Line integrals from graphs Determine whether ∫ c F · d r along the paths C 1 and C 2 shown in the following vector fields is positive or negative. Explain your reasoning. a. ∫ C 1 F ⋅ d r b. ∫ C 2 F ⋅ d r 48.
Line integrals from graphs Determine whether ∫ c F · d r along the paths C 1 and C 2 shown in the following vector fields is positive or negative. Explain your reasoning. a. ∫ C 1 F ⋅ d r b. ∫ C 2 F ⋅ d r 48.
Line integrals from graphs Determine whether ∫cF · dr along the paths C1 and C2 shown in the following vector fields is positive or negative. Explain your reasoning.
a.
∫
C
1
F
⋅
d
r
b.
∫
C
2
F
⋅
d
r
48.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
You are on a rollercoaster, and the path of your body is modeled by a vector function r(t),
where t is in seconds, the units of distance are in feet, and t = 0 represents the start of the
ride. Assume the axes represent the standard cardinal directions and elevation (x is E/W, y
is N/S, z is height). Explain what the following would represent physically, being as specific
as possible. These are all common roller coaster shapes/behaviors and can be explained in
specific language with regard to units:
a. r(0)=r(120)
b. For 0 ≤ t ≤ 30, N(t) = 0
c. r'(30) = 120
d. For 60 ≤ t ≤ 64, k(t) =
40
and z is constant.
e.
For 100 ≤ t ≤ 102, your B begins by pointing toward positive z, and does one full
rotation in the normal (NB) plane while your T remains constant.
A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by
v = (x - y, z + y +7,z²) and the net is decribed by the equation y = √1-x²-2², y 20, and oriented in the positive
y-direction.
(Use symbolic notation and fractions where needed.)
1.45-1
yas
Using a graphic calculator, plot the vector field F(r, y) = e
j. Explain briefly with words why the plot has this shape.
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