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Chapter 15: Problem 12

Use the divergence theorem to evaluate \(\iint_{1} \mathbf{F} \cdot \mathbf{n} d \mathcal{S}\), where \(\mathbf{n}\) is the outer unit normal vector to the surface \(S .\) \(\mathbf{F}=\mathbf{r} /|\mathbf{r}| .\) where \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}: \quad S\) is the sphere \(\rho=2\) of radius 2 centered at the origin.

Short Answer

Expert verified
The integral is zero due to symmetry and divergence.

Step by step solution

01

Understand the Problem

We need to evaluate \(\iint_{S} \mathbf{F} \cdot \mathbf{n} \, d \mathcal{S}\) for the vector field \(\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|}\), where \(\mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k}\), and \(S\) is a sphere of radius 2 centered at the origin. The divergence theorem will be used for this purpose.
02

Recall the Divergence Theorem

The divergence theorem states: \[ \iint_{S} \mathbf{F} \cdot \mathbf{n} \, d \mathcal{S} = \iiint_{V} abla \cdot \mathbf{F} \, dV \] where \(V\) is the volume inside the surface \(S\). Our task is to find \(abla \cdot \mathbf{F}\) and evaluate the volume integral over the sphere of radius 2.
03

Calculate \(\nabla \cdot \mathbf{F}\)

Since \(\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|} = \frac{x \mathbf{i} + y \mathbf{j} + z \mathbf{k}}{\sqrt{x^2 + y^2 + z^2}}\), we find \(abla \cdot \mathbf{F}\). First, notice that \(\mathbf{F}\) can be expressed with the radial unit vector \(\mathbf{F} = \frac{\mathbf{r}}{\rho}\), where \(\rho = \sqrt{x^2 + y^2 + z^2}\). It turns out that the divergence of \(\frac{\mathbf{r}}{\rho}\) is zero everywhere except at the origin (a singularity).
04

Evaluate the Volume Integral

Due to the symmetry of \(\mathbf{F}\) and since \(abla \cdot \mathbf{F} = 0\) except at the origin, the volume integral over any volume not containing the origin is zero. Since the origin is inside our volume (the sphere of radius 2), the volume integral results in zero because the contribution around the singularity is effectively ignored by the divergence theorem when applied correctly to infinite or undefined values at points.
05

