Calculus III - Lecture Notes and Videos

Math 273 (Fall 2014)  - Lecture Notes and Videos on Calculus III (Calculus of functions of several variables)

Scribes from all lectures so far (as a single big file)

Lec # Date Topic(s) Scribe Tegrity
1 Aug 26 syllabus, functions of several variables, domain, range, interior, boundary, open and closed sets scribe video
2 Aug 28 (un)bounded sets, level curves and surface of $$f(x,y)$$, limits and continuity in high dim., partial derivatives scribe video
3 Sep   2 $$\frac{\partial f}{\partial x}$$ as tangent in one plane to $$z=f(x,y)$$, implicit partial differentiation, 2nd order partial derivatives scribe video
4 Sep   4 mixed derivative theorem: $$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$$, chain rule for one independent and one intermediate variable scribe video
5 Sep   9 application of chain rule, chain rule for $$f(x(t),y(t),z(t))$$, branch diagrams, other instances of chain rule scribe video
6 Sep 11 chain rule in implicit differentiation, more chain rule problems, intuition for directional derivative scribe video
7 Sep 16 gradient vector $$\nabla f$$, $$~(D_{\mathbf{\hat{u}}} f)_{P_0} = (\nabla f)_{P_0} \cdot \mathbf{\hat{u}},~$$ derivative of $$f$$ at $$P_0$$ in the direction of a vector $$\mathbf{u}$$ scribe video
8 Sep 18 direction of largest increase and decrease, tangent line to level curve, find $$\hat{\mathbf{u}}$$ along which $$(D_{\hat{\mathbf{u}}} f)_{P_0}=d$$ scribe video
9 Sep 23 Find $$(D_{\mathbf{w}} f)_{P_0}$$ given $$(D_{\mathbf{u}} f)_{P_0}, (D_{\mathbf{v}} f)_{P_0}$$, tangent plane and normal line to surface $$f(x,y,z)=c$$ at $$P_0$$ scribe video
10 Sep 25 tangent plane and normal line, tangent line to curve of intersection of two surfaces, plot 3D surfaces scribe video
11 Sep 30 estimating change in specific direction, review for exam 1 scribe video
12 Oct   2 exam 1
13 Oct   7 linearization of $$f(x,y)$$, total differential $$df = f_x dx + f_y dy$$, change in temperature wrt space and time scribe video
14 Oct   9 wind chill factor exercise (Matlab/Octave session), application of total differential, local maxima/minima scribe video
15 Oct 14 local extrema, first derivative test, critical points, saddle point, second derivative test, Hessian $$f_{xx} f_{yy} - f_{xy}^2$$ scribe video
16 Oct 16 more on seocnd derivative test, critical point where $$f_x,f_y$$ are undefined, finding absolute extrema in a region scribe video
17 Oct 21 more problems on absolute extrema in a region $$R$$, critical points in interior of $$R$$ and along boundaries of $$R$$ scribe video
18 Oct 23 limits of an integral that give absolute maximum, multiple integral over rectangular domain as volume sum scribe video
19 Oct 28 examples of double integrals over rectangular regions, volume under surface and above the $$xy$$-plane scribe video
20 Oct 30 double integrals over general domains, region of integration, limits using vertical and horizontal cross sections scribe video
21 Nov  4 sketching the region of integration $$R$$, reversing order of integration, splitting $$R$$ into simpler regions scribe video
22 Nov  6 properties of double integrals-- sum,domination, additivity, volume of region bounded by surface and $$R$$ scribe video
23 Nov 11 Veteran's Day (no class); review for exam 2 (flipped lecture) scribe video
Nov 13 exam 2
24 Nov 18 area of closed region in plane by double integration, average value of $$f(x,y)$$ over $$R$$ scribe video
25 Nov 20 double integrals in polar coordinates, finding limits of $$r,\theta$$, area of $$R$$ in polar coordinates $$A = \iint\limits_R r dr d\theta$$ scribe video
26 Dec   2 line integrals, curve C: $$\mathbf{r}(t) = g(t)\mathbf{i} + h(t)\mathbf{j} + \ell(t)\mathbf{k}, a \leq t \leq b$$, $$\int\limits_C f(x,y,z) ds = \int\limits_a^b f(g(t),h(t),\ell(t))|\mathbf{v}(t)| dt$$ scribe video
27 Dec   4 line intergals over vector fields, $$\int\limits_C \mathbf{F}\cdot\mathbf{T} ds = \int\limits_a^b ( \mathbf{F}(\mathbf{r}(t))\cdot\left( \frac{d\mathbf{r}}{dt} \right) dt$$, work done in moving along $$C$$ in field $$\mathbf{F}$$ scribe video
28 Dec   9 simple closed curve $$C$$, circulation around $$C$$, flux across $$C = \int\limits_C \mathbf{F} \cdot \hat{\mathbf{n}} ds = \int\limits_C M dx - N dy$$ for $$\mathbf{F} = M \mathbf{i} + N \mathbf{j}$$ scribe video
29 Dec 11 flux/circulation density, Green's theorem $$\oint\limits_C \mathbf{F} \cdot \hat{\mathbf{n}} ds = \iint\limits_R \left(\frac{\partial M}{\partial x} + \frac{\partial N}{\partial y}\right) dA$$, $$\oint\limits_C \mathbf{F} \cdot \hat{\mathbf{T}} ds = \iint\limits_R \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right) dA$$ scribe video
30 Dec 14 review for the final exam - problems from the practice final scribe video