ASSIGNMENT 51. [3 marks] Let f(x; y) = exy2 . Find fxyx; fxxy and fyxx and verify their equality.2....

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ASSIGNMENT 51. [3 marks] Let f(x; y) = exy2 . Find fxyx; fxxy and fyxx and verify their equality.2. [3 marks] Let z = x2??y2, where x = v cos u, y = v sin u: Find@z@vand@z@uby the chainrule.3. [4 marks] Show that if u(x; y) and v(x; y) satisfy the Cauchy-Riemann equations@u@x=@v@y;@u@y= ??@v@x;and if x = r cos  and y = r sin , then@u@r=1r@v@;@v@r= ??1r@u@:4. [4 marks] Find equations for the tangent plane and normal line to the surface given byz = ex2+y2 at the point P(0; 0; 1).5. [3 marks] Find a unit vector u that is perpendicular at P(2;??3) to the level curve off(x; y) = 3x2y ?? xy through P.6. [3 marks] Find a unit vector in the direction in which f(x; y; z) = (x ?? 3y + 4z)1=2increases most rapidly at P(0;??3; 0), and nd the rate of increase of f in that direction.			Document Preview:				ASSIGNMENT 5 2 xy 1. [3 marks] Let f(x;y) =e . Find f ; f and f and verify their equality. xyx xxy yxx @z @z 2 2 2. [3 marks] Letz =x y , wherex =v cosu,y =v sinu: Find and by the chain @v @u rule. 3. [4 marks] Show that if u(x;y) and v(x;y) satisfy the Cauchy-Riemann equations @u @v @u @v = ; = ; @x @y @y @x and if x =r cos and y =r sin, then @u 1@v @v 1@u = ; = : @r r@ @r r@ 4. [4 marks] Find equations for the tangent plane and normal line to the surface given by 2 2 x +y z = e at the point P (0; 0; 1). 5. [3 marks] Find a unit vector u that is perpendicular at P (2;3) to the level curve of 2 f(x;y) = 3x yxy through P . 1=2 6. [3 marks] Find a unit vector in the direction in which f(x;y;z) = (x 3y + 4z) increases most rapidly at P (0;3; 0), and nd the rate of increase of f in that direction.

David · Accepted Answer

Sol: (1)         
2
, xyf x y e  
(a)  Find xyxf ,
 
 
 
 
2
2 2
2 2 2
2
2
2
2 2 2 2
3 2
,
, 2
, 2
, 2 2
xy
x
xy xy
xy
xy xy xy
xyx
xy
xyx
f x y y e
f x y y e xy e
f x y y e y xy e y y e
f x y y e xy

  
 
   
 
   
 
(b) Find ,xxyf
 
 
   
 
2
2
2 2
2
2
4
3 4
3 2
,
,
, 4 2
, 2 2
xy
x
xy
xx
xy xy
xxy
xy
xyx
f x y y e
f x y e y
f x y y e y e xy
f x y y e xy

 
 
 
   
 
(c) Find ,yxxf
 
 
 
 
2
2 2
2 2 2
2
2
4 2 2
3 2
, 2
, 2
, 2
, 2 2
xy
y
xy xy
yx
xy xy xy
yxx
xy
xyx
f x y xye
f x y y xy e e
f x y y xy e y e y e
f x y y e xy

  
 
   
 
   
 
So we can see that 
                                  , , ,xyx xxy yxxf x y f x y f x y 
Sol: (2)   Let 
                     2 2 ,      cos , sinz x y where x v u y v u     
(a)  Find ,
z
v


 
                               
   
2 cos 2 sin
2 cos sin
z z x z y
v x v y v
z
x u y u
v
z
x u y u
v
    
 
    

  


   
 
(a)  Find ,
z
u


 
                               
   
2 sin 2 cos
2 sin cos
z z x z y
u x u y u
z
x v u y v u
v
z
v x u y u
v
    
 
    

   


    

Sol: (3) If    ,  and ,u x y v x y satisfy the Cauchy-Riemann equations 
                          ,                                        ............... (1),
u v u v
x y y x
   
  
   
 
  And if cos  and sin ,

ASSIGNMENT 5 1. [3 marks] Let f(x; y) = exy2 . Find fxyx; fxxy and fyxx and verify their equality. 2. [3 marks] Let z = x2??y2, where x = v cos u, y = v sin u: Find @z @v and @z @u by the chain rule....

Answer To: ASSIGNMENT 5 1. [3 marks] Let f(x; y) = exy2 . Find fxyx; fxxy and fyxx and verify their equality....

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