Problem 77 - Calculus: One and Several Variables

Question

Show that if \(f\) and \(g\) have continuous sceond derivatives and \(f(a)=g(a)=f(b)=g(b)=0,\) then $$\int_{a}^{b} f(x) y^{\prime \prime}(x) d x=\int_{a}^{b} g(x) f^{\prime \prime}(x) d x$$

Step-by-Step Solution

Answer

By applying Green's First Identity and the boundary conditions, it is clear that the original equation \(\int_{a}^{b} f(x) y^{\prime\prime}(x) dx = \int_{a}^{b} g(x) f^{\prime\prime}(x) dx \) holds true.

Step 1: Understand the components

We have two functions \(f\) and \(g\) which have continuous second derivatives. The boundaries \(a\) and \(b\) of the function also have given conditions: \(f(a)=g(a)=f(b)=g(b)=0\) . It is needed to show that \(\int_{a}^{b} f(x) y^{\prime\prime}(x) dx = \int_{a}^{b} g(x) f^{\prime\prime}(x) dx \).

Step 2: Apply the Green's First Identity

According to Green's First Identity, the above equation holds true when both functions have continuous second derivatives and the provided boundary conditions hold. The identity states that: \(\int_{a}^{b} f(x) g^{\prime\prime}(x) dx = [f(x)g'(x)]_{a}^{b} - \int_{a}^{b} g(x) f^{\prime\prime}(x) dx\). Substitute \(g'\) with \(y'\) and \(g^{\prime\prime}\) with \(y^{\prime\prime}\) in the above equation.

Step 3: Apply the integral and boundary conditions

Next, apply the original conditions \(f(a)=g(a)=f(b)=g(b)=0.\) This will make the term [f(x)g'(x)]_{a}^{b} equal to zero, since either f(a), g(a), f(b) or g(b) is equal to zero. The equation now becomes \(\int_{a}^{b} f(x) y^{\prime\prime}(x) dx = \int_{a}^{b} g(x) f^{\prime\prime}(x) dx \), which completes the proof.

Key Concepts

Continuous Functions

Continuous functions are functions that have no breaks, jumps, or holes in their domain. This is essential when working with calculus, as it ensures that the function behaves predictably across its range. Here are key points to understand about continuous functions:

• They are smooth and unbroken throughout their domain, making them ideal for analysis.
• You should be able to draw the graph of a continuous function without lifting your pencil from the paper.
• In the context of Green's First Identity, continuous functions ensure that integration techniques, such as integration by parts, can be effectively applied.

In calculus, dealing with functions that are not continuous can lead to difficulties in both analysis and computation. This is why proving properties like those in Green’s First Identity requires continuity to make use of the predictable behavior of the functions involved.

Second Derivatives

The second derivative of a function gives us valuable information about the curvature and concavity of its graph. It is essentially the derivative of the derivative, providing deeper insights into the function's behavior.

• The second derivative, denoted as \( f''(x) \), measures how the rate of change of a function is changing.
• It helps in determining whether a function is concave up or concave down at a particular point: if \( f''(x) > 0 \), the function is concave up, and if \( f''(x) < 0 \), it is concave down.
• In Green's First Identity, continuous second derivatives imply differentiability, and thus predictability, over the interval \([a, b]\).

Understanding the role of second derivatives is crucial in fields such as physics, where they describe acceleration, or in economics, where they highlight changes in rate of growth or decline.

Boundary Conditions

Boundary conditions refer to the constraints or fixed values that a function must satisfy at the boundaries of its domain. They are particularly relevant in problems involving differential equations or integrals, such as in Green's First Identity.

• These conditions specify the values of the function, and often its derivatives, at the endpoints \( a \) and \( b \).
• They are crucial for solving many types of differential equations, where without such constraints, multiple solutions could satisfy the equations.
• In the given problem, \( f(a) = g(a) = f(b) = g(b) = 0 \) are the boundary conditions. These ensure that terms such as \([f(x)g'(x)]_{a}^{b}\) simplify, often reducing a lot of the complexity in the calculations.

Boundary conditions work hand in hand with the properties of continuous and differentiable functions to ensure a well-defined solution.