Problem 33 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

3-33. If \(f:[a, b] \times[c, d] \rightarrow \mathbf{R}\) is continuous and \(D_{2} f\) is continuous, define \(F(x, y)=\int_{a}^{x} f(t, y) d t\) (a) Find \(D_{1} F\) and \(D_{2} F\). (b) If \(G(x)=\int_{a}^{\theta(x)} f(t, x) d t\), find \(G^{\prime}(x)\).

Step-by-Step Solution

Answer

For part (a), \(D_1 F(x, y) = f(x, y)\) and \(D_2 F(x, y) = \int_{a}^{x} D_2 f(t, y) dt\). For part (b), \(G^{\backprime}(x) = f(\theta(x), x) \theta^{\backprime}(x) + \int_{a}^{\theta(x)} D_2 f(t, x) dt\).

Step 1 - Recalling Definitions and Assumptions

Consider the given function \(f(x, y)\) defined on the domain \([a, b] \times [c, d]\), and the functions \(F(x, y)\) and \(G(x)\). Also, remember that \(f\) and \(D_2 f\) are continuous.

Step 2 - Finding \(D_1 F\)

Define \(F(x, y) = \int_{a}^{x} f(t, y) dt\). To find \(D_1 F\), we differentiate \(F\) with respect to \(x\). Since \(F(x, y)\) is the integral of \(f(t,y)\) with respect to \(t\) from \(a\) to \(x\), using the Fundamental Theorem of Calculus, we get \(D_1 F(x, y) = f(x, y)\).

Step 3 - Finding \(D_2 F\)

To find \(D_2 F\), we need to differentiate \(F(x, y)\) with respect to \(y\). Use Leibniz's rule for differentiation under the integral sign: \(D_2 F(x, y) = \int_{a}^{x} D_2 f(t, y) dt\).

Step 4 - Defining Function \(G(x)\)

Define \(G(x)= \int_{a}^{\theta(x)} f(t, x) dt\).

Step 5 - Finding \(G^{\backprime}(x)\) Using Chain Rule

Differentiate \(G(x)\) with respect to \(x\) using the chain rule. Note that \(G(x)\) is composed of two parts in terms of \(x\): the upper limit of integration \(\theta(x)\), and the function \(f(t,x)\) itself. Therefore, we apply Leibniz's rule combined with the chain rule: \[G^{\backprime}(x) = f(\theta(x), x) \theta^{\backprime}(x) + \int_{a}^{\theta(x)} D_2 f(t, x) dt\].

Key Concepts

Continuous Functions

Partial derivatives and integrals often involve continuous functions. In our example, we start by noting that the function \(f(x, y)\) is continuous. A continuous function is one where small changes in the input result in small changes in the output. This concept is vital because it ensures no sudden jumps or gaps in the function.
In more technical terms, for \(f:[a, b] \times [c, d] \rightarrow \text{R}\) to be continuous, for all \( (x_0, y_0) \) in this domain and every \( \text{ε} > 0 \), there must exist a \( \text{δ} > 0 \) such that whenever \( |(x, y) - (x_0, y_0)| < \text{δ} \), it follows that \( |f(x, y) - f(x_0, y_0)| < \text{ε} \).
This is essential in calculus because it allows for the use of various theorems like the Fundamental Theorem of Calculus and Leibniz's Rule, which hinge on continuity for their validity and application.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concept of differentiation and integration. It consists of two parts:
The first part states that if \( f \) is continuous on \([a, b]\), then the function \( F \) defined by \( F(x) = \int_{a}^{x} f(t) \, dt \) is continuously differentiable, and \( F'(x) = f(x) \).
In our exercise, this theorem is used to find \( D_{1} F\). Given that \( F(x, y) = \int_{a}^{x} f(t, y) dt \), by applying the Fundamental Theorem of Calculus, we see that we can directly differentiate the integral's upper limit:\(D_1 F(x, y) = f(x, y)\).
This simple yet powerful theorem allows us to connect the integral's evaluation to its antiderivative, providing intuitive and direct insights into complex calculations.

Leibniz's Rule

Leibniz's Rule, also known as the differentiation under the integral sign, is a useful method for dealing with integrals that have variable limits or integrands that depend on parameters. The rule states:
\[\frac{\text{d}}{\text{d}x} \bigg( \begin{array}{c} \bigg( \begin{array}{c} \bigg(\text{integral from a to θ(x)} f(t, x) dt\bigg)\bigg)\bigg)\bigg)\bigg)= \big[ d x \times θ(x) \bigg] = f( ex(x), x)aθ'(x)+ d_p(x)F(x, t) d t.\bigg)\]
We apply this rule in our solution. To find the partial derivative \( D_2 F(x, y) \), we use Leibniz's rule since the integrals within \( G(x) \) have variable upper limits that depend on \( y \). Thus:\( D_2 F(x, y) = \bigg( \text{integral from a to x}\bigg( D_2 f(t, y) dt \bigg).\)
Similarly, for \( G'(x)\), we have:\( G'(x)= \bigg( f (θ(x) , x) \bigg(\tfrac{dy}{dx}\bigg)+ \bigg( \text{integral from a to t}(D2f(t, x) dt) \bigg).\).
This rule simplifies differentiation in integrals with variable limits, making them easier to handle analytically.