8-7. Find the minimum distance between the lines defined by the parametric
equations
L 1 : x=τ− 1 , y =−τ+ 16, z = 2τ− 2
L 2 : x=−t, y = 2t, z = 3t
8-8. Find the minimum distance from the origin to the plane x+y+z= 1
8-9. The special symbol dφdn is used to denote the normal derivative of a function
φon the boundary of a region R. The normal derivative is defined
dφ
dn = grad φ·ˆen=∇φ·ˆen,
where ˆenis the unit exterior normal vector to the boundary of the region. Find the
normal derivative of φ=x^3 y+xy^2 on the boundary of the regions given.
(i) The unit circle x^2 +y^2 = 1
(ii) The ellipse x
2
a^2 +
y^2
b^2 = 1
(iii) The square with vertices (0,0),(1 ,0),(1,1),(0 ,1)
8-10. Find the critical points associated with the given functions and test for
relative maxima and minima.
(i) z= (x−2)^2 + (y−3)^2
(ii) z= (x−2)^2 −(y−3)^2
(iii) z=−(x−2)^2 −(y−3)^2
8-11. Let u(x, y, z)denote a scalar field which is continuous and differentiable. Let
x=x(t), y =y(t)and z=z(t)denote the position vector of a particle moving through
the scalar field. Show that on the path of the particle one finds
du
dt
= (grad u)·dr
dt
.
8-12. Let f(x, y, z, t)denote a scalar field which is changing with time as well as
position. Let x=x(t), y =y(t) and z=z(t)denote the position vector of a particle
moving through the scalar field. Show that on the path of the particle
df
dt = (grad f)·
dr
dt +
∂f
∂t.
In hydrodynamics, where dxdt ,dydt ,dzdt represents the velocity of the particle, the
above derivative
df
dt is called a material derivative and is represented using the nota-
tion Df
Dt
.Note that the material derivative represents the change of fas one follows
the motion of the fluid.
