If u(t) = < sin(5t),
cos(5t), t > and
v(t) = < t, cos(5t),
sin(5t) >, use the formula below to find the given
derivative.
d/dt[ u(t) * v(t)] = u'(t) * v(t) + u(t)* v'(t)
d/dt [ u(t) x v(t)] = ?
Let Zt = U sin(2*pi*t) + V cos(2*pi*t), where U and V are
independent random variables, each with
mean 0 and variance 1.
(a) Is Zt strictly stationary?
(b) Is Zt weakly stationary?
Consider a fruit fly flying a room with velocity v(t) =
< -sin(t), cos(t), 1 >
a. if the z = 1 + 2(pi) is the room's ceiling, where
will the fly hit the ceiling?
b. if the temperature in the room is T(z) = 65
+ (1/2)z2 how quickly is the temperature increasing for
the fly at time t = 2.
c. from the velocity, find the location of the fruit fly at time
t if at t =...
Given the parametrized curve r(u) = a cos u(1 − cos u)ˆi + a sin
u(1 − cos u)ˆj, u ∈ [0, 2π [ , (with a being a constant)
i) Sketch the curve (e.g. by constructing a table of values or
some other method)
ii) Find the tangent vector r 0 (u). What is the tangent vector
at u = 0? And at u = 2π? Explain your result.
iii) Is the curve regularly parametrized? Motivate your answer
using...
P(u,v)=(f(v)cos(u),f(v)sin(v),g(v))
Find formulas for the Christoffel symbols, the second
fundamental form, the shape operator, the Gaussian curvature and
the mean curvature.
Calculus dictates that
(∂U/∂V) T,Ni = T(∂S/∂V)T,Ni – p = T(∂p/∂T)V,Ni – p
(a) Calculate (∂U/∂V) T,N for an ideal gas [ for which p = nRT/V
]
(b) Calculate (∂U/∂V) T,N for a van der Waals gas
[ for which p = nRT/(V–nb) – a (n/V)2 ]
(c) Give a physical explanation for the difference between the
two.
(Note: Since the mole number n is just the particle number N
divided by Avogadro’s number, holding one constant is equivalent...
Using Matlab, Write a script that validates the relationship
between sin u, cos u, and tan u by evaluating these functions at
suitably chosen values. Please screenshot Matlab screen. Thank
you!
Consider the helix
r(t)=(cos(2t),sin(2t),−3t)r(t)=(cos(2t),sin(2t),−3t).
Compute, at t=π/6
A. The unit tangent vector T=T= ( , , )
B. The unit normal vector N=N= ( , , )
C. The unit binormal vector B=B= ( , , )
D. The curvature κ=κ=