Problem 23 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

2-22. \(^{*}\) If \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) and \(D_{2} f=0\), show that \(f\) is independent of the second variable. If \(D_{1} f=D_{2} f=0\), show that \(f\) is constant.

Step-by-Step Solution

Answer

\(f\) is independent of \(y\) when \(D_{2} f = 0\) and constant when \(D_{1} f = D_{2} f = 0\).

Step 1: Understand the Problem

Given a function \(f: \mathbf{R}^{2} \rightarrow \mathbf{R}\) with partial derivatives \(D_{2} f = 0\), the task is to show that \(f\) is independent of the second variable. Additionally, if \(D_{1} f = D_{2} f = 0\), show that \(f\) is constant.

Step 2: Analyze the First Condition \(D_{2} f = 0\)

Since \(D_{2} f = \frac{\partial f}{\partial y} = 0\), the partial derivative of \(f\) with respect to \(y\) is zero. This means \(f\) does not change as \(y\) changes, which implies \(f\) is independent of \(y\). Therefore, \(f(x, y) = g(x)\) for some function \(g\) dependent only on \(x\).

Step 3: Analyze the Second Condition \(D_{1} f = 0\)

Now consider the additional condition \(D_{1} f = \frac{\partial f}{\partial x} = 0\). This indicates that \(f\) does not change as \(x\) changes, meaning \(f\) is also independent of \(x\). Since \(f\) is independent of both \(x\) and \(y\), \(f\) must be a constant function.

Step 4: Combine the Conditions

Combining the steps, \(f = g(x)\) is independent of \(y\). With \(g'(x) = 0\), \(g\) is independent of \(x\) as well. This implies \(f(x, y) = c\) where \(c\) is a constant.

Key Concepts

Function of Two Variables

A function of two variables, usually written as \(f(x, y)\), is a rule that assigns to every pair of values \((x, y)\), within a certain domain, a single value \(f(x, y)\). This type of function is essential in fields like physics and engineering because it allows for the modeling of systems with two independent variables.
For example, consider the height of a landscape at any given point. The height can depend on both the horizontal position \(x\) and the vertical position \(y\). Therefore, the height is a function of \(x\) and \(y\), and can be represented as \(h(x, y)\).
When dealing with functions of two variables, we often use partial derivatives to understand how the function changes with respect to one variable when the other variable is held constant. This helps in analyzing and manipulating the function in its broader context.

Constant Function

A constant function is a function that always returns the same value, no matter the input. For a function \(f(x, y)\) to be constant, the value must stay the same regardless of changes in \(x\) and \(y\).
Mathematically, if both partial derivatives \(D_{1} f\) and \(D_{2} f\) are zero, then the function does not change with either variable. This implies:

• \(D_{1} f = \frac{\partial f}{\partial x} = 0\)
• \(D_{2} f = \frac{\partial f}{\partial y} = 0\)

If these conditions hold, it means \(f\) does not vary with changes in \(x\) or \(y\). Hence, it must be a constant value \(c\). This makes the function \(f(x, y) = c\) where \(c\) is some constant.
Understanding constant functions is important because they represent systems or scenarios where regardless of the inputs, the outcome is unaffected by any changes.

Independence of Variables

When we say a function is independent of a certain variable, it means that the function's value does not change when that variable changes. For instance, if \(D_{2} f = 0\), it implies that the function \(f\) does not change when \(y\) changes. Therefore, \(f\) is independent of \(y\).
To illustrate, let’s take a function \(f(x, y) = g(x)\), where \(g(x)\) is a function dependent only on \(x\). Here, no matter what \(y\) value is inputted, \(f\) remains the same. So, the function \(f\) with respect to \(y\) doesn't change, proving independence from \(y\).
Similarly, if \(D_{1} f = 0\), \(f\) isn't affected by changes in \(x\), leading to the conclusion that \(f\) is dependent solely on \(y\), or it is a combination where changes in \(x\) don't alter the function value.
Recognizing the independence of variables in a function allows for simplifying and understanding the behavior and characteristics of multidimensional spaces, especially in complex system modeling.