Conclusion

Since the divergence of the field is zero everywhere inside the volume except at an isolated singularity, and since there is no net flux out of a symmetric field on a closed surface, by the divergence theorem, the flux through the surface \(S\) is zero.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Calculus
Vector calculus is a branch of mathematics concerned with differentiation and integration of vector fields. It deals with curves, surfaces, and volumes in three dimensions and explores how these multivariable calculus techniques apply to vectors. In our exercise involving the Divergence Theorem, vector calculus helps analyze vector fields and surfaces, ensuring that we can compute properties like flux.
Here are some important aspects of vector calculus:
  • Vector Functions: These are functions that have vectors as outputs, crucial when dealing with physics-related problems where quantities like force and velocity are represented as vectors.
  • Gradient, Divergence, and Curl: These operations give us insight into the behavior of vectors in fields. For the divergence theorem, we are particularly interested in divergence, which measures a vector field's tendency to 'spread outwards' from a point.
  • Line and Surface Integrals: Integrals can be taken over paths or surfaces, helping us compute quantities like work done by a field or flux through a surface.
Understanding these basics provides the foundation for solving more complex problems such as those involving the divergence theorem.
Surface Integrals
Surface integrals are a type of integral where we sum up the values of a function over a surface. In this context, the function is often a dot product of a vector field and a normal vector to that surface, essentially measuring the "flow" across it.
To compute a surface integral, follow these steps:
  • Parameterize the Surface: Describe the surface using a set of parameters, normally two, since a surface is 2-dimensional.
  • Calculate Normal Vectors: Determine the unit normal vectors to the surface, which are essential for the dot product with the vector field.
  • Integrate: Sum the contributions over the entire surface using the appropriate limits for your chosen parameters.
For our sphere of radius 2, the surface integral considers the flow of the vector field hrough the sphere's boundary.
Vector Fields
Vector fields assign a vector to every point in a space, illustrating how vectors vary over that space. They're often used in physics to describe things like gravitational or magnetic fields. In our exercise, the vector field is \(\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|}\), which indicates a radial field pointing outward from the origin with a constant magnitude of 1.
Key points to remember about vector fields include:
  • Vectors at Every Point: Each vector has direction and magnitude, showing how the field affects that specific location.
  • Field Lines: Imaginary lines that represent the direction the vectors point in. In our problem, these lines radiate from the origin due to the radial nature of the vector field.
  • Divergence and Curl: These concepts help determine behaviors like expanding (divergence) or swirling (curl) of the field.
Understanding vector fields' nature helps in applying and understanding the Divergence Theorem.
Spherical Coordinates
Spherical coordinates are a curvilinear coordinate system often used in three-dimensional space in which location points are defined relative to an origin and angles. They are particularly beneficial for dealing with problems having spherical symmetry, like the one in our exercise.
Key components of spherical coordinates include:
  • Radial Distance (\( \rho \)): The distance from the origin to the point. For a sphere of radius 2, \( \rho \) is constant at 2.
  • Polar Angle (\( \theta \)): The angle measured from the positive z-axis toward the point.
  • Azimuthal Angle (\( \phi \)): The angle measured in the xy-plane from the positive x-axis.
These coordinates are especially useful when analyzing spheres or circular paths as they simplify expressions and computations, making integration more manageable.

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Most popular questions from this chapter

Let \(\mathbf{F}(x, y)=(-y \mathbf{i}+x \mathbf{j}) /\left(x^{2}+y^{2}\right)\) for \(x\) and \(y\) not both zero. Calculate the values of $$\int_{C} \mathbf{F} \cdot \mathbf{T} d s$$ along both the upper and the lower halves of the circle \(x^{2}+y^{2}=1\) from \((1,0)\) to \((-1.0)\). Is there a function \(f=f(x, y)\) defined for \(x\) and \(y\) not both zero such that \(\boldsymbol{r} f=\mathbf{F}\) ? Why?

The given curve \(C\) joins the points \(P\) and \(Q\) in the \(x y\) -plane. The point \(P\) represents the top of a ten-story building. and \(Q\) is a point on the ground \(100 \mathrm{ft}\) from the base of the building. A \(150-\) lb person slides down a frictionless slide shaped like the curve \(C\) from \(P\) to \(Q\) under the influence of the gravitational force \(\mathbf{F}=-150 \mathbf{j} .\) In each problem show that F does the same amount of work on the person, \(W=15000 \mathrm{ft} \cdot \mathrm{lb}\), as if he or she had dropped straight down to the ground. \(C\) is the straight line segment \(y=x\) from \(P(100,100)\) to \(Q(0,0)\)

Apply Green's theorem to evaluate the integral $$\oint_{C} P d x+Q d y$$ around the specified closed curve \(C\). \(P(x, y)=x^{2}, Q(x, y)=-y^{2} ; \quad C\) is the cardioid with polar equation \(r=1+\cos \theta\)

Determine whether the vector fields are conservative. Find potential functions for those that are conservative (either by inspection or by using the method of Example 4 ). \(\mathbf{F}(x, y)=(4 x-y) \mathbf{i}+(6 y-x) \mathbf{j}\)

Evaluate the line integral $$\int_{C} \mathbf{F} \cdot d \mathbf{r}=\int_{C} \mathbf{F} \cdot \mathbf{T} d s$$ along the indicated path \(C\). \(\mathbf{F}(x, y, z)=y \mathbf{i}-x \mathbf{j}+z \mathbf{k}: \quad x=\sin t, y=\cos t . z=2 t\) \(0 \leqq y \leqq \pi\)
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