Problem 248
Question
For the following exercises, use this information: A function \(f(x, y)\) is said to be homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) For all homogeneous functions of degree \(n, \quad\) the following equation is true: \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\) . Show that the given function is homogeneous and verify that \(x \frac{\partial f}{\partial x}+y \frac{\partial f}{\partial y}=n f(x, y)\) $$ f(x, y)=3 x^{2}+y^{2} $$
Step-by-Step Solution
Verified Answer
\( f(x, y) = 3x^2 + y^2 \) is homogeneous of degree 2 and satisfies Euler's theorem.
1Step 1: Verify Homogeneity
We need to verify that the function is homogeneous of degree \( n \) using the definition. Substitute \( (tx, ty) \) for \( (x, y) \) in \( f(x, y) \):\[f(tx, ty) = 3(tx)^2 + (ty)^2 = 3t^2x^2 + t^2y^2.\]We can factor out \( t^2 \), so:\[f(tx, ty) = t^2(3x^2 + y^2).\]Thus, \( f(tx, ty) = t^2 f(x, y) \), showing that \( f(x, y) \) is homogeneous of degree 2.
2Step 2: Partial Derivatives
Calculate the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \):- \( \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (3x^2 + y^2) = 6x \).- \( \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (3x^2 + y^2) = 2y \).
3Step 3: Verify Euler's Theorem
Using the calculated partial derivatives, verify Euler's theorem:\[x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = x(6x) + y(2y) = 6x^2 + 2y^2.\]Since \( f(x, y) = 3x^2 + y^2 \), we multiply it by 2 to match the degree of homogeneity:\[2f(x, y) = 2(3x^2 + y^2) = 6x^2 + 2y^2.\]Euler's theorem is satisfied because:\[x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y) = 2 f(x, y).\]
Key Concepts
Euler's theorempartial derivativeshomogeneity degree
Euler's theorem
Euler's theorem is a fascinating concept used in calculus. It relates a particular type of function, known as a homogeneous function, to its partial derivatives.
Here, the theorem states that for a homogeneous function of degree \( n \), the equation \[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x,y) \] holds true. This equation is quite powerful. It helps us confirm the degree of homogeneity of a function.
In simple terms, Euler's theorem tells us that if you take a function, calculate its partial derivatives, and combine them in a certain way, you can find the degree of homogeneity as a constant factor. This is pivotal in scenarios where checking homogeneity directly is complex.
In the context of the exercise, we see this theorem in action by verifying that \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} \) equals \( 2 f(x, y) \), showing the function does indeed have homogeneity degree 2.
Here, the theorem states that for a homogeneous function of degree \( n \), the equation \[ x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x,y) \] holds true. This equation is quite powerful. It helps us confirm the degree of homogeneity of a function.
In simple terms, Euler's theorem tells us that if you take a function, calculate its partial derivatives, and combine them in a certain way, you can find the degree of homogeneity as a constant factor. This is pivotal in scenarios where checking homogeneity directly is complex.
In the context of the exercise, we see this theorem in action by verifying that \( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} \) equals \( 2 f(x, y) \), showing the function does indeed have homogeneity degree 2.
partial derivatives
Partial derivatives are an extension of the basic concept of derivatives into the realm of functions with more than one variable. When dealing with functions of two variables such as \( f(x, y) \), partial derivatives measure how the function changes as one of the variables changes, while keeping the other constant.
For example, the partial derivative of \( f(x, y) = 3x^2 + y^2 \) with respect to \( x \) is found by treating \( y \) as a constant, leading to \( \frac{\partial f}{\partial x} = 6x \). Similarly, the partial derivative with respect to \( y \) is found by treating \( x \) as a constant, giving \( \frac{\partial f}{\partial y} = 2y \).
Why are these partial derivatives useful? They show how the function's output changes in relation to each variable independently. In the original exercise, calculating these derivatives is crucial to verify Euler's theorem. Understanding how each variable alone influences a function allows deeper analysis of multi-variable functions used in various fields like physics and economics.
For example, the partial derivative of \( f(x, y) = 3x^2 + y^2 \) with respect to \( x \) is found by treating \( y \) as a constant, leading to \( \frac{\partial f}{\partial x} = 6x \). Similarly, the partial derivative with respect to \( y \) is found by treating \( x \) as a constant, giving \( \frac{\partial f}{\partial y} = 2y \).
Why are these partial derivatives useful? They show how the function's output changes in relation to each variable independently. In the original exercise, calculating these derivatives is crucial to verify Euler's theorem. Understanding how each variable alone influences a function allows deeper analysis of multi-variable functions used in various fields like physics and economics.
homogeneity degree
The degree of homogeneity is a core concept in handling and classifying functions. A function is considered homogeneous if, when all its inputs are scaled by a constant \( t \), the output is scaled by \( t^n \). The power \( n \) is known as the degree of homogeneity.
To determine if a function is homogeneous, the exercise used the definition: substitute \( (tx, ty) \) for \( (x, y) \) in the function and check if you can factor out \( t^n \). If you can, the function is homogeneous.
In the example \( f(x, y) = 3x^2 + y^2 \), substituting \( (tx, ty) \) yields \( t^2(3x^2 + y^2) \). Here, since \( t^2 \) can be factored out, the function is homogeneous of degree 2, which matches the result from applying Euler's theorem.
Knowing a function's homogeneity degree is fundamental in theoretical mathematics and applied sciences, contributing to simplifying complex problems through classification and scalability.
To determine if a function is homogeneous, the exercise used the definition: substitute \( (tx, ty) \) for \( (x, y) \) in the function and check if you can factor out \( t^n \). If you can, the function is homogeneous.
In the example \( f(x, y) = 3x^2 + y^2 \), substituting \( (tx, ty) \) yields \( t^2(3x^2 + y^2) \). Here, since \( t^2 \) can be factored out, the function is homogeneous of degree 2, which matches the result from applying Euler's theorem.
Knowing a function's homogeneity degree is fundamental in theoretical mathematics and applied sciences, contributing to simplifying complex problems through classification and scalability.
Other exercises in this chapter
Problem 246
Find \(\frac{\partial w}{\partial s}\) if \(w=4 x+y^{2}+z^{3}, x=e^{r s^{2}}, y=\ln \left(\frac{r+s}{t}\right)\) and \(z=r s t^{2}\)
View solution Problem 247
If \(w=\sin (x y z), x=1-3 t, y=e^{1-t}\) and \(z=4 t, \quad\) find \(\frac{\partial w}{\partial t}\)
View solution Problem 249
For the following exercises, use this information: A function \(f(x, y)\) is said to be homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) For all h
View solution Problem 250
For the following exercises, use this information: A function \(f(x, y)\) is said to be homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) For all h
View solution