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

Last modified: Sun Dec 14 13:29:01 PST 2014