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Chapter 10: Problem 3

Let \(S\) be the silo-shaped closed surface consisting of: \- the cylinder \(x^{2}+y^{2}=4,0 \leq z \leq 3\), \- a hemispherical cap of radius 2 centered at (0,0,3) , i.e., \(x^{2}+y^{2}+(z-3)^{2}=4, z \geq 3\), and the disk \(x^{2}+y^{2} \leq 4, z=0,\) at the base, all oriented by the normal pointing away from the silo. Find \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) if \(\mathbf{F}(x, y, z)=\) \(\left(3 x z^{2}, 2 y, x+y^{2}-z^{3}\right)\)

Short Answer

Expert verified
The value of the surface integral is \(\frac{104\pi}{3}\).

Step by step solution

01

Understand the surface S

The surface S consists of three parts: the cylindrical part defined by \(x^2 + y^2 = 4, 0 \leq z \leq 3\), a hemispherical cap given by \(x^2 + y^2 + (z-3)^2 = 4, z \geq 3\), and a circular disk at the base \(x^2+y^2 \leq 4, z=0\). The surface is oriented such that the outward normal always points away from the center.
02

Verify the application of Gauss' Divergence Theorem

Gauss' Divergence Theorem states that \( \iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V abla \cdot \mathbf{F} \; dV \), where V is the volume bounded by S. First, check if \(\mathbf{F}\) is well-behaved: it's a polynomial vector field, so it satisfies necessary conditions across all space.
03

Calculate the divergence of F

Calculate \(abla \cdot \mathbf{F}\) by taking the partial derivatives: \[ abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(3xz^2) + \frac{\partial}{\partial y}(2y) + \frac{\partial}{\partial z}(x+y^2-z^3) \]. Compute the derivatives: \(3z^2 + 2 - 3z^2 = 2\), hence \(abla \cdot \mathbf{F} = 2\).
04

Setup and perform the volume integral over V

Since \(abla \cdot \mathbf{F} = 2\), integrate over volume V, which is the silo:\[ \iiint_V abla \cdot \mathbf{F} \, dV = \iiint_V 2 \, dV = 2 \times \text{Vol}(V) \]. Calculate Vol(V): base is the cylinder with height 3 and cap is a hemisphere of radius 2.
05

Calculate the volume of V

Volume of the cylinder is \( \pi \times 4 \times 3 = 12\pi\). Volume of the hemisphere is half the volume of a sphere: \(\frac{1}{2} \times \frac{4}{3}\pi (2)^3 = \frac{16 \pi}{3}\). Total volume \(= 12\pi + \frac{16\pi}{3} = \frac{52\pi}{3} \).
06

Calculate the surface integral

From the divergence theorem, we have:\[ \iint_S \mathbf{F} \cdot d \mathbf{S} = 2 \times \text{Vol}(V) = 2 \times \frac{52\pi}{3} = \frac{104\pi}{3} \].

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Key Concepts

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

Surface Integrals
Surface integrals help us understand how a vector field interacts with a surface. In this exercise, the surface integral \( \iint_{S} \mathbf{F} \cdot d \mathbf{S} \) represents the flow of the vector field \( \mathbf{F} \) through the surface \( S \). Surface \( S \) includes:
  • A cylindrical side \( x^2 + y^2 = 4, 0 \leq z \leq 3 \)
  • A hemispherical top with center (0,0,3) and radius 2
  • A flat disk base \( x^2+y^2 \leq 4, z=0 \)
The direction of the surface normal is crucial. It should point outward from the surface. To solve the integral, apply the Divergence Theorem, which converts surface integrals into volume integrals.
Vector Fields
A vector field like \( \mathbf{F}(x, y, z) = (3 x z^{2}, 2 y, x+y^{2}-z^{3}) \) assigns a vector to every point in space. This exercise uses the properties of such a vector field, addressing its divergence. The function's components:
  • \( 3 x z^2 \) in the \( x \)-direction
  • \( 2 y \) in the \( y \)-direction
  • \( x+y^2-z^3 \) in the \( z \)-direction
The polynomial form ensures that the vector field is smooth and behaves well everywhere. Understanding vector fields allows for deeper insights into phenomena like fluid flow and electromagnetic fields.
Volume Calculations
For the Divergence Theorem, it's necessary to calculate the volume \( V \) enclosed by \( S \). This volume includes the cylindrical portion and the hemispherical cap at the top.To find the volume:
  • Calculate cylinder volume: Area of base \( \times \) height = \( \pi \times 4 \times 3 = 12\pi \)
  • Calculate hemisphere volume: Half the volume of a full sphere with radius 2, \( \frac{1}{2} \times \frac{4}{3}\pi (2)^3 = \frac{16\pi}{3} \)
  • Total Volume = \( 12\pi + \frac{16\pi}{3} = \frac{52\pi}{3} \)
This computation is fundamental to solving the surface integral using the Divergence Theorem.
Partial Derivatives
The concept of partial derivatives is key when working with vector fields like \( \mathbf{F}(x, y, z) = (3 x z^{2}, 2 y, x+y^{2}-z^{3}) \). These derivatives measure changes in \( \mathbf{F} \) with respect to each variable, holding others constant. To find the divergence \( abla \cdot \mathbf{F} \):
  • \( \frac{\partial}{\partial x}(3xz^2) = 3z^2 \)
  • \( \frac{\partial}{\partial y}(2y) = 2 \)
  • \( \frac{\partial}{\partial z}(x+y^2-z^3) = -3z^2 \)
Summing these gives \( abla \cdot \mathbf{F} = 2 \), simplifying the volume integral calculation. Understanding partial derivatives also empowers you to grasp how functions change in multidimensional spaces.

