Question

Determining if a Function Is Homogeneous In Exercises \(67-74,\) determine whether the function is homogeneous, and if it is, determine its degree. A function \(f(x, y)\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y)\) $$ f(x, y)=x^{3}-4 x y^{2}+y^{3} $$

Short answer

The function \(f(x, y)=x^{3}-4 x y^{2}+y^{3}\) is homogeneous of degree 3.

Step-by-step solution

Substitute the Variables

Replace \(x\) and \(y\) with \(t x\) and \(t y\) in the original function \(f(x, y)=x^{3}-4 x y^{2}+y^{3} \). After substitution the function \(f\) becomes \(f(t x, t y)=(t x)^3-4 (t x) (t y)^2+(t y)^3 = t^3x^3-4t^3x y^2+t^3y^3\)

Factor out the Power of t

Factor out \(t^3\) from \(f(t x, t y)\) to get \(f(t x, t y)= t^3(x^3-4 x y^{2}+y^3)= t^3 f(x, y)\)

Determine if the Function is Homogeneous and its Degree

The function \(f(t x, t y)= t^{3} f(x, y)\) shows that the function is homogeneous since it fits the definition of a homogeneous function \(f(tx, ty) = t^n f(x, y)\). The degree of the homogeneous function is 3, which is the exponent of \(t\) in the equation.

Key concepts

Homogeneous Function Degree

Understanding the degree of a homogeneous function is crucial in the realm of algebra and calculus. A function is said to be homogeneous if it can be expressed in a way such that all of its terms scale identically when the input variables are scaled. Specifically, a function of two variables, here represented as homogeneous. The degree of the homogeneous function is then the power of and denotes the degree to which the function scales when its input variables are multiplied by a scalar value.

For instance, consider the function from the given exercise. When we substitute with and with respectively, and then factor out the common term, we obtain indicating this function is homogeneous and its degree is This degree tells us that multiplying both variables and by a scalar will result in the output of the function being scaled by Using this concept, problems in advanced algebra and calculus, such as scaling differential equations and economic models, can be approached with a clearer strategy.

Algebraic Function Properties

Algebraic functions play a pivotal role in numerous mathematical realms due to their diverse properties. These functions are formed by algebraic expressions using a finite number of arithmetic operations such as addition, subtraction, multiplication, division, and taking roots of the variables. One intriguing property of algebraic functions is their behavior in relation to the manipulation of their variables.

An essential property to consider is how algebraic functions respond to scaling of their variables, specifically in the context of homogeneous functions. The function given in our exercise, can also be analyzed for its symmetry, periodicity, and the behavior near critical points. Understanding the algebraic properties provides deep insights into the function's graph and potential applications, which might include physics and engineering contexts where such functions describe systems following specific rules.

Multivariable Calculus

When we step into the world of multiple variables, the complexity and beauty of calculus unfold further. Multivariable calculus extends the concepts of single-variable calculus to functions with several variables. The function in our exercise is a fantastic primer into this world, as it is defined in terms of two variables, and Understanding these multivariable functions is essential not only for academic purposes but also for real-world applications in fields like engineering, economics, and the sciences.

Through multivariable calculus, we consider partial derivatives, gradients, multiple integrals, and vector calculus. In the context of our exercise, one would extend the analysis to evaluate how the function changes in different directions through partial derivatives, identify local and global extremums, and even describe the function's level curves. This rigorous toolkit allows us to deeply appreciate and precisely quantify phenomena in multiple dimensions.