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Homework 3
Extrema; Differential Geometry; Vector Operators
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Find the extrema and their type for the functions
f1 = x2 − y 2
f2 = x2 + y 2
f3 = (x + y − 3)2 + (x2 + y 2 − 9)2
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A point moves in space as a function of time along a path described, in parametric form by
t
r = xi + yj + zk = (1 + cos(t))i + sin(t)j + 2 sin( )k
2
a) Determine the velocity and acceleration vectors as well as the tangential, normal and
binormal unit vectors.
b) Compute the numerical values of all vectors as well as the curvature and the torsion
of the path for the time when t = 1.0.
1
3
A force field is given by the expression
F = y 2i + 2(xy + z)j + 2yk
a) Determine ∇ × F.
b) Determine
H
C
F · dr.
c) Determine (if it exists) a scalar function φ such that
F · dr = dφ =
∂φ
∂φ
∂φ
dx +
dy +
dz
∂x
∂y
∂z
d) Determine ∇φ and compare with F.
e) Determine ∇ · ∇φ = ∇2φ.
f) Determine ∇ · F and verify that
Z Z Z
∇ · FdV =
Ω
I I
F · ndS
∂Ω
where Ω is the unit sphere.
g) Let z = 0 and determine the extrema of φ inside the square x ∈ [−1..1], y ∈ [−1..1].
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