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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
The line through the points P = (a 1 , b1 , c1) and Q = (a2 , b2 , c2) vector parametrization: parametric equation:
r(t) = (1 - t) \* **OP** + t \* **OQ** x = a1 + (a2 - a1) \* t ; y = b1 + (b2 - b1) \* t ; z = c1 + (c2 - c1) \* t
26
Equation of a line through origin that consists of the multiples of the non-zero vector **v** = vector parametrization: parametric equation:
**r0** = t \* **v** = ## Footnote x = a \* t ; y = b \* t ; z = c \* t
27
unit vector cross products
**i** X **j** = **k** **j ** X **k** = **i** **k** X **i** = **j** **i** X **i** = **j** X **j** = **k** X **k** = 0
28
parallelogram spanned by **v** and **w** has area:
||**v** X **w**|| ||**v**|| \* ||**w**|| \* sin Ø
29
parallelepiped spanned by **u** , **v** , and **w** has volume
| **u** \* (**v** X **w**) | ||**v** X **w**|| \* ||**u**|| \* |cos Ø|
30
vector triple product
**u** \* (**v** X **w**) = the determinant of [**u v w**]
31
||**v** X **w**||2
||**v**||2 \* ||**w**||2 - (**v** \* **w**)2
32
right hand rule
fingers curl from x-axis to y-axis thumb is in the positive z direction
33
coordinate planes
defined by setting one the coordinates equal to 0
34
plane in **R**3
defined by a\*x + b\*y + c\*z = d
35
normal vector **n**
a non-zero vector orthogonal to a point P0 = (x0 , y0 , z0) determining a plane passing through a point P0
36
P lies on a plane if
**n** \* **P0P** = 0 or **n** \* **OP** = **n** \* **OP0**
37
equation of a plane through point P0 = (x0 , y0 , z0) where normal vector = **n** = \< a , b , c \>
vector form: **n** \* \< x , y , z \> = d scalar form: a\*(x - x0) + b\*(y - y0) + c\*(z - z0) = 0
38
parallel planes
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
collinear
points that lie on the same line three points determine a plan if they are not collinear
40
trace
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
ellipsoids ![http://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Ellipsoid_tri-axial_abc.svg/200px-Ellipsoid_tri-axial_abc.svg.png](http://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Ellipsoid_tri-axial_abc.svg/200px-Ellipsoid_tri-axial_abc.svg.png)
(x / a)2 + (y / b)2 + (z / c)2 = 1
42
hyperboloids of one sheet ![http://upload.wikimedia.org/wikipedia/commons/7/7f/HyperboloidOfOneSheet.png](http://upload.wikimedia.org/wikipedia/commons/7/7f/HyperboloidOfOneSheet.png)
(x / a)2 + (y / b)2 - (z / c)2 = 1
43
hyperboloids of two sheets ![http://upload.wikimedia.org/wikipedia/commons/9/91/HyperboloidOfTwoSheets.png](http://upload.wikimedia.org/wikipedia/commons/9/91/HyperboloidOfTwoSheets.png)
(x / a)2 + (y / b)2 - (z / c)2 = -1
44
elliptic cone ![http://www.mhhe.com/math/calc/smithminton2e/cd/folder_structure/text/chap10/section06/figure_1058b.gif](http://www.mhhe.com/math/calc/smithminton2e/cd/folder_structure/text/chap10/section06/figure_1058b.gif)
(x / a)2 + (y / b)2 = (z / c)2
45
elliptic paraboloid ![http://mathinsight.org/media/image/image/elliptic_paraboloid_circular_domain.png](http://mathinsight.org/media/image/image/elliptic_paraboloid_circular_domain.png)
z = (x / a)2 + (y / b)2
46
hyperbolic paraboloid ![http://erikdemaine.org/hypar/maple_hypar_plot.gif](http://erikdemaine.org/hypar/maple_hypar_plot.gif)
z = (x / a)2 - (y / b)2
47
orthogonal projection of vector **u** obtained from a vector **v**
**u** - **u**|| = **u**perp
48
**PQ** = ( (x1 - x2)2 + (y1 - y2)2 + (z1 - z2)2 )1/2
Distance Formula for (x,y,z)
49
||**v**|| = ( a2 + b2 + c2 )1/2
magnitude of **v**
50
**PQ** = \< d - a , e - b , f - c\>
vector components P = point (a,b,c) Q = point (d,e,f)
51
(a,b,c) + 1/2 \* **PQ** or
midpoint formula
52
**v** \* **w** = a1\*a2 + b1\*b2 + c1\*c2 **v** \* **w** = ||**v**|| \* ||**w**|| \* cos Ø **v \* v** = ||**v**||2
dot product
53
**v** / ||**v**||
unit vector = **e**v
54
**u|| **= ( (**u \* v)** / (||**v**||2) ) \* **v** **u||** = (**u \* ev**) \* **ev** **u||** = ( (**u \* v**) / (**v \* v**) ) \* **v**
vector projection of **u** onto **v** **u||**
55
if lines through **w** and **v** are parrallel or **w** = C**v** for scalar C not equal to 0
parallel vectors
56
(x - a)2 + (y - b)2 + (z - c)2 = R2
equation of a sphere of radius R centered at (a, b, c)
57
(x - a)2 + (y - b)2 = R2
equation of a cylinder of radius R whose center line is the vertical axis through (a, b, 0)
58
||**u||**|| = ||**u**|| \* cos Ø
magnitude of projection of **u** along **v**
59
orthogonal to **v** and **w** **v** X **w** = - (**v** X **w**) **v** X **v** = 0 **v** X **w** = 0 iff **w** = C**v** or **v** = 0
Cross Product **v** X **w**
60
||**v** X **w**|| = ||**v**|| \* ||**w**|| \* sinØ ||**v** X **w**||2 = ||**v**||2 \* ||**w**||2 - (**v \* w**)2
