Lecture 18: 7.2 Surface Integrals Suppose we want to nd the total volume
of water in the oceans of the earth. At each point of the surface the depth of the
ocean is given by a function f .
To measure the total volume we divide up the surface S into smalle
(
Lecture 5: Section 2.3 Cont. If F : R3 R3 is given by F(x) = f1 (x1 , x2 , x3 ), f2 (x1 , x2 , x3 ), f3 (x
then the derivative matrix of partial derivatives is dened by
f1 f1 f1
x1 x2 x3 [
]
f
F F F
2 f2 f2
DF =
=
x1 x2 x3
x1 x2 x3
f
f3 f3
3
x1
Math 211 honors Multivariable Calculus
Lecture 1: Overview+Review of vector operations.
What is the multivariable calculus course about?
Curves in space (x(t), y (t), z (t), e.g. path of a particle.
Vectors and vector operations, e.g. the dot and the cros
Section 1.7: Cylindrical and Spherical Coordinates. Recall that in the plane
it is sometimes useful to introduce polar coordinates. There are two possible natural
and useful generalizations of this to space:
Cylindrical coordinates (r, , z ) of a point P
Lecture 2: Derivation of the geometric formulas for the dot and cross
products from the algebraic expressions. Let be the angle between the
vectors a and b. Then
a b = a1 b1 + a2 b2 + a3 b3 = a b cos
Here we take the rst equality as the denition and we m
Lecture 6: 2.5 The Chain Rule.
The Chain rule in one variable: suppose that y = g (x), and z = f (y ), i.e.
z = h(x), where h(x) = f (g (x) = f g (x) then
dz
dz dy
=
dx
dy dx
h (x) = f (y ) g (x)
The intuitive way to understand this is through the linear
Lecture 8: Section 3.4. We are now about to take derivatives of a vector eld.
Since the vector eld have three components and each depends on three variables
we are faced with trying to interpret the meaning of nine dierent derivatives.
However, it turns o
Lecture 11: 4.3 Lagrange multipliers.
From previous lecture: In particular in one dimension. If we want to nd the
maximum of f (x) over the interval I = [a, b] = cfw_x; a x b, then we rst nd
all the critical points f (ci ) = 0, i = 1, ., N and we check th
Lecture 10: 4.2: Maximum and minimum values.
Def. A function f : Rn R has a local maximum at a if f (x) f (a) when x
is near a. Similarly, f has a local minimum at a if f (x) f (a) when x is near a.
If the inequalities hold for all x in the domain of f th
Lecture 7: 2.6 The implicit function theorem. A surface can be described
as a graph:
z = f (x, y )
or as a level surface
F (x, y, z ) = C
It is clear that a graph can always be written as a level surface with F (x, y, z ) =
z f (x, y ). The question is if
Lecture 12: 5.1-2 Double Integrals.
Areas and integrals If f 0 the integral is intuitively dened by the area:
b
f (x) dx = Area below the graph y = f (x) and above the x axis, from a to b.
a
The area of a region is dened by lling in the region with many s
Lecture 9: 4.1 Taylors formula in several variables.
Recall Taylors formula for f : R R:
f
f (k) (a)
(a)(x a)2 + . +
(x a)k + Rk (x a, a)
2
k!
where the remainder or error tends to 0 faster than the previous terms when x a:
M
(2)
|Rk (x a, a)|
|x a|k+1
,
Lecture 15: 6.1 Path integrals and Line Integrals. Let us rst recall the
denition of arc-length of a parameterized curve c(t), a t b. One starts by
dividing the curve up into smaller curves, a = t0 < t1 < . < tn = b, tk = a + k t,
t = (b a)/n, with endpoi
Lecture 14: 5.5 The change of variable theorem on the line. Suppose that
x(u), a u b is a change of variables. In order for it to be invertible we assume
that dx(u)/du > 0, when a u b. Then we can change variables in the integral:
x(b)
b
dx
(1)
f (x) dx
Lecture 13: 5.2 Double integrals over more general regions.
If D is a region of type I: D = cfw_(x, y ); a x b, g1 (x) y g2 (x) then
b
[
]
g2 (x)
f (x, y ) dA =
f (x, y ) dy dx
D
a
g1 (x)
In fact by the slice method
b
f (x, y ) dA =
A(x) dx,
D
g2 (x)
wher
Lecture 16: 6.2 Greens theorem. Suppose that D is a domain in the plane
with boundary curve C going in positive direction, i.e. walking in the direction of
C the domain D should be on your left. Greens theorem says that
(
Q P )
P dx + Qdy =
dxdy
x
y
C
D
Lecture 17: 7.1 Parameterized surface. A Parameterized surface is given
in terms of two parameters
x = x(u, v ),
y = y (u, v ),
z = z (u, v ),
or
T(u, v ) = x i + y j + z k
A particular example of a parameterized surface is a graph:
z = f (x, y ),
or
T(x,
Math 211 Practice Midterm 2, Fall 11, Lindblad.
1. (a) Use Lagranges method to nd the point on the plane x + 2y + 3z = 1 closest
to the point (2, 1, 0).
(b) Find the critical points of f (x, y ) = x2 + y 2 + x + y + 1 and determine if they
are local max m
Math 211 Practice Midterm 1, Fall 2011, Lindblad.
(0, 0, 0) = 2,
(0, 0, 0) = 3 and
(0, 0, 0) = 4.
x
y
z
(
)
dw
(a) Let w(t) = c(t) , where c(t) = t i + t2 j + 3tk is a curve. Find
(0)!
dt
(b) In which direction is the rate of increase of largest at the po
Lecture 20: 7.3 Stokes theorem. Let S be a surface with unit normal n and
positively oriented boundary C , i.e. if you walk in the direction of the curve on the
side of the normal then the surface should be on your left. Stokes theorem says
F ds =
curl F
Lecture 19: 7.2 Derivation of the ow integral. Let S be a closed surface
and for each point (x, y, z ) on the surface let f (x, y, z ) be the rate of ow of uid
out through the surface per unit surface area and unit time, i.e. the ow out of a
small area S
Lecture 4: 2.2 Limits.
Def. Suppose that f : Rn \ cfw_a Rm . We say that the limit of f (x) as x
approaches a is L and write
lim f (x) = L
x a
if for every > 0 there is a > 0 such that f (x) L < whenever 0 < x a < .
Def A function f : Rn Rm is called cont