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Most popular questions from this chapter

Find \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) if \(\mathbf{F}(x, y, z)=\left(x^{3} y^{3} z^{4}, x^{2} y^{4} z^{4}, x^{2} y^{3} z^{5}\right)\) and \(S\) is the boundary of the cube \(W=[0,1] \times[0,1] \times[0,1],\) oriented by the normal pointing away from \(W\)

Determine whether the given vector field \(\mathbf{F}\) is a curl field. If it is, find a vector field \(\mathbf{G}\) whose curl is \(\mathbf{F}\). If not, explain why not. $$ \mathbf{F}(x, y, z)=(y, z, x) $$

Let \(\mathbf{F}\) be a vector field defined on all of \(\mathbb{R}^{3},\) except at the two points \(\mathbf{p}=(2,0,0)\) and \(\mathbf{q}=(-2,0,0) .\) Let \(S_{1}, S_{2},\) and \(S\) be the following spheres, centered at \((2,0,0),(-2,0,0),\) and (0,0,0) , respectively, each oriented by the outward normal. $$ \begin{array}{lc} S_{1}: & (x-2)^{2}+y^{2}+z^{2}=1 \\ S_{2}: & (x+2)^{2}+y^{2}+z^{2}=1 \\ S: & x^{2}+y^{2}+z^{2}=25 \end{array} $$ Assume that \(\nabla \cdot \mathbf{F}=0 .\) If \(\iint_{S_{1}} \mathbf{F} \cdot d \mathbf{S}=5\) and \(\iint_{S_{2}} \mathbf{F} \cdot d \mathbf{S}=6,\) what is \(\iint_{S} \mathbf{F} \cdot d \mathbf{S} ?\)

Consider the vector field \(\mathbf{F}(x, y, z)=\left(e^{x+y}-x e^{y+z}, e^{y+z}-e^{x+y}+y e^{z},-e^{z}\right)\) (a) Is \(\mathbf{F}\) a conservative vector field? Explain. (b) Find a vector field \(\mathbf{G}=\left(G_{1}, G_{2}, G_{3}\right)\) such that \(G_{2}=0\) and the curl of \(\mathbf{G}\) is \(\mathbf{F}\). (c) Find \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) if \(S\) is the hemisphere \(x^{2}+y^{2}+z^{2}=4, z \geq 0,\) oriented by the outward normal. (d) Find \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) if \(S\) is the hemisphere \(x^{2}+y^{2}+z^{2}=4, z \leq 0,\) oriented by the outward normal. (e) Find \(\iint_{S} \mathbf{F} \cdot d \mathbf{S}\) if \(S\) is the cylinder \(x^{2}+y^{2}=4,0 \leq z \leq 4,\) oriented by the outward normal.

Determine whether the given vector field \(\mathbf{F}\) is a curl field. If it is, find a vector field \(\mathbf{G}\) whose curl is \(\mathbf{F}\). If not, explain why not. $$ \mathbf{F}(x, y, z)=\left(x+2 y+3 z, x^{4}+y^{5}+z^{6}, x^{7} y^{8} z^{9}\right) $$
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