Partial Derivatives

Introduction to Partial Differentiation • There exists some functions which depends upon more than one variable. • Eg. Area of a rectangle depends upon its length and breadth, hence we can say that area is the function of two variables i.e. its length and breadth. • Thus, z is called a function of two variables of x and y if z has one definite value for every pair of x and y i.e. z = f ( x, y )

Partial Differentiation (Limits) • Definition: The function f ( x, y ) is said to tend to limit l as x → a and y → b iff the limit l is independent of the path followed by the point ( x, y ) as x → a and y → b then lim

( x , y ) →( a ,b )

f ( x, y ) = l

Partial Derivative (w.r.t. ‘x’) • If z = f ( x, y ) be a function of two variables x and y . • The partial derivative of f ( x, y) with respect to ‘ x ’ at a point ( x0 , y0 ) is  ∂f  d  = f ( x, y 0 )     ∂x ( x0 , y0 )  dx ( x =x0 ) f ( x0 + h, y0 ) − f ( x0 , y0 ) = lim h →0 h

provided the limit exists.

Partial Derivative (w.r.t. ‘y’) • If z = f ( x, y ) be a function of two variables x and y . • The partial derivative of f ( x, y) with respect to ‘y ’ at a point ( x0 , y0 ) is  ∂f  d     = f ( x0 , y )   ∂y    ( x0 , y0 )  dy ( y =y0 ) = lim h →0

f ( x0 , y0 + h) − f ( x0 , y0 ) h

provided the limit exists.

Partial Derivative versus Continuity • A function f ( x, y) can have partial derivatives w.r.t. both x and y at a point without being continuous there. • While, it is different in case of functions of a single variable. • Conversely, if partial derivatives of f ( x, y) exist and are continuous throughout a disk centered at ( x0 , y0 ) , then f is continuous at ( x0 , y0 ) .

Chain Rule • For functions of two independent variables: • If w = f ( x, y ) is differentiable and x and y are differentiable functions of ‘t ’, then w is a differentiable function of t and dw ∂f dx ∂f dy = + dt ∂x dt ∂y dt

• ||ly, for three independent variables: dw ∂f dx ∂f dy ∂f dz = + + dt ∂x dt ∂y dt ∂z dt

Chain Rule • Two independent and three intermediate variables: • Let w = f ( x, y, z ) , x = g (r , s) , y = h(r , s) and z = k (r , s ) and all the four functions are differentiable

then we have ∂w ∂w ∂x ∂w ∂y ∂w ∂z = + + and ∂r ∂x ∂r ∂y ∂r ∂z ∂r ∂w ∂w ∂x ∂w ∂y ∂w ∂z = + + ∂s ∂x ∂s ∂y ∂s ∂z ∂s

Extension of Chain Rule • Functions of many variables: ∂w ∂w ∂x ∂w ∂y ∂w ∂v = + + ...... + ∂p ∂x ∂p ∂y ∂p ∂v ∂p

Euler’s Theorem • The Mixed Derivative Theorem: If f ( x, y ) and its partial derivatives

f x , f y , f xy and f yx are defined throughout an open region containing a point (a, b) and are all continuous at (a, b) then

f xy (a, b) = f yx (a, b)

Euler’s Theorem • If ‘ u ’ is a homogeneous function of degree ‘ n ’ in x and y then, ∂u ∂u x

∂x

+y

∂y

= nu

Implicit Function Theorem • Let us consider a function F ( x, y ) which is differentiable and that the equation F ( x, y ) = 0 defines y as a differentiable function of ‘ x’. Then, at any point where F ≠ 0, y

Fx dy =− . dx Fy

Total Derivatives • One variable: y = f (x) , dy = f ( x)dx ∂f ∂f z = f ( x , y ) • Two variable: , df = dx + '

dy

∂x ∂y = f x ( x, y ) dx + f y ( x, y ) dy

• Three variables: w = f ( x, y, z )

∂f ∂f ∂f df = dx + dy + dz ∂x ∂y ∂z = f x ( x, y, z ) dx + f y ( x, y, z ) dy + f z ( x, y, z ) dz

Change of Variables • If u =

f ( x, y ) where

x =φ( s, t ) and y =ψ( s, t ) then using chain rule, we get ∂u ∂u ∂x ∂u ∂y = * + * ∂s ∂x ∂s ∂y ∂s ∂u ∂u ∂x ∂u ∂y = * + * , on solving these equations, ∂t ∂x ∂t ∂y ∂t we get ' s' and ' t' in terms of x and y, hence ∂u ∂u ∂s ∂u ∂t = * + * ∂x ∂x ∂x ∂t ∂y ∂u ∂u ∂s ∂u ∂t = * + * ∂y ∂s ∂y ∂t ∂y

Jacobians • Jacobian (or Jacobian Determinant) of the coordinate transformation x = g (u, v) , y = h(u, v) ∂x ∂x is ∂u ∂v ∂x ∂y ∂y ∂x J (u , v) = = − . ∂y ∂y ∂u ∂v ∂u ∂v ∂u ∂v also denoted by, ∂ ( x, y ) J(u, v) = ∂ (u , v) and so on for more number of variables.

