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Chapter 16: Problem 26

Flux of a constant field Show that the outward flux of a constant vector field \(\mathbf{F}=\mathbf{C}\) across any closed surface to which the Divergence Theorem applies is zero.

Short Answer

Expert verified
The outward flux of a constant vector field across any closed surface is zero due to zero divergence.

Step by step solution

01

Understand the Divergence Theorem

The Divergence Theorem states that for a vector field \( \mathbf{F} \), the flux across a closed surface \( S \) is equal to the volume integral of the divergence of \( \mathbf{F} \) over the volume \( V \) enclosed by \( S \). Mathematically: \[ \int_{S} \mathbf{F} \cdot d\mathbf{A} = \int_{V} (abla \cdot \mathbf{F}) \, dV \].
02

Calculate the Divergence of the Constant Field

For a constant vector field \( \mathbf{F} = \mathbf{C} \), where \( \mathbf{C} \) is a constant vector, the divergence is zero. This is because the divergence of a constant vector field is: \( abla \cdot \mathbf{C} = 0 \).
03

Apply the Divergence Theorem to the Constant Field

Substitute the divergence of the constant field (which is zero) into the Divergence Theorem: \[ \int_{S} \mathbf{C} \cdot d\mathbf{A} = \int_{V} 0 \, dV \].
04

Evaluate the Integral

Evaluate the volume integral: \( \int_{V} 0 \, dV = 0 \). Hence, the outward flux \( \int_{S} \mathbf{C} \cdot d\mathbf{A} \) across the closed surface \( S \) is also zero.

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Key Concepts

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

Flux
Flux is a key concept in vector calculus. It represents the amount of a field passing through a surface. Imagine a stream of water. This stream can be thought of as vectors flowing through a net. The net is the surface, and the amount of water pouring through tells us the flux. Mathematically, if we have a vector field \( \mathbf{F} \), and a surface \( S \), the flux through \( S \) is given by \( \int_{S} \mathbf{F} \cdot d\mathbf{A} \),
  • \( \mathbf{F} \) is your vector field, any kind of directional flow.
  • \( d\mathbf{A} \) is a tiny piece of surface area on \( S \), with its own direction.
  • The dot product \( \mathbf{F} \cdot d\mathbf{A} \) measures the field passing perpendicularly through each piece.
Ultimately, flux helps us measure how much of something like fluid or force crosses a surface. It's particularly important for understanding field behaviors.
Constant Vector Field
A constant vector field is one where each vector, regardless of location, points in the same direction and has the same magnitude. Think of wind blowing in one constant direction. No matter where you measure the wind, it has the same directional and speed characteristics. In mathematical terms, it's denoted as \( \mathbf{F} = \mathbf{C} \), where \( \mathbf{C} \) is a constant vector. This means,
  • The divergence, which measures how much a vector field spreads out from a point, is zero for constant fields. Essentially, the field doesn't change as you move around.
  • For closed surfaces, this means the net flux is zero, as there are no changes in the vector field within the surface.
Understanding constant vector fields is crucial in applying the Divergence Theorem and predicting behaviors like field consistency.
Vector Calculus
Vector calculus is a branch of mathematics focusing on vector fields. These fields can describe various natural phenomena like electricity, gravity, and fluid dynamics. In vector calculus, we're interested in phenomena like how fields move (flux) and how they accumulate or dissipate (divergence and curl). Core operations in vector calculus include:
  • Gradient - Measures how a field changes at a point. Think of it as finding the steepness at any location.
  • Divergence - Shows how much a field expands from a point, helping identify sources or sinks.
  • Curl - Describes rotation of the field, indicating spin or vortex strength.
Vector calculus links to the Divergence Theorem, providing tools for evaluating integrals and effective field behaviors over spaces.
Closed Surface
A closed surface contains a 3D volume with no gaps or edges. Think of a balloon. It's entirely sealed, with no openings. In mathematics, closed surfaces are significant since they allow us to use the Divergence Theorem. Characteristics of closed surfaces:
  • They enclose a volume: Everything inside is entirely surrounded.
  • They have no boundary, making them continuous like spheres.
  • Flux through them can be evaluated using the Divergence Theorem, which calculates net flows based solely on inside changes.
Closed surfaces are prominent in physics for describing systems where field interactions and flux computations are analyzed over enclosed volumes.

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Most popular questions from this chapter

In Exercises \(47-52,\) use a CAS to perform the following steps for finding the work done by force \(\mathbf{F}\) over the given path: a. Find \(d \mathbf{r}\) for the path \(\mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+k(t) \mathbf{k}\) b. Evaluate the force \(\mathbf{F}\) along the path. c. Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) $$ \begin{array}{l}{\mathbf{F}=(y+y z \cos x y z) \mathbf{i}+\left(x^{2}+x z \cos x y z\right) \mathbf{j}+} \\ {(z+x y \cos x y z) \mathbf{k} ; \quad \mathbf{r}(t)=(2 \cos t) \mathbf{i}+(3 \sin t) \mathbf{j}+\mathbf{k}} \\ {0 \leq t \leq 2 \pi}\end{array} $$

In Exercises \(35-44,\) use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the given direction. Cone frustum \(\mathbf{F}=-x \mathbf{i}-y \mathbf{j}+z^{2} \mathbf{k}\) outward (normal away from the \(z\) -axis ) through the portion of the cone \(z=\sqrt{x^{2}+y^{2}}\) between the planes \(z=1\) and \(z=2\)

Centroid Find the centroid of the portion of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) that lies in the first octant.

In Exercises \(47-52,\) use a CAS to perform the following steps for finding the work done by force \(\mathbf{F}\) over the given path: a. Find \(d \mathbf{r}\) for the path \(\mathbf{r}(t)=g(t) \mathbf{i}+h(t) \mathbf{j}+k(t) \mathbf{k}\) b. Evaluate the force \(\mathbf{F}\) along the path. c. Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) $$ \begin{array}{l}{\mathbf{F}=(2 y+\sin x) \mathbf{i}+\left(z^{2}+(1 / 3) \cos y\right) \mathbf{j}+x^{4} \mathbf{k}} \\ {\mathbf{r}(t)=(\sin t) \mathbf{i}+(\cos t) \mathbf{j}+(\sin 2 t) \mathbf{k}, \quad-\pi / 2 \leq t \leq \pi / 2}\end{array} $$

Spherical shells a. Find the moment of inertia about a diameter of a thin spherical shell of radius \(a\) and constant density \(\delta .\) (Work with a hemispherical shell and double the result.) b. Use the Parallel Axis Theorem (Exercises 15.5\()\) and the result in part (a) to find the moment of inertia about a line tangent to the shell.
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