Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by Ca

Projection of one vector on another
Projection of one vector on another
Use of cosine to find adjacent side
cos =
a
c
a = c cos
Use of cosine to find a projection
|proju v| = |v| cos
Definition of Dot Product
u v = |u| |v| cos
Relationship of Projectio

(x, y)
(x, y) can be represented as a point in the plane
A pair of numbers can be represented by an arrow.
A quantity with both magnitude and direction is
called a vector
The vector v from the origin to the point (3, 1) is
denoted by:
v = 3, 1
The length

d = distance between u = (u1 , u2 ) and v = (v1 , v2 )
d2 = (v1 u1 )2 + (v2 u2 )2
d = (v1 u1 )2 + (v2 u2 )2
u = (u1 , u2 , u3 )
v = (v1 , v2 , v3 )
d = distance between u and v
d2 = d21 + d22
d2 = d21 + d22 = d21 + (v3 u3 )2
d21 = (v1 u1 )2 + (v2 u2 )2
d2

f (x, y) = 4 x2 y 2
f (x, y) = 4 x2 y 2
f (x, y) = 4 x2 y 2 = C
f (x, y) = 4 x2 y 2 = C
with f
drawn
f (x, y) = 1 x + y
f (x, y) = 1 x + y = C
f (x, y) = 1 x + y = C
with f
drawn
f (x, y) = 1 x2 + y 2
f (x, y) = 1 x2 + y 2
f (x, y) = 1 x2 + y 2 = C
f (x,

Find a vector perpendicular to two given vectors
= r F
= r p
L
B = qv B
F
There are many vectors perpendicular to two given vectors.
= w1 , w2 , w3 that is perpendicular to both
Find a vector w
x = x1 , x2 , x3 and y = y1 , y2 , y3
x w
=0
y w
=0
x

z = z0 +
z
z
(x0 , y0 ) (x x0 ) +
(x0 , y0 ) (y y0 )
x
y
z = z0 +
z
z
(x x0 ) +
(y y0 )
x
y
z = z0 +
z
z
(x x0 ) +
(y y0 )
x
y
z z0 =
z
z
(x x0 ) +
(y y0 )
x
y
dz =
z
z
dx +
dy
x
y

Let T be the triangle in the xy plane with vertices (0, 0), (1, 2)
and (0, 4).
(3 3x) dA =
T
(3 3x) dy dx
Let R be the semicircular
region in the xy plane bounded by
the y-axis and x = 1 y 2
(3 3x) dA =
R
(3 3x) dy dx
Let R be the semicircular
region in t

f (x, y) = 100 x2 y
D~i f (x, y)
D~i f (x, y)
This is the slope of the tangent plane when (x, y) is moved in
the direction of ~i
f
D~i f (x, y) =
x
D~j f (x, y)
This is the slope of the tangent plane when (x, y) is moved in
the direction of ~j
f
D~j f (x,

A set of points (x, y) is a level set of z = f (x, y) if
f (x, y) = constant
for all points in this set.
A level set is also called a contour
f (x, y) = 1 (y 2x)2
f (x, y) = 1 (y 2x)2

Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by Ca

Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by Ca

Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Scanned by Ca

1
f (x, y) = x2 y 2 + 1
2
1
f (x, y) = x2 y 2 + 1
2
f
f (x + h, y) f (x, y)
= lim
x h0
h
Other notations:
f
x
fx
f
f (x, y + h) f (x, y)
= lim
h0
y
h
Other notations:
f
y
fy
f
f (x + h, y) f (x, y)
= lim
x h0
h
f
f (x + h, y) f (x, y)
= lim
x h0
h
f
f (x,