Jacobians (Particular Case) If u1 , u 2 .........u n are function of x1 , x 2 ,......., x n , Like, u1 = f1 (x) u 2 = f 2 (x1 , x 2 ) ........................ u n = f n (x1 , x 2 ........, x n ). Then

∂ (u1 , u 2 ,......, u n ) ∂u1 ∂u 2 ∂u 3 ∂u n = . . ...... . ∂ (x1 , x 2 ,......, x n ) ∂x1 ∂x 2 ∂x 3 ∂x n

Jacobians (Function of Function) • Property I: If u1 , u2 , u3 ,....., un are functions of the set of the variables y1 , y2 , y3 ,......, yn and y1 , y2 , y3 ,......, yn are themselves functions of x1 , x2 , x3 ,....., xn ∂(u1,u2 ,......,un ) then, ∂(u1,u2 ,......,un ) ∂(y1,y2 ,......,yn ) = × ∂(x1,x2 ,......,xn ) ∂(y1,y2 ,......,yn ) ∂(x1,x2 ,......,xn )

Jacobians • Property II : • If ‘J ’ is the Jacobian of the system u,v with ' J regard to x,y and the Jacobian of x,y with regard to u,v then JJ ' = I

Jacobians • Property III: Implicit functions : if u1 , u2 , u3 ,...., un and x1 , x2 , x3 ,..., xn are connected as f1 (u1 , u2 , u3 ,...., un , x1 , x2 , x3 ,..., xn ) = 0 f 2 (u1 , u2 , u3 ,...., un , x1 , x2 , x3 ,..., xn ) = 0 ........................................................... f n (u1 , u2 , u3 ,...., un , x1 , x2 , x3 ,..., xn ) = 0 Then, ∂ (u1 , u2 , u3 ,...., un ) n ∂ ( f1 , f 2 , f 3 ,...., f n ) ∂ ( f1 , f 2 , f 3 ,...., f n ) = (− 1) * ∂ ( x1 , x2 , x3 ,..., xn ) ∂ ( x1 , x2 , x3 ,..., xn ) ∂ (u1 , u2 , u3 ,...., un )

Jacobians Property IV : If u1 , u2 , u3 be the functions of x1 , x2 , x3 then necessary and sufficient condition for the existence of a functional relationship of the form f (u1 , u2 , u3 ) = 0 , is  u1 , u 2 , u3   = 0 J   x1 , x2 , x3 

Extreme Values •

Definitions: Let f ( x, y ) be defined on region R containing the point (a, b) . Then • f (a, b) is the local maximum value off if f (a, b) ≥ f ( x, y ) for all domain points (x, y) in an open disk centered at (a, b). • f (a, b)is a local minimum value of f if f (a, b) ≤ f ( x, y ) for all domain points (x, y) in an open disk centered at (a, b).

Local Extreme Values • First Derivative Test: If f ( x, y ) has a local maximum or minimum value at an interior point (a, b) of its domain, and if the first partial derivatives exist there, then f x (a, b) = 0 and

f y ( a, b) = 0

Critical Value • An interior point of the domain of a function f ( x, y ) where both f x and f y are zero or where one or both f x and f y do not exist is a critical point of f

Saddle Point • A differentiable function f ( x, y ) has a saddle point at a critical point (a, b) if in every open disk centered at (a, b) there are domain points ( x, y ) where f ( x, y ) > f (a, b) or f ( x, y ) < f (a, b) . The corresponding point ( a, b, f (a, b)) on the surface z = f ( x, y ) is called a saddle point of the surface.

Saddle Points

Local Extreme Values • Second Derivative Test: Suppose f(x,y) and its first and second partial derivatives are continuous throughout a disk centered at (a, b) and f x (a, b) = f y (a, b) = 0.Then 1. f has a local maximum at (a, b) if f xx < 0 2

and f xx f yy − f xy > 0 at (a, b). 2. f has a local minimum at (a, b) if f xx > 0 2

and f xx f yy − f xy > 0 at (a, b).

Local Extreme Values • Cont:3. f has a saddle point at (a, b) if 2

f xx f yy − f xy < 0 at (a, b). 2

4. Inconclusive at (a, b) if f xx f yy − f xy = 0 at (a, b). The behaviour is determined by some other method.

Lagrange Multipliers • It is a powerful method for finding extreme values of constrained functions. • It is used to solve max-min problems in geometry. • This method is important in economics, in engineering (ex. In designing multistage rockets) and in mathematics.

Method of Lagrange Multipliers • Suppose that f ( x, y, z ) and g ( x, y, z ) are differentiable. To find the local maxima and minima values of f subject to the constraint g ( x, y, z ) = 0 , find the values of x, y, z and λthat simultaneously satisfy the equations ∇f = λ∇g and g ( x, y, z ) = 0 • For functions of two independent variables equations are ∇f = λ∇g and g ( x, y ) = 0

Lagrange Multipliers(Two Constraints) • In the case of two constraints, g1 ( x, y, z ) = 0 and g 2 ( x, y, z ) = 0

Where these are differentiable, with ∇g1 not parallel to ∇g 2 , we have the equations ∇f = λ1∇g1 + λ2∇g 2 , g1 ( x, y, z ) = 0, g 2 ( x, y, z ) = 0

Where λ1 and λ2 are Lagrange Multipliers.