Problem 32 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

3-32. \({ }^{*}\) Let \(f:[a, b] \times[c, d] \rightarrow \mathbf{R}\) be continuous and suppose \(D_{2} f\) is continuous. Define \(F(y)=\int_{a}^{b} f(x, y) d x .\) Prove Leibnitz's rule: \(F^{\prime}(y)\) \(=\int_{a}^{b} D_{2} f(x, y) d x . \quad\) Hint \(: F(y)=\int_{a}^{b} f(x, y) d x=\int_{a}^{b}\left(\int_{c}^{v} D_{2} f(x, y) d y+\right.\) \(f(x, c)) d x\). (The proof will show that continuity of \(D_{2} f\) may be replaced by considerably weaker hypotheses.)

Step-by-Step Solution

Answer

We proved that if \( f \) and \( D_2 f \) are continuous, then \( F'(y) = \int_{a}^{b} D_2 f(x, y) \, dx \) for the function \( F(y) = \int_{a}^{b} f(x, y) \, dx \).

Step 1: Understand the Problem and Definitions

Let’s start by understanding the definition of the function and the problem statement.Given a continuous function:\( f:[a, b] \times [c, d] \rightarrow \mathbf{R} \),and \( D_2 f \) is continuous, to prove the Leibnitz's rule for:\( F(y) = \int_{a}^{b} f(x, y) \, dx \), we need to show that\( F'(y) = \int_{a}^{b} D_2 f(x, y) \, dx \).

Step 2: Express F Using a Hint

According to the hint provided:\( F(y) = \int_{a}^{b} f(x, y) \, dx = \int_{a}^{b} \left( \int_{c}^{v} D_2 f(x, y) \, dy + f(x, c) \right) dx \). This expression will be useful in showing the derivative properties and confirming Leibnitz's rule.

Step 3: Differentiate Under the Integral Sign

We'll now apply differentiation under the integral sign on the expression for \( F(y) \):Using Leibnitz's rule for differentiation under the integral sign, we have:\( F'(y) = \frac{d}{dy} \int_{a}^{b} f(x, y) \, dx = \int_{a}^{b} \frac{\partial}{\partial y} f(x, y) \, dx \).

Step 4: Use Given Condition

Given \( D_2 f \) is continuous: Therefore, \( \frac{\partial}{\partial y} \left( f(x, y) \right) = D_2 f(x, y) \) as \( D_2 f(x, y) \) is just another notation for the partial derivative with respect to y.

Step 5: Combine and Simplify

Substituting \( D_2 f(x, y) \) in place of \( \frac{\partial}{\partial y} f(x, y) \):\( F'(y) = \int_{a}^{b} \frac{\partial}{\partial y} f(x, y) \, dx = \int_{a}^{b} D_2 f(x, y) \, dx \).Thus, we have shown that:\( F'(y) = \int_{a}^{b} D_2 f(x, y) \, dx \).

Step 6: Conclusion

Hence, we have proven that for a continuous function \( f \) with a continuous partial derivative \( D_2 f \):\( F(y) = \int_{a}^{b} f(x, y) \, dx \) implies \( F'(y) = \int_{a}^{b} D_2 f(x, y) \, dx \).This completes the proof of Leibnitz's rule.

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, often used for functions with multiple variables. In essence, a partial derivative represents how a function changes as one specific variable is varied, keeping all other variables constant. For instance, consider a function \( f(x, y) \), which depends on two variables, x and y. The partial derivative of \( f \) with respect to \( y \) is denoted as \( D_2 f(x, y) \) or \( \frac{\partial f}{\partial y} \). This measures the rate of change of \( f \) as \( y \) changes, while \( x \) remains fixed.
The ability to compute partial derivatives is crucial when working with functions of several variables as they reveal how each variable uniquely influences the overall function. This concept underpins many advanced applications, such as in physics, engineering, and economics. Understanding partial derivatives is foundational for grasping more complex topics like gradient vectors, optimization, and, indeed, the Leibnitz's rule.

Differentiation Under the Integral Sign

Differentiation under the integral sign is an important technique in advanced calculus. It allows us to interchange the operations of differentiation and integration. This technique is particularly useful when dealing with integrals that have parameters, like in the function presented in the exercise: \( F(y)= \int_{a}^{b} f(x, y) \, dx \).
To apply this technique, certain conditions must be met, including the continuity of the function and its partial derivatives. In our case, we have a function \( f \) that is continuous, and its partial derivative with respect to \( y \) (denoted \( D_2 f \)) is also continuous. Given these conditions, we can say:
\[ F'(y) = \int_{a}^{b} \frac{\partial}{\partial y} f(x, y) \, dx \].
This equation illustrates the power of differentiation under the integral sign—transforming a potentially complex differentiation problem into a manageable integral. This approach forms the crux of Leibnitz's rule, seamlessly connecting the concepts of integration and partial differentiation. Understanding this rule is essential for tackling many problems in mathematical analysis.

Continuity

Continuity is another vital mathematical concept that ensures functions behave predictably across their domains. A function \( f(x, y) \) is continuous if small changes in \( x \) and \( y \) result in small changes in \( f(x, y) \). In simpler terms, there are no abrupt jumps or breaks in the function.
For Leibnitz's rule, the continuity of both the function \( f \) and its partial derivative \( D_2 f \) are crucial. The continuity of \( f \) across the region \( [a, b] \times [c, d] \) ensures that the integral \( F(y) \) depends smoothly on \( y \). Furthermore, the continuity of \( D_2 f \) guarantees that the differentiation under the integral sign is valid. Without these continuity conditions, the interchange of integration and differentiation might not be justified, leading to incorrect results.
Therefore, verifying the continuity of involved functions and their derivatives is an indispensable step in proving many theorems, including Leibnitz's rule. Sustaining this property allows for proper analytical manipulation and application of various calculus techniques.