North Maharashtra University 2008 B.Sc Mathematics S.YBSc MTH-211 (Calculus of Several Variables) - exam paper

2) If w =f(u,v) is a differentiable function and u= F( x, y ), v= ?( x, y )
are differentiable functions then prove that,
x
v
v
w
y
u
u
w
x
w
?
?
?
?
+
?
?
?
?
=
?
?
3) If w =f(u,v) is a differentiable function and u= F( x, y ), v= ?( x, y )
are differentiable functions then prove that,
y
v
v
w
y
u
u
w
y
w
?
?
?
?
+
?
?
?
?
=
?
?
4) State and prove Euler’s theorem for homogenous function in x and y.
5) If, u = }
x
y f x { G n ??
?
??
-1 ? and G1( u ) ? 0 Then prove that
( )
G (u)
n G u
y
y u
x
x u one =
?
?
+
?
?
Hence obtain ,
y
y u
x
x u
?
?
+
?
?
if u= ?
??
?
? ??
?
+
- +
x y
sin x y
2 2
1
6) If u = f(x,y) is homogenous function of degree n in x and y then prove that,
x2 n( n )u.
y
y u
x y
xy u
x
u two one 2
2
2
2
2
2
= -
?
?
+
? ?
?
+
?
?
7) State and prove mean value theorem for the function f(x,y).
8) obtain
dt
du if u = x3 + y3 , where x = a cost, y = b sint
9) obtain ifz xy x y
dt
dz = two + 2
Where , x = a t two , y =2at
10) If u = f (e y-z ,ez-x ,ex- y )
7
Prove that, = 0
?
?
+
?
?
+
?
?
z
u
y
u
x
u
11) If u = f ?
??
?
? ??
?
x
, z
z
, y
y
x show that, x = 0
?
?
+
?
?
+
?
?
z
z u
y
y u
x
u
12) If u = f ( y-z, z-x, x-y) , prove that = 0
?
?
+
?
?
+
?
?
z
u
y
u
x
u
13) If z is a function f x and y where x = eu + e-v and y = e-u - ev
Show that,
y
y z
x
x z
v
u
u
z
?
?
-
?
?
=
?
?
-
?
?
14) Let z = f (u,v) where u = 2x-3y and v = x +2y, prove that,
u
z
v
z
y
z
x
z
?
?
-
?
?
=
?
?
+
?
?
3
15) If u =
x
tan-1 y where x = et - e-t , y =
dt
et + e-t , obtain du
16) If z = f (x,y) =
y
tan-1 x , x = u+v , y = u-v , show that
u2 v2
u v
v
z
u
z
+
-
=
?
?
+
?
?
17) If z =f(x,y), x = uv, y =
u v
u v
-
+
then prove that,
2x
v
v z
u
u z
x
z
?
?
+
?
?
=
?
?
18) If z = f(u,v) ,u = x2 - y2 - 2xy,v = y prove that the formula
= 0
?
?
+ -
?
?
+
y
( x y ) z
x
( x y ) z is equivalent to = 0
?
?
v
z
19) Verify Euler’s Theorem for function, f(x,y) = x3 + y3 - 3x2 y
20) If u =
2
1
3
1
3
1
2
1
2
1
1
? ? ?
?
?
? ? ?
?
?
+
- +
x y
y x ec cos show ??
?
??
?
= +
?
?
+
? ?
?
+
?
?
12 12
13
12
2
2
2
2
2
2
2
2
2 tanu tan u
y
y u
x y
xy u
x
x u
21) If u =
5
1
2
1 2
? ??
?
? ??
?
-
- +
x y
sin x xy obtain the values of x 2
2
2
2
2
2
2 2
y
y u
x y
xy u
x
andx u
y
y u
x
u
?
?
+
? ?
?
+
?
?
?
?
+
?
?
22) If u = log ( x3 + y3 - x2 y - xy2 ) prove that,
x 2
2
2
2
2
2
2 2
y
y u
x y
xy u
x
andx u
y
y u
x
u
?
?
+
? ?
?
+
?
?
?
?
+
?
?
= -3
23) If u = sin-1 x + y2 prove that 2
2
2
2
2
2
2 2
y
y u
x y
xy u
x
x u
?
?
+
? ?
?
+
?
? = tan3 u
24) If u = ??
?
??
? + ?
??
?
? ??
?
F
x
f y
y
x then prove that, 2
2
2
2
2
2
2 2
y
y u
x y
xy u
x
x u
?
?
+
? ?
?
+
?
?
= 0
25) If u = ( )5
1
sin-1 x2 + y two , prove that, ,
8
2
2
2
2
2
2
2 2
y
y u
x y
xy u
x
x u
?
?
+
? ?
?
+
?
?
= (2 3)
25
2 tanu tan2 u -
26) If u = ?
??
?
? ??
?
-
- +
x y
tan x y
3 3
1 show that
i) sin u
y
y u
x
x u = 2
?
?
+
?
?
ii) 2
2
2
2
2
2
2 2
y
y u
x y
xy u
x
x u
?
?
+
? ?
?
+
?
?
= sin2u (1- four sin2 u)
27) Verify Euler’s theorem for the function f(x,y) =
x
x4 log y
28) If f(x,y) = x3 - xy2 show that ? used in the mean value theorem applied to the points (2,1) and
(4,1) satisfies the quadratic formula 3? two + 6? - four = 0
29) If f(x,y) = x2 y + 2xy2 show that the value of ? in the expression of the mean value theorem
applied to the line segment joining the point (1,2) to (3,3) satisfies the formula
12? two + 30? -19 = 0
30) If u =
x
tan-1 y show that xx xy yy x2u + 2xyu + y 2u = 0
IV) Compulsory Examples ( two marks each)
1) If z = f(x,y) where, x = r cos? , y = r sin? then prove that,
y
sin z
x
cos z
r
z
?
?
+
?
?
=
?
?
? ?
2) If z = f(x,y) where x = rcos? , y = rsin? then prove that
y
r cos z
x
z r sin z
?
?
+
?
?
= -
?
?
? ?
?
3) Verify Euler’s theorem for the function f(x,y) =
x
tan-1 y
4) Verify Eulers theorem for the function f(x,y) = ( )2
1
x2 + y 2
5) If u = ?
?
?
?
? ?
?
?
+
- +
x y
tan one x y prove that sin u
y
y u
x
x u 2
4
= 1
?
?
+
?
?
6) If u = ( )5
1
sin-1 x2 + y two prove that tanu
y
y u
x
x u
5
= 2
?
?
+
?
?
7) If u = x+y and x = et , , y = t three then obtain
dt
du at t =1
8) If u = ?
?
?
?
? ?
?
?
+
- +
x y
sin one x y prove that tanu
y
y u
x
x u
2
= 1
?
?
+
?
?

Earning:   Approval pending.