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Calculus III > Vector Calculus > Flashcards

Vector Calculus Flashcards

1
Q

Distance Formula for (x,y,z)

A

PQ = ( (x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2 )1/2

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2
Q

magnitude of v

A

||v|| = ( a2 + b2 + c2 )1/2

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3
Q

vector components

P = point (a,b,c)

Q = point (d,e,f)

A

PQ = < d - a , e - b , f - c>

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4
Q

midpoint formula

A

(a,b,c) + 1/2 * PQ

or

< (a1 + 2)/2 , (b1 +b2)/2 , (c1 + c2)/2 >

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5
Q

dot product

A

v * w = a1*a2 + b1*b2 + c1*c2

v * w = ||v|| * ||w|| * cos Ø

v * v = ||v||2

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6
Q

unit vector = ev

A

v / ||v||

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7
Q

vector projection of u onto v

u||

A

u|| = ( (u * v) / (||v||2) ) * v

u|| = (u * ev) * ev

u|| = ( (u * v) / (v * v) ) * v

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8
Q

parallel vectors

A

if lines through w and v are parrallel

or

w = Cv

for scalar C not equal to 0

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9
Q

equation of a sphere

of radius R

centered at (a, b, c)

A

(x - a)2 + (y - b)2 + (z - c)2 = R2

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10
Q

equation of a cylinder

of radius R

whose center line is the vertical axis through (a, b, 0)

A

(x - a)2 + (y - b)2 = R2

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11
Q

magnitude of projection

of u along v

A

||u|||| = ||u|| * cos Ø

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12
Q

Cross Product

v X w

A

orthogonal to v and w

v X w = - (v X w)

v X v = 0

v X w = 0 iff w = Cv or v = 0

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13
Q

length of v X w

A

||v X w|| = ||v|| * ||w|| * sinØ

||v X w||2 = ||v||2 * ||w||2 - (v * w)2

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14
Q

scalar projection of u onto v

or

the component of u along v

A

u * ev

where ev = v / ||v||

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15
Q

a vector v is orthogonal to w

A

if and only if

v * w = 0

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16
Q

the angle Ø between v and w is obtuse/acute

A

v * w < 0 is obtuse

v * w > 0 is acute

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17
Q

the decomposition of u with respect to v

A

u = u|| + uperp

where uperp is orthogonal to v

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18
Q

C * v X w = v X (C * w) =

A

= C (v X w)

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19
Q

(u + v) X w =

A

= u X w + v X w

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20
Q

translation

A

a vector v undergoes a translation when it is moved parallel to itself

without changing length or direction

w is the translate

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21
Q

equivalent

A

v and w are equivalent if w is a translate of v

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22
Q

linear combination

(if v and w are not parallel)

A

u = r * v + s * w

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23
Q

parallelogram spanned

A

parallelogram whose vertices are the origin and the terminal

points of v, w, and w + v

consists of linear combination

where 0

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24
Q

The line L through P0 = (x0 , y0 , z0) in the direction

of (parallel to) **v; ** where O is origin

vector parametrization :

parametric equation:

A

r(t) = OP0 + v = < x0 , y0 , z0> + t *

x = x0 + at ; y = y0 + bt ; z = z0 + zt

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25
Q

The line through the points P = (a 1 , b1 , c1) and Q = (a2 , b2 , c2)

vector parametrization:

parametric equation:

A

r(t) = (1 - t) * OP + t * OQ

x = a1 + (a2 - a1) * t ; y = b1 + (b2 - b1) * t ; z = c1 + (c2 - c1) * t

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26
Q

Equation of a line through origin that consists of the multiples

of the non-zero vector v =

vector parametrization:

parametric equation:

A

r0 = t * v =

x = a * t ; y = b * t ; z = c * t

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27
Q

unit vector cross products

A

i X j = k

**j ** X k = i

k X i = j

i X i = j X j = k X k = 0

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28
Q

parallelogram spanned by v and w

has area:

A

||v X w||

||v|| * ||w|| * sin Ø

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29
Q

parallelepiped spanned by u , v , and w

has volume

A

||v X w|| * ||u|| * |cos Ø|

u * (v X w) |

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30
Q

vector triple product

A

u * (v X w)

