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SHORT REVIEW FOR MATH 231 FINAL
Geometry
: dot products, cross products, scalar triple product ; interpretation as area of
parallelogram, volume of parallelpiped.
Euclidean/Cartesian, Polar, Cylindrical, Spherical Coordinates and their volume elements
dxdydz,rdrdθ,ρ
2
sin(
φ
)
dρdφdθ
.
Parametric equations of curves
C
:
r
(
t
): (1) line through P in direction v [
P
+
tv
], (2) line
segment from P to Q [(1

t
)
P
+
tQ
]. Tangent line to curve at P. Level curves
f
(
x,y
) =
k.
Parametric equations of surfaces
S
:
R
(
u,v
) =
x
(
u,v
)
,y
(
u,v
)
,z
(
u,v
)): (1) plane through
P
0
with normal vector
n
[(
P

P
0
)
•
n
= 0], (2) sphere
ρ
=
a
in spherical, (3) cylinder
r
=
k
in
cylindrical, (4) graph
z
=
g
(
x,y
) or
y
=
g
(
x,z
).
Tangent plane to surface at
P
0
=
r
(
u
0
,v
0
):
n
=
r
u
×
r
v
; graph
z
=
g
(
x,y
) has
n
=
(

g
x
,

g
y
,
1). Level surfaces
f
(
x,y,z
) =
k.
Area of surface
A
(
S
) =
R R
S
1
d
S
; surface integral of function
R R
S
f
(
x,y,z
)
d
S
=
R R
D
f
(
r
(
u,v
)
r
u
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This note was uploaded on 04/24/2009 for the course CALC 2401 taught by Professor Stojanovic during the Spring '09 term at Georgia Institute of Technology.
 Spring '09
 Stojanovic
 Geometry, Scalar, Dot Product

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