Assignment 3 FSV (MAL102)

Visvesvaraya National Institute of Technology, Nagpur
Department of Mathematics
Assignment-3 (Functions of several variables)
Subject: MAL-102
1. If u = (x2 + y 2 + z 2 )m/2 find m 6= 0 such that
2
2
∂2u
∂x2
+
∂2u
∂y 2
+
∂2u
∂z 2
= 0.
2
∂ u
+ y 2 ∂∂yu2 , where u(x, y) = x2 φ( xy ) + yψ( xy ) and φ, ψ are arbitrary func2. Find x2 ∂∂xu2 + 2xy ∂x∂y
tions.
3. Let u = F (x − y, y − z, z − x), then show that
∂u
∂x
+
∂u
∂y
+
∂u
∂z
= 0.
4. If f (x, y) = φ(u, v) and u = x2 − y 2 , v = 2xy then prove that
2
2
y 2 ) ∂∂uφ2 + ∂∂vφ2 .
5. If u = f
x y z
, ,
y z x
6. If u = tan
−1
7. If u = csc
−1
+
∂2f
∂y 2
= 4(x2 +
then find x ∂u
+ y ∂u
+ z ∂u
.
∂x
∂y
∂z
x3 + y 3
then show that
x−y
2
2
∂2u
= sin(2u) and (ii) x2 ∂∂xu2 + 2xy ∂x∂y
+ y 2 ∂∂yu2 = sin(4u) − sin(2u)
(i) x ∂u
+ y ∂u
∂x
∂y
∂2f
∂x2
1
2
1
2
1
1
x +y
x3 + y 3
! 21
then show that
2
x2 ∂∂xu2
+
∂2u
2xy ∂x∂y
∂2u
∂x2
8. If u = f (x, y) where x = r cos θ, y = r sin θ. Show that
9. If z = ex sin y, where x = st2 and y = s2 t. Find
∂z
∂s
and
+
2
y 2 ∂∂yu2
=
+
∂2u
∂y 2
∂2u
∂r 2
=
tan u
12
+
13
12
+
1 ∂2u
r 2 ∂θ 2
tan2 u
12
+
.
1 ∂u
.
r ∂r
∂z
.
∂t
10. If z = f (x, y) where f is differentiable, x = g(t) and y = h(t). Given g(3) = 2, g ′(3) =
when t = 3.
5, h(3) = 7, h′ (3) = −4, fx (2, 7) = 6 and fy (2, 7) = −8, then find dz
dt
11. If u = f (x, y) where x = es cos t and y = es sin t, show that
h
h 2 ∂u 2
i
∂u 2
∂2u
∂2u
∂u 2
−2s ∂ 2 u
−2s
+
+
,
(ii)
+
=
e
+
=
e
(i) ∂u
∂x
∂y
∂s
∂t
∂x2
∂y 2
∂s2
12. Find the equation of the tangent plane to the surfaces at the specified points
(i) z = 9x2 + y 2 + 6x − 3y + 5, (1, 2, 18) (ii) z = y cos(x − y), (2, 2, 2).
∂2u
∂t2
i
13. Find the equation of the tangent plane and normal line, (i) at the point (−2, 1, −3) to the
2
2
ellipsoid x4 + y 2 + z9 = 3 and (ii) at the point (1, 0, 0) to surface z + 1 = xey cos(z).
14. Find the total differential of the functions
(i) z = x2 + 3xy − y 2 (ii) v = y cos(xy) (iii) u = e−t sin(s + 2t)
1

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