Test 2 (2.1-3.5)

Question 1

How do you write the equation of a parametric line?

Answer 1

r(t) = P + tV
r(t) = point (a,b,c) + t<vector></vector>

Question 2

How are circles/ellipses parameterized?

Answer 2

r(t) = (a + Acosta, b + Bsint)

(a, b) is center, <A, B> gives stretching

Question 3

If you are given a parameterized curve r(t), a<= t <= b, how do you find the reverse path?

Answer 3

r{superscript of -}(t) = r(a + b - t),
a<= t <= b

Question 4

What is a path integral? (And what is it also called?)

Answer 4

Aka line integral

Like arc length but w/ function, check exam reference sheet (the one on there technically is just arc length) or pg 96 in workbook, section 2.2

Question 5

How to find speed?

Answer 5

Magnitude of velocity, velocity is r’(t) in parametric

Question 6

What is scalar line integral / what does it represent?

Answer 6

Represents area of “curtain” above/below curve

If f(x,y) = 1, then line integral gives length of curve
(Pg 102, 2.2.5)

Question 7

How to find theta for spherical coordinates? (Not on equation sheet directly)

Answer 7

Tan(theta) = y/x

^in polar coordinates box, also applies for spherical

Question 8

How to find r for spherical coordinates (not explicitly written anywhere on formula sheet)

Answer 8

r = (rho)sin(phi)
r = ρsinφ

How to:
x = rcosθ
x = ρsinφcosθ
r = ρsinφ