Consider the vector function given below.
r(t) =
2t, 3 cos(t), 3 sin(t)
(a) Find the unit tangent and unit normal vectors T(t) and
N(t).
T(t) =
N(t) =
(b) Use this formula to find the curvature.
κ(t) =
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=
a. 1 1 cos(x)cos(y) = -cos(x-y) + -cos(x + y) 1 l
sin(x)sin(y) = -cos(x-y)--cos(x+ y) 1 l sin(x)cos(y) =—sin(x-y)
+-sin(x + y) A DSB-FC (double sideband-full carrier) signal s(t) is
given by, s(t) = n cos(2rr/cf)+ cos(2«-/mt)cos(2«-fct) What is the
numeric value for the AM index of modulation, m, fors(f) ?
If u(t) = < sin(8t), cos(4t), t > and v(t) = < t,
cos(4t), sin(8t) >, 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)] =
<.______ , _________ , _______>
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)] = ?
Based on Euler’s formula: e i = cos +i sin where is a real
number.
1a.) What is cos in terms of e i and its complex conjugate?
1b.) What is sin in terms of e i and its complex conjugate?
1c.)Use Euler’s formula on e i (A + B) to develop the trig
addition formulas for (A+B)and sin(A+B)
Show
that in 2D, the general orthogonal transformation as matrix A given
by
{{cos, sin}, {-sin, cos}}. Verify that det[A] = 1 and that the
transpose of A equals its inverse. Let Tij be a tensor in this
space. Write down in full the transformation equations for all its
components and deduce that Tii is an invariant.