7-8. If r =r (t)denotes a space curve, show that the torsion can be calculate from
the relation
V = r
′·(r ′′ ×r ′′′)
(r ′·r ′)(r ′′ ·r ′′)−(r ′·r ′′)^2
where prime ′=dtd always denotes differentiation with respect to the argument of
the function. Hint: Show dr
ds
·
(
d^2 r
ds^2
×d
(^3) r
ds^3
)
=κ^2 V
7-9.
(a) Find the curvature of the straight line r =r (t) = ˆe 1 +ˆe 2 +ˆe 3 + (7 ˆe 1 + 2 ˆe 2 − 3 ˆe 3 )t
(b) Find the torsion of the plane curve r =r (x) = xˆe 1 +x^2 ˆe 2
7-10. For r =r (t)the position vector of a curve, show that
|r ′×r ′′|=κ|r ′|^3 , where ′= d
dt
7-11. Find the directional derivative of φin the specified direction, at the given
point.
(i) φ=y^2 x^2 z+x^3 z, P (1, 1 ,1), 3 ˆe 1 − 2 ˆe 2 + 6 ˆe 3
(ii) φ=xyz, P (2, 1 ,−1) 5 ˆe 1 − 4 ˆe 2 + 20 ˆe 3
(iii) φ=xy^2 +yz^3 , P (1,− 1 ,0), 2 ˆe 1 − 5 ˆe 2 − 14 ˆe 3
(iv) φ=x^2 y^2 +yz^3 x, P (1, 1 ,1), ˆe 1 + 2 ˆe 2 + 2 ˆe 3
7-12.
(i) Let φ=x^2 ydefine a two-dimensional scalar field. Find the directional derivative
of φat the point (2 ,
√
3) in the direction eˆα= cos αˆe 1 + sin αˆe 2
(ii) In what direction αis the directional derivative a maximum?
(iii) In what direction αis the directional derivative a minimum?
7-13. Show that d
dt
[
r ·
(
dr
dt
×d
(^2) r
dt^2
)]
=r ·
(
dr
dt
×d
(^3) r
dt^3
)
7-14. Prove that A×(B×C) + B×(C×A) + C×(A×B) = 0
7-15. Discuss the critical points of the function
z=z(x, y ) =^1
3
x^3 +^1
3
y^3 +^1
2
x^2 −^3
2
y^2 − 2 x+ 2y
7-16. Show that the Frenet-Serret formulas may be expressed in the form
dˆet
ds
=ω׈et, dˆeb
ds
=ω׈eb, dˆen
ds
=ω׈en