= the determinant of [u v w]

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31
Q

||v X w||2

A

||v||2 * ||w||2 - (v * w)2

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32
Q

right hand rule

A

fingers curl from x-axis to y-axis

thumb is in the positive z direction

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33
Q

coordinate planes

A

defined by setting one the

coordinates equal to 0

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34
Q

plane in R3

A

defined by

a*x + b*y + c*z = d

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35
Q

normal vector

n

A

a non-zero vector orthogonal to a point

P0 = (x0 , y0 , z0) determining

a plane passing through a point P0

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36
Q

P lies on a plane if

A

n * P0P = 0

or

n * OP = n * OP0

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37
Q

equation of a plane

through point P0 = (x0 , y0 , z0)

where normal vector = n = < a , b , c >

A

vector form:

n * < x , y , z > = d

scalar form:

a*(x - x0) + b*(y - y0) + c*(z - z0) = 0

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38
Q

parallel planes

A

the planes are parallel if they have a common normal vector

obtained by choosing normal vector n and varying

constant d in equation

a*x + b*y + c*z = d

39
Q

collinear

A

points that lie on the same line

three points determine a plan if they are

not collinear

40
Q

trace

A

the intersection of a plane with a coordinate plane

or a plane parallel to a coordinate plane

trace is a line unless the plane is parrallel to coordinate plane

(then trace is empty or is the plane itself)

obtained by setting x , y , or z = 0

41
Q

ellipsoids

A

(x / a)2 + (y / b)2 + (z / c)2 = 1

42
Q

hyperboloids of one sheet

A

(x / a)2 + (y / b)2 - (z / c)2 = 1

43
Q

hyperboloids of two sheets

A

(x / a)2 + (y / b)2 - (z / c)2 = -1

44
Q

elliptic cone

A

(x / a)2 + (y / b)2 = (z / c)2

45
Q

elliptic paraboloid

A

z = (x / a)2 + (y / b)2

46
Q

hyperbolic paraboloid

A

z = (x / a)2 - (y / b)2

47
Q

orthogonal projection of vector u obtained from a vector v

A

u - u|| = u<strong>perp</strong>

48
Q

PQ = ( (x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2 )1/2

A

Distance Formula for (x,y,z)

49
Q

||v|| = ( a2 + b2 + c2 )1/2

A

magnitude of v

50
Q

PQ = < d - a , e - b , f - c>

A

vector components

P = point (a,b,c)

Q = point (d,e,f)

51
Q

(a,b,c) + 1/2 * PQ

or

A

midpoint formula

52
Q

v * w = a1*a2 + b1*b2 + c1*c2

v * w = ||v|| * ||w|| * cos Ø

v * v = ||v||2

A

dot product

53
Q

v / ||v||

A

unit vector = ev

54
Q

u|| = ( (u * v) / (||v||2) ) * v

u|| = (u * ev) * ev

u|| = ( (u * v) / (v * v) ) * v

A

vector projection of u onto v

u||

55
Q

if lines through w and v are parrallel

or

w = Cv

for scalar C not equal to 0

A

parallel vectors

56
Q

(x - a)2 + (y - b)2 + (z - c)2 = R2

A

equation of a sphere

of radius R

centered at (a, b, c)

57
Q

(x - a)2 + (y - b)2 = R2

A

equation of a cylinder

of radius R

whose center line is the vertical axis through (a, b, 0)

58
Q

||u|||| = ||u|| * cos Ø

A

magnitude of projection

of u along v

59
Q

orthogonal to v and w

v X w = - (v X w)

v X v = 0

v X w = 0 iff w = Cv or v = 0

A

Cross Product

v X w

60
Q

||v X w|| = ||v|| * ||w|| * sinØ

||v X w||2 = ||v||2 * ||w||2 - (v * w)2

A

length of v X w

61
Q

u * ev

where ev = v / ||v||

A

scalar projection of u onto v

or

the component of u along v

62
Q

if and only if

v * w = 0

A

a vector v is orthogonal to w

63
Q

v * w < 0 is obtuse

v * w > 0 is acute

A

the angle Ø between v and w is obtuse/acute

64
Q

u = u|| + uperp

where uperp is orthogonal to v

A

the decomposition of u with respect to v

65
Q

= C (v X w)

