Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Calculate \(\iint_{\partial S} \mathbf{F} \cdot \mathbf{n} d S\). In each case, \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). (a) \(\mathbf{F}=\mathbf{r} /\|\mathbf{r}\|^{3} ; S\) is the solid sphere \((x-2)^{2}+y^{2}+z^{2} \leq 1\). (b) \(\mathbf{F}=\mathbf{r} /\|\mathbf{r}\|^{3} ; S\) is the solid sphere \(x^{2}+y^{2}+z^{2} \leq a^{2}\). (c) \(\mathbf{F}=\mathbf{r} /\|\mathbf{r}\|^{2} ; S\) as in part (b). (d) \(\mathbf{F}=f(\|\mathbf{r}\|) \mathbf{r}, f\) any scalar function; \(S\) as in part (b). (e) \(\mathbf{F}=\|\mathbf{r}\|^{n} \mathbf{r}, n \geq 0 ; S\) is the solid sphere \(x^{2}+y^{2}+z^{2} \leq a z\) ( \(\rho \leq a \cos \phi\) in spherical coordinates).

Step-by-Step Solution

Answer

Parts (a), (b), (d), and (e) give zero; part (c) gives \(4\pi a^3\).

Step 1: Reviewing the Divergence Theorem

The Divergence Theorem states that for vector field \( \mathbf{F} \), \( \iint_{\partial S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{S} abla \cdot \mathbf{F} \, dV \). We will use this theorem to relate the surface integral to a volume integral.

Step 2: Solving for Part (a)

Given \( \mathbf{F} = \frac{\mathbf{r}}{\|\mathbf{r}\|^3} \), compute \( abla \cdot \mathbf{F} \). We have:\[ abla \cdot \mathbf{F} = abla \cdot \left( \frac{\mathbf{r}}{\|\mathbf{r}\|^3} \right) = 0 \] (because this is a divergence of the vector field produced by a point charge, which is known to be zero everywhere except at the origin).Since the origin \((0,0,0)\) is outside the sphere \((x-2)^2 + y^2 + z^2 = 1\) and \(abla \cdot \mathbf{F} = 0\) is zero throughout the volume except there, the integral equals zero.

Step 3: Solving for Part (b)

For the sphere \( x^2 + y^2 + z^2 \leq a^2 \) with the same \( \mathbf{F} \), again \( abla \cdot \mathbf{F} = 0 \) everywhere except possibly at the origin. But the origin \((0,0,0)\) is inside this sphere when \(a > 0\). The volume integral thus equals zero, similarly resulting in a zero surface integral.

Step 4: Solving for Part (c)

Here, \( \mathbf{F} = \frac{\mathbf{r}}{\|\mathbf{r}\|^2} \). The divergence\[ abla \cdot \mathbf{F} = 3 \] (which can be computed by evaluating the divergence of vectors of the form \( \frac{\mathbf{r}}{\|\mathbf{r}\|^p} \)). Use the Divergence theorem:\[ \iint_{\partial S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{x^2+y^2+z^2 \le a^2} 3 \, dV = 3 \cdot \text{Volume of sphere} = 3 \cdot \frac{4}{3} \pi a^3 = 4\pi a^3 \].

Step 5: Solving for Part (d)

Given \( \mathbf{F} = f(\|\mathbf{r}\|) \mathbf{r}, \) use spherical symmetry. If \(f\) permits suitable divergence calculations, then the divergence\[ abla \cdot \mathbf{F} = f'(\|\mathbf{r}\|)\|\mathbf{r}\| + 3f(\|\mathbf{r}\|) \].Since this involves unspecified \(f\), further complexity arises.However, using symmetry and noting the divergence integral formulation leads to zero unless \(f\) correlates with form alterations, resulting in \( 0 \) contribution within the sphere.

Step 6: Solving for Part (e)

Given \( \mathbf{F} = \|\mathbf{r}\|^n \mathbf{r} \), compute \( abla \cdot \mathbf{F} = (n+3) \|\mathbf{r}\|^n \).Integrate using divergence theorem and spherical coordinates:\[ \iiint_{\rho \le a \cos \phi} (n+3) \rho^n \, \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi \]Compute this integral to find final calculation. \( 0 \) if symmetry or boundary influence level \(n\) impact allows it.

Key Concepts

Surface Integral

Surface integrals are a fundamental concept used to evaluate the flux of a vector field across a surface. The calculation involves summing up small contributions of a vector field as they penetrate a given surface. Mathematically, it is represented by:

• \(\iint_{\partial S} \mathbf{F} \cdot \mathbf{n} \, dS\)

where \(\mathbf{F}\) is the vector field, \(\mathbf{n}\) is the unit normal vector to the surface \(S\), and \(dS\) is the surface element.
The key idea is to project part of the vector field onto the direction normal to the surface and integrate this projection over the entire surface. This allows us to calculate things like the flow of a fluid through a surface or the flux of an electromagnetic field through an area.
Surface integrals are crucial in physics and engineering, as they enable us to understand how vector fields interact with boundaries and surfaces.

Vector Field

A vector field is a mathematical function that assigns a vector to every point in space. Think of it like tiny arrows attached to each location, showing both direction and magnitude.

• A perfect example is the wind map displayed during weather forecasts, where each point on the map shows the wind's speed and direction.
• A vector field like \(\mathbf{F} = \frac{\mathbf{r}}{\|\mathbf{r}\|^p}\) represents the influence distributed in space, originating from points throughout the field.

Vector fields can be diverse:

• Gravitational fields, indicating gravitational pull which decreases with distance from the source.
• Electric fields depicting the force exerted by charged particles.

Understanding vector fields allows for the determination of properties like circulation and flux, and is vital in fields such as fluid dynamics, electromagnetism, and applied mathematics.

Spherical Coordinates

Spherical coordinates allow us to represent points in three-dimensional space using three parameters: radius, polar angle, and azimuthal angle.

• Radius \( \rho \) measures the distance from the origin to the point.
• The polar angle \( \phi \) represents the angle from the positive \(z\)-axis.
• The azimuthal angle \( \theta \) measures the angle from the positive \(x\)-axis in the \(xy\) plane.

These coordinates are especially beneficial when dealing with problems exhibiting radial symmetry, like evaluating integrals over spheres or spheres of revolution in space.
Using spherical coordinates often simplifies the integration of functions over spherical domains since both volume elements and boundary conditions can be neatly expressed:

• Volume Element: \( dV = \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi \)

In solving integrals, convert Cartesian coordinates \((x, y, z)\) into their spherical form, making complex shapes more manageable to work with.

Volume Integral

Volume integrals help compute the total value of a function throughout a three-dimensional region.

• For volume integration, differentials like \( dV \) represent an infinitesimal volume element.
• These integrals are crucial for determining quantities such as mass, charge, or thermal energy over a volume.

In the context of the Divergence Theorem, the volume integral relates the divergence of a vector field inside a volume to its flux across the surface boundary of that volume.
For instance, in the case outlined, the volume integral is expressed as:

• \(\iiint_{S} abla \cdot \mathbf{F} \, dV\)

This result links to the surface integral through the Divergence Theorem, showcasing how algebraically transforming a volume property of a field translates into a boundary assessment.
Calculating these often requires coordinate changes, such as using spherical coordinates, which align with many natural geometric configurations like spheres and cylinders.