Question

Write the explicit function \(z=x y^{2}+x^{2} y-10\) in the implicit form \(F(x, y, z)=0\)

Short answer

Answer: The implicit form of the given function is \(F(x, y, z) = z - xy^2 - x^2y + 10 = 0\).

Step-by-step solution

Write down the given equation

The given explicit function is \(z = xy^2 + x^2y - 10\).

Rewrite the explicit function in the form \(F(x, y, z) = 0\)

To rewrite the given equation as an implicit function, we need to obtain the function \(F(x, y, z) = z - (xy^2 + x^2y - 10)\) and set it to 0.

Present the implicit form of the function

The final implicit form of the given function is: \(F(x, y, z) = z - xy^2 - x^2y + 10 = 0\) Now, the explicit function \(z = xy^2 + x^2y - 10\) is transformed into its implicit form \(F(x, y, z) = 0\).

Key concepts

Explicit Functions

An explicit function is a type of mathematical expression where the dependent variable is isolated on one side of the equation. This makes the relationship between the dependent and independent variables very clear. For example, in the explicit function given in the exercise, we have

• Dependent variable: \( z \)
• Independent variables: \( x \) and \( y \)

The explicit equation, \( z = xy^2 + x^2y - 10 \), clearly outlines how \( z \) changes based on \( x \) and \( y \). It's straightforward and makes it easy to calculate \( z \) for any given \( x \) and \( y \) values. The clarity and simplicity of explicit functions make them a preferred choice in many mathematical problems, especially when quick calculations are necessary.

Function Transformation

Function transformation refers to the process of converting one form of a function into another. In this exercise, we transformed an explicit function into an implicit one. This transformation often helps in analyzing relationships between variables that aren't immediately obvious.When dealing with complex equations, sometimes it's more useful to express a function implicitly because:

• It can offer insights into the relationship between multiple variables.
• It can simplify the manipulation of the function for calculus-based applications.

In our case, we started with the explicit form \( z = xy^2 + x^2y - 10 \) and transformed it into its implicit counterpart \( F(x, y, z) = z - xy^2 - x^2y + 10 = 0 \). This step was crucial to see the equation in terms of all involved variables and pave the way for potential derivative calculations or finding roots related to \( x \), \( y \), and \( z \).

Equation Manipulation

Equation manipulation is a mathematical skill that involves altering an equation to achieve a particular form or solve a problem. In this exercise, we manipulated the explicit form of a function to convert it into an implicit one.The process of equation manipulation often includes:

• Rearranging terms
• Performing operations like addition, subtraction, or distribution
• Equating to zero for implicit forms

Specifically, we took the explicit function \( z = xy^2 + x^2y - 10 \) and subtracted it from \( z \) to achieve \( F(x, y, z) = 0 \). By reorganizing the equation terms this way, we make it easier to apply further mathematical techniques, such as symmetry analysis or solving systems of equations.