A

C * v X w = v X (C * w) =

66
Q

= u X w + v X w

A

(u + v) X w =

67
Q

a vector v undergoes a translation when it is moved parallel to itself

without changing length or direction

w is the translate

A

translation

68
Q

v and w are equivalent if w is a translate of v

A

equivalent

69
Q

u = r * v + s * w

A

linear combination

(if v and w are not parallel)

70
Q

parallelogram whose vertices are the origin and the terminal

points of v, w, and w + v

consists of linear combination

where 0

A

parallelogram spanned

71
Q

r(t) = OP0 + v = < x0 , y0 , z0> + t *

x = x0 + at ; y = y0 + bt ; z = z0 + zt

A

The line L through P0 = (x0 , y0 , z0) in the direction

of (parallel to) **v; ** where O is origin

vector parametrization :

parametric equation:

72
Q

r(t) = (1 - t) * OP + t * OQ

x = a1 + (a2 - a1) * t ; y = b1 + (b2 - b1) * t ; z = c1 + (c2 - c1) * t

A

The line through the points P = (a 1 , b1 , c1) and Q = (a2 , b2 , c2)

vector parametrization:

parametric equation:

73
Q

r0 = t * v =

x = a * t ; y = b * t ; z = c * t

A

Equation of a line through origin that consists of the multiples

of the non-zero vector v =

vector parametrization:

parametric equation:

74
Q

i X j = k

**j ** X k = i

k X i = j

i X i = j X j = k X k = 0

A

unit vector cross products

75
Q

||v X w||

||v|| * ||w|| * sin Ø

A

parallelogram spanned by v and w

has area:

76
Q

||v X w|| * ||u|| * |cos Ø|

u * (v X w) |

A

parallelepiped spanned by u , v , and w

has volume

77
Q

u * (v X w)

= the determinant of [u v w]

A

vector triple product

78
Q

||v||2 * ||w||2 - (v * w)2

A

||v X w||2

79
Q

fingers curl from x-axis to y-axis

thumb is in the positive z direction

A

right hand rule

80
Q

defined by setting one the

coordinates equal to 0

A

coordinate planes

81
Q

defined by

a*x + b*y + c*z = d

A

plane in R3

82
Q

a non-zero vector orthogonal to a point

P0 = (x0 , y0 , z0) determining

a plane passing through a point P0

A

normal vector

n

83
Q

n * P0P = 0

or

n * OP = n * OP0

A

P lies on a plane if

84
Q

vector form:

n * < x , y , z > = d

scalar form:

a*(x - x0) + b*(y - y0) + c*(z - z0) = 0

A

equation of a plane

through point P0 = (x0 , y0 , z0)

where normal vector = n = < a , b , c >

85
Q

the planes are parallel if they have a common normal vector

obtained by choosing normal vector n and varying

constant d in equation

a*x + b*y + c*z = d

A

parallel planes

86
Q

points that lie on the same line

three points determine a plan if they are

not collinear

A

collinear

87
Q

the intersection of a plane with a coordinate plane

or a plane parallel to a coordinate plane

trace is a line unless the plane is parrallel to coordinate plane

(then trace is empty or is the plane itself)

obtained by setting x , y , or z = 0

A

trace

88
Q

(x / a)2 + (y / b)2 + (z / c)2 = 1

A

ellipsoids

89
Q

(x / a)2 + (y / b)2 - (z / c)2 = 1

A

hyperboloids of one sheet

90
Q

(x / a)2 + (y / b)2 - (z / c)2 = -1

A

hyperboloids of two sheets

91
Q

(x / a)2 + (y / b)2 = (z / c)2

A

elliptic cone

92
Q

z = (x / a)2 + (y / b)2

A

elliptic paraboloid

93
Q

z = (x / a)2 - (y / b)2

A

hyperbolic paraboloid

94
Q

u - u|| = u<strong>perp</strong>

A

orthogonal projection of vector u obtained from a vector v

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