| Lec | Date | Topic(s) | Scribe |
| 1 |
Aug 26 |
syllabus,
functions of several variables, domain, range, interior,
boundary, open and closed sets
|
Scb1
|
| 2 |
Aug 28 |
(un)bounded sets, level curves and surface of \(f(x,y)\),
limits and continuity in high dim., partial derivatives
|
Scb2
|
| 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
|
Scb3
|
| 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
|
Scb4
|
| 5 |
Sep 9 |
application of chain rule, chain rule for \(f(x(t),y(t),z(t))\),
branch diagrams, other instances of chain rule
|
Scb5
|
| 6 |
Sep 11 |
chain rule in implicit differentiation, more chain rule
problems, intuition for directional derivative
|
Scb6
|
| 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 \(\mathbf{u}\)
|
Scb7
|
| 8 |
Sep 18 |
direction of largest increase and decrease, tangent line to
level curve, find \(\hat{\mathbf{u}}\) along which
\((D_{\hat{\mathbf{u}}} f)_{P_0}=d\)
|
Scb8
|
| 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 \(P_0\)
|
Scb9
|
| 10 |
Sep 25 |
tangent plane and normal line, tangent line to curve of
intersection of two surfaces, plot 3D surfaces
|
Scb10
|
| 11 |
Sep 30 |
estimating change in specific direction, review
for exam
1
|
Scb11
|
| 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
|
Scb13
|
| 14 |
Oct 9 |
wind chill factor exercise (Matlab/Octave
session), application of total differential, local
maxima/minima
|
Scb14
|
| 15 |
Oct 14 |
local extrema, first derivative test, critical points, saddle
point, second derivative test, Hessian \(f_{xx} f_{yy} - f_{xy}^2\)
|
Scb15
|
| 16 |
Oct 16 |
more on seocnd derivative test, critical point where \(f_x,f_y\)
are undefined, finding absolute extrema in a region
|
Scb16
|
| 17 |
Oct 21 |
more problems on absolute extrema in a region \(R\), critical
points in interior of \(R\) and along boundaries of \(R\)
|
Scb17
|
| 18 |
Oct 23 |
limits of an integral that give absolute maximum, multiple
integral over rectangular domain as volume sum
|
Scb18
|
| 19 |
Oct 28 |
examples of double integrals over rectangular regions, volume
under surface and above the \(xy\)-plane
|
Scb19
|
| 20 |
Oct 30 |
double integrals over general domains, region of integration,
limits using vertical and horizontal cross sections
|
Scb20
|
| 21 |
Nov 4 |
sketching the region of integration \(R\), reversing order of
integration, splitting \(R\) into simpler regions
|
Scb21
|
| 22 |
Nov 6 |
properties of double integrals-- sum,domination, additivity,
volume of region bounded by surface and \(R\)
|
Scb22
|
| 23 |
Nov 11 |
Veteran's Day (no class); review for exam 2 (flipped lecture)
|
Scb23
|
| |
Nov 13 |
exam
2
|
|
| 24 |
Nov 18 |
area of closed region in plane by double integration, average
value of \(f(x,y)\) over \(R\)
|
Scb24
|
| 25 |
Nov 20 |
double integrals in polar coordinates, finding limits of
\(r,\theta\), area of \(R\) in polar coordinates \(A =
\iint\limits_R r dr d\theta\)
|
Scb25
|
| 26 |
Dec 2 |
line integrals, curve C: \(\mathbf{r}(t) = g(t)\mathbf{i} +
h(t)\mathbf{j} + \ell(t)\mathbf{k}, a \leq t \leq b\),
\(\int\limits_C f(x,y,z) ds = \int\limits_a^b
f(g(t),h(t),\ell(t))|\mathbf{v}(t)| dt\)
|
Scb26
|
| 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
\(\mathbf{F}\)
|
Scb27
|
| 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 \(\mathbf{F} = M \mathbf{i} + N
\mathbf{j}\)
|
Scb28
|
| 29 |
Dec 11 |
flux/circulation density, Green's theorem \(\oint\limits_C
\mathbf{F} \cdot \hat{\mathbf{n}} ds = \iint\limits_R
\left(\frac{\partial M}{\partial x} + \frac{\partial N}{\partial
y}\right) dA\), \(\oint\limits_C \mathbf{F} \cdot
\hat{\mathbf{T}} ds = \iint\limits_R \left(\frac{\partial
N}{\partial x} - \frac{\partial M}{\partial y}\right) dA\)
|
Scb29
|
| 30 |
Dec 14 |
review for the final exam - problems from
the practice
final
|
Scb30
|