Coordinate Systems Flashcards Preview

► Calculus > Coordinate Systems > Flashcards

Flashcards in Coordinate Systems Deck (34):
1

Polar axis

X-axis in the coordinate system

2

The radial coordinate (r)

The distance from the origin

3

Angular coordinate (θ)

The angle measure from the polar axis to the radial coordinate

4

Calculate 'x' from 'r' and θ

x=r*cos(θ)

5

Calculate 'y' from 'r' and θ

y=r*sin(θ)

6

Calculate 'r' from 'x' and θ

r=x/(cos θ)

7

Calculate 'r' from 'y' and θ

r=y/(sin θ)

8

Calculate 'r' from 'x' and 'y'

r=√(x^2+y^2)

9

Calculate 'x' from 'r' and 'y'

x=√(r^2-y^2)

10

Calculate 'y' from 'x' and 'r'

y=√(r^2-x^2)

11

Calculate θ from 'y' and 'x'

θ=tan^(-1)(y/x)

12

Calculate θ from 'y' and 'r'

θ=sin^-1(y/r)

13

Calculate θ from 'x' and 'r'

θ=cos^-1(x/r)

14

Calculate 'x' from 'y' and θ

x=y/tanθ

15

Calculate 'y' from 'x' and 'θ'

y=x*tanθ

16

Cardioid

Graph of the radius with a change in θ

17

Lemniscate

Graph of the (radius)^2 with a change in θ

18

Cartesian Coordinates in a Plane

Describe the location of a point in terms of (x,y)

19

Polar Coordinates

Describe the location of a point in terms of (r,θ)

20

Slope of a tangent line in polar coordinates

mTan=[f'(θ)sinθ+f(θ)cosθ]/[f'(θ)cos-f(θ)sinθ]

21

Using polar coordinates to approximate integrals

∫1/2*(f(θ)^2-g(θ)^2)dθ
On the interval from [θi,θf]
See pg 660

22

Cartesian coordinates for horizontal ellipses

1=(x^2/a^2)+(y^2/b^2)

b- √(a^2-c^2)
a- vertices at (0,±a) or (±a,0)
c- foci at (0,±c) or (±c,0)

23

Cartesian coordinates for vertical ellipses

1=(y^2/a^2)+(x^2/b^2)

b- √(a^2-c^2)
a- vertices at (0,±a) or (±a,0)
c- foci at (0,±c) or (±c,0)

24

Cartesian coordinates for horizontal hyperbolas

1=(x^2/a^2)-(y^2/b^2)

b- √(c^2-a^2)
a- vertices at (0,±a) or (±a,0)
c- foci at (0,±c) or (±c,0)

25

Cartesian coordinates for vertical hyperbolas

1=(y^2/a^2)-(x^2/b^2)

b- √(c^2-a^2)
a- vertices at (0,±a) or (±a,0)
c- foci at (0,±c) or (±c,0)

26

Polar equation for parabolic curves of d>(x=0)

r=ε*d/(1+ε*cosθ)

27

Polar equation for parabolic curves of d

r=ε*d/(1-ε*cosθ)

28

Polar equation for parabolic curves of d>(y=0)

r=ε*d/(1+ε*sinθ)

29

Polar equation for parabolic curves of d

r=ε*d/(1-ε*sinθ)

30

Cartesian coordinates for parabola

y=x^2/(4r)

31

Arc Length for polar curves

L=∫√[f(θ)^2+f'(θ)^2]dθ

32

xy-Coordinates for the center of mass

X-coordinate from zero position=(∫∫x*p(x,y)/(∫∫p(x,y)
Y-coordinate from zero position=(∫∫y*p(x,y)/(∫∫p(x,y)

Density functions, and the product of their linear position

33

Center of mass in a 3d coordinate plane

X-coordinate from zero position=(∫∫∫x*p(x,y,z)/(∫∫∫p(x,y,z)
Y-coordinate from zero position=(∫∫∫y*p(x,y,z)/(∫∫∫p(x,y,z)
Z-coordinate from zero position=(∫∫∫z*p(x,y,z)/(∫∫∫p(x,y,z)


Density functions, and the product of their linear position

34

Pole

Origin of the coordinate system