
State and prove Euler’s theorem for homogeneous function.
Answer
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Hint:
Here, we are required to state and prove the Euler’s theorem for homogeneous function. Euler’s theorem is used to establish a relationship between the partial derivatives of a function and the product of the function with its degree. Here, we will first write the statement pertaining to the mathematical expression of the Euler’s theorem, and explain it. We will then differentiate the function and manipulate it to obtain the required result.
Complete step by step solution:
Statement –
Euler’s theorem states that if is a homogeneous function of degree of the variables ; then –
, where, is the partial derivative of the function w.r.t , is the partial derivative of the function w.r.t and is the partial derivative of the function w.r.t .
Proof –
Let be a homogenous function of degree of the variables .
…………
Now, we know that –
…………
This is because when is a function of , then it becomes a function of because are a function of .
This means that –
Now, we will consider the above equations for degree 1-
………
……….
……….
where, is any arbitrary parameter.
Now, we will differentiate the equation partially w.r.t .
……….
Again, we will differentiate the equation partially w.r.t .
…………
We will differentiate the equation partially w.r.t .
…………
Now, we will substitute in equation .
Multiplying the terms, we get
…………….
We will again substitute in equation .
Multiplying the terms, we get
…………….
We will now substitute in equation .
Multiplying the terms, we get
………….
We will now differentiate the equation partially w.r.t .
………….
Partial differentiation of w.r.t is 0.
Now, we will differentiate the equation partially w.r.t .
………..
Partial differentiation of w.r.t is 0.
Again, we will differentiate the equation partially w.r.t .
………..
Partial differentiation of w.r.t is 0.
Now, we will differentiate the equation partially w.r.t .
Since, is a function of ,and are a function of , and , thus apply the chain rule of partial differentiation to differentiate.
………..
We will substitute for , for , for , for , for , for in equation .
Thus, the equation becomes –
Using the result of equations , , and , we can write above equation as
……….
We will again substitute and for in the equation , we get
Hence, the Euler’s equation is finally proved.
Note:
A homogeneous function of degree of the variables can be defined as a function in which all terms have degree . For example, consider the following function –
We can see that the degree of all the terms in the above function is three. Since, the degree is constant for all the terms, hence, the above function is a homogeneous function of degree 3.
Here, we are required to state and prove the Euler’s theorem for homogeneous function. Euler’s theorem is used to establish a relationship between the partial derivatives of a function and the product of the function with its degree. Here, we will first write the statement pertaining to the mathematical expression of the Euler’s theorem, and explain it. We will then differentiate the function and manipulate it to obtain the required result.
Complete step by step solution:
Statement –
Euler’s theorem states that if
Proof –
Let
Now, we know that –
This is because when
This means that –
Now, we will consider the above equations for degree 1-
where,
Now, we will differentiate the equation
Again, we will differentiate the equation
We will differentiate the equation
Now, we will substitute
Multiplying the terms, we get
We will again substitute
Multiplying the terms, we get
We will now substitute
Multiplying the terms, we get
We will now differentiate the equation
Partial differentiation of
Now, we will differentiate the equation
Partial differentiation of
Again, we will differentiate the equation
Partial differentiation of
Now, we will differentiate the equation
Since,
We will substitute
Thus, the equation
Using the result of equations
We will again substitute
Hence, the Euler’s equation is finally proved.
Note:
A homogeneous function of degree
We can see that the degree of all the terms in the above function is three. Since, the degree is constant for all the terms, hence, the above function is a homogeneous function of degree 3.
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