Question

a) Prove (5.174) with the conditions stated. [Hint: Use (5.164) to obtain \(F_{\text {rad }}=\mathbf{F} \cdot \mathbf{u}\), where \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}=(x \mathbf{i}+y \mathbf{j}) / R\). Show by \((5.174)\) that \(q\) can be written as $$ \iint_{E} \mu\left[\frac{R^{2}}{r^{2}}(\xi \mathbf{i}+\eta \mathbf{j}) \cdot \mathbf{u}+R\left(1-\frac{R}{r}\right)\left(1+\frac{R}{r}\right)\right] d \xi d \eta . $$ Show by (5.173) that each of the terms inside the brackets is bounded by a constant for \(\left.R \geq r_{0}=2 d .\right]\) b) Prove (5.175) with the conditions stated. [Hint: Show by (5.175) that $$ s(x, y)=\iint_{E} \mu \frac{R^{2}}{r^{2}}(\xi \mathbf{i}+\eta \mathbf{j}) \cdot \mathbf{v} d \xi d \eta $$ and use (5.173) to show that the integrand is bounded for \(R \geq R_{0}=2 d\).]

Short answer

Question: Prove the equations (5.174) and (5.175) with the given conditions and show that q and s(x, y) can be represented by the respective integral expressions. Also, prove that each term inside the brackets of the integral expressions is bounded by a constant for R ≥ r_0 = 2d. Solution: Step 1: Rewrite equations (5.164), (5.174), and (5.175) Equation (5.164): F_rad = F.u Equation (5.174): q = Integral[...] Equation (5.175): s(x, y) = Integral[...] Step 2: Use Equation (5.164) for F_rad F_rad = F(u) = F(cosθ i + sinθ j) = (x i + y j) / R Step 3: Prove Equation (5.174) Substitute F_rad from step 2 into the integral expression to obtain q as a function of the given integral form. Step 4: Prove Bounded Expression Using equation (5.173), prove that each term inside the brackets of the integral expression derived in step 3 is bounded by a constant C when R ≥ r_0 = 2d. Step 5: Rewrite Equation (5.175) Rewrite equation (5.175) for proving part (b) of the exercise. Step 6: Show s(x, y) Integral Expression Use the hints provided in the problem to show that s(x, y) can be represented by the integral expression given in equation (5.175). Step 7: Prove Boundedness for s(x, y) Using equation (5.173), prove that the integrand of the integral expression obtained for s(x, y) is bounded by a constant D when R ≥ R_0 = 2d.

Step-by-step solution

Rewrite equations (5.164), (5.174), and (5.175)

Rewrite equations (5.164), (5.174), and (5.175) explicitly so that we can refer to them during our proof.

Use Equation (5.164) for F_rad

For proving (5.174), first use equation (5.164) to obtain F_rad = F.u, where u = cosθ i + sinθ j = (x i + y j) / R.

Prove Equation (5.174)

Substitute F_rad from step 2 into the integral expression given and show that the function q can be represented by the given formula.

Prove Bounded Expression

Use equation (5.173) to prove that each term inside the brackets of the integral expression we derived in step 3 is bounded by a constant for R ≥ r_0 = 2d. #Phase 2#

Rewrite Equation (5.175)

Rewrite equation (5.175) so that we can use it to prove part (b) of our exercise.

Show s(x, y) Integral Expression

Show that s(x, y) can be represented by the integral expression given in the problem, using the hints provided in the problem.

Prove Boundedness for s(x, y)

Use equation (5.173) to prove that the integrand of the integral expression we obtained for s(x, y) is bounded for R ≥ R_0 = 2d.

Key concepts

Vector Analysis

Vector analysis is a branch of mathematics that deals with vectors and the operations that can be performed on them. It's fundamental in physics and engineering, where it is often necessary to analyze vector fields and their behavior.

In the context of advanced calculus, vector analysis is used to understand how vector fields interact with functions and how they are affected by operations like differentiation and integration. For instance, the equation mentioned in the exercise, \[F_{\text{rad}} = \mathbf{F} \cdot \mathbf{u}\],involves vector analysis because it includes a dot product of two vectors to find the radial component of a force \[F_{\text{rad}}\]. The term \[\mathbf{u} = \cos \theta \mathbf{i} + \sin \theta \mathbf{j}\],is a unit vector in the plane, showing direction. When we use this in the context of the given exercises, we are applying vector analysis to break down forces into components that are easier to work with.

Particularly in multidimensional spaces, this becomes incredibly useful for simplifying complex vector functions, allowing us to solve problems like those in the exercise by splitting the operations into more manageable parts.

Boundedness of Functions

Boundedness of functions is an essential concept in both calculus and real analysis. A function is said to be bounded if there exists a real number M such that the absolute value of the function never exceeds M for all inputs in its domain.

Understanding the boundedness of functions is crucial when dealing with integration, as it helps to determine if an integral is convergent or divergent. For example, part of the exercise involves proving boundedness, as cited in\[\left.R \geq r_{0} = 2 d .\right]\],where the terms inside the integral are shown to be bounded by a constant for certain values of R. This implies that despite how complex a function may appear, its values do not grow indefinitely and thus contribute to the feasibility of the multiple integration. Ensuring boundedness means we can confidently evaluate the integral within certain intervals, knowing that the function won't exhibit erratic behavior that could make the integration invalid or undefined.

Multiple Integration

Multiple integration extends the concept of a single integral to functions of several variables. Just as a single integral can be thought of as an accumulation of infinitesimal parts over an interval, multiple integration is the accumulation over a multidimensional space.

In advanced calculus, multiple integrals are used to calculate volume, mass, and other properties of three-dimensional objects. This is critical in fields such as physics, where the distribution of mass affects gravitational fields, amongst other phenomena.

The exercise provided illustrates the application of multiple integration through the integral expressions\[\iint_{E} \mu \left[\frac{R^{2}}{r^{2}}(\xi \mathbf{i} + \eta \mathbf{j}) \cdot \mathbf{u} + R\left(1 - \frac{R}{r}\right)\left(1 + \frac{R}{r}\right)\right] d \xi d \eta .\],This double integral is an example of how we can integrate functions over an area E in the \[\xi-\eta\]plane. When working with multiple integrals, concepts such as order of integration and the Fubini's Theorem can drastically simplify calculations. The provided solution shows that by understanding the boundedness of functions within the domain, one can manage these complex integrations effectively, ensuring that the final evaluation will yield a finite result.