length of **v** X **w**
61
**u \*** **ev** where **ev** = **v /** ||**v**||
scalar projection of **u** onto **v** or the component of **u** along **v**
62
if and only if **v \* w** = 0
a vector **v** is orthogonal to **w**
63
**v \* w** \< 0 is obtuse **v \* w** \> 0 is acute
the angle Ø between **v** and **w** is obtuse/acute
64
**u** = **u**|| + **uperp** ## Footnote where **uperp **is orthogonal to **v**
the decomposition of **u** with respect to **v**
65
= C (**v** X **w**)
C \* **v** X **w** = **v** X (C \* **w**) =
66
= **u** X **w** + **v** X **w**
(**u** + **v**) X **w** =
67
a vector **v** undergoes a translation when it is moved parallel to itself without changing length or direction **w** is the translate
translation
68
**v** and **w** are equivalent if **w** is a translate of **v**
equivalent
69
**u** = r \* **v** + s \* **w**
linear combination (if **v** and **w** are not parallel)
70
parallelogram whose vertices are the origin and the terminal points of **v, w,** and **w + v** consists of linear combination where 0
parallelogram spanned
71
r(t) = **OP0** + **v** = \< x0 , y0 , z0\> + t \* x = x0 + at ; y = y0 + bt ; z = z0 + zt
The line L through P0 = (x0 , y0 , z0) in the direction of (parallel to) **v; ** where O is origin vector parametrization : parametric equation:
72
r(t) = (1 - t) \* **OP** + t \* **OQ** x = a1 + (a2 - a1) \* t ; y = b1 + (b2 - b1) \* t ; z = c1 + (c2 - c1) \* t
The line through the points P = (a 1 , b1 , c1) and Q = (a2 , b2 , c2) vector parametrization: parametric equation:
73
**r0** = t \* **v** = ## Footnote x = a \* t ; y = b \* t ; z = c \* t
Equation of a line through origin that consists of the multiples of the non-zero vector **v** = vector parametrization: parametric equation:
74
**i** X **j** = **k** **j ** X **k** = **i** **k** X **i** = **j** **i** X **i** = **j** X **j** = **k** X **k** = 0
unit vector cross products
75
||**v** X **w**|| ||**v**|| \* ||**w**|| \* sin Ø
parallelogram spanned by **v** and **w** has area:
76
| **u** \* (**v** X **w**) | ||**v** X **w**|| \* ||**u**|| \* |cos Ø|
parallelepiped spanned by **u** , **v** , and **w** has volume
77
**u** \* (**v** X **w**) = the determinant of [**u v w**]
vector triple product
78
||**v**||2 \* ||**w**||2 - (**v** \* **w**)2
||**v** X **w**||2
79
fingers curl from x-axis to y-axis thumb is in the positive z direction
right hand rule
80
defined by setting one the coordinates equal to 0
coordinate planes
81
defined by a\*x + b\*y + c\*z = d
plane in **R**3
82
a non-zero vector orthogonal to a point P0 = (x0 , y0 , z0) determining a plane passing through a point P0
normal vector **n**
83
**n** \* **P0P** = 0 or **n** \* **OP** = **n** \* **OP0**
P lies on a plane if
84
vector form: **n** \* \< x , y , z \> = d scalar form: a\*(x - x0) + b\*(y - y0) + c\*(z - z0) = 0
equation of a plane through point P0 = (x0 , y0 , z0) where normal vector = **n** = \< a , b , c \>
85
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
parallel planes
86
points that lie on the same line three points determine a plan if they are not collinear
collinear
87
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
trace
88
(x / a)2 + (y / b)2 + (z / c)2 = 1
ellipsoids ![http://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Ellipsoid_tri-axial_abc.svg/200px-Ellipsoid_tri-axial_abc.svg.png](http://upload.wikimedia.org/wikipedia/commons/thumb/5/50/Ellipsoid_tri-axial_abc.svg/200px-Ellipsoid_tri-axial_abc.svg.png)
89
(x / a)2 + (y / b)2 - (z / c)2 = 1
hyperboloids of one sheet ![http://upload.wikimedia.org/wikipedia/commons/7/7f/HyperboloidOfOneSheet.png](http://upload.wikimedia.org/wikipedia/commons/7/7f/HyperboloidOfOneSheet.png)
90
(x / a)2 + (y / b)2 - (z / c)2 = -1
hyperboloids of two sheets ![http://upload.wikimedia.org/wikipedia/commons/9/91/HyperboloidOfTwoSheets.png](http://upload.wikimedia.org/wikipedia/commons/9/91/HyperboloidOfTwoSheets.png)
91
(x / a)2 + (y / b)2 = (z / c)2
elliptic cone ![http://www.mhhe.com/math/calc/smithminton2e/cd/folder_structure/text/chap10/section06/figure_1058b.gif](http://www.mhhe.com/math/calc/smithminton2e/cd/folder_structure/text/chap10/section06/figure_1058b.gif)
92
z = (x / a)2 + (y / b)2
elliptic paraboloid ![http://mathinsight.org/media/image/image/elliptic_paraboloid_circular_domain.png](http://mathinsight.org/media/image/image/elliptic_paraboloid_circular_domain.png)
93
z = (x / a)2 - (y / b)2
hyperbolic paraboloid ![http://erikdemaine.org/hypar/maple_hypar_plot.gif](http://erikdemaine.org/hypar/maple_hypar_plot.gif)
94
**u** - **u**|| = **u**perp
orthogonal projection of vector **u** obtained from a vector **v**

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