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Chapter: Mathematics - Functions of several Variables

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Functions of several Variables

· Problems based on Partial Derivatives · Problems based on Euler`s Theorem · Problems based on Total Derivatives-Differentiation of Implicit Function · Problems based on Jacobian · Problems based on Taylor`s and Laurent Series · Problems based on Maxima and Minima for Functions of Two Variables · Problems based on Lagrangian Multiplier

Functions of several Variables

 

· Problems based on Partial Derivatives

· Problems based on Euler`s Theorem

· Problems based on Total Derivatives-Differentiation of Implicit Function

· Problems based on Jacobian

· Problems based on Taylor`s and Laurent Series

· Problems based on Maxima and Minima for Functions of Two Variables

· Problems based on Lagrangian Multiplier

 

Partial Derivatives

Partial Derivatives: Let z=f(x,y) be a function of two Variables x and y, If we keep y as a constant and Vary x alone , then z is a function of x only ,

The derivative of z w.r.to x, treating y as a constant is called the partial derivatives w.r.to x and it is denoted by the symbols



Successive Partial Differentiation:




 


Euler`s Theorem for Homogeneous Function

Euler`s Theorem: If u be a homogeneous function of degree n an x and y then


 




Total Derivatives-Differentiation of Implicit Function

Total Derivative:















 










 

Maxima and Minima and Lagrangian Multiplier

 

 Defn: Maximum Value

if f(a,b) is a maximum value of (x,y) if their exists some neighbourhood of the point (a,b) such that for every point (a+h,b+k) of the neighbourhood

f(a,b)>f(a+h,b+k)

 

 Defn: Minimum Value

if f(a,b) is a maximum value of (x,y) if their exists some neighbourhood of the point (a,b) such that for every point (a+h,b+k) of the neighbourhood

f(a,b)<f(a+h,b+k)

 

 Defn: Extremum Value

if f(a,b) is said to be an extremum value o if f(a,b) it is maximum or minimum

 

 Defn: Lagrangian Multiplier

 Suppose we require to find the maximum and minimum values of (x,y,z)  where x,y,z are subject to a constraint equation g(x,y,z)=0

We define a function F(x,y,z) = f(x,y,z) + λg(x,y,z) where λ is called Lagrangian Multiplier which is independent of x,y,z.

 

Problems:























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