring a #2 pencil.

Wednesday, Mar 14, Lecturer 26, Week 10

Review.

Friday, Mar 16, Lecturer 27, Week 10

Review.

petersen@math.oregonstate.edu

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