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Chapter 5: Problem 38

Compute \(\iint_{S} \mathbf{F} \cdot \mathbf{N} d S\), where \(\mathbf{F}(x, y, z)=2 y z \mathbf{i}+\left(\tan ^{-1} x z\right) \mathbf{j}+e^{x y} \mathbf{k}\) and \(\mathbf{N}\) is an outward normal vector \(S\), where S is the surface of sphere \(x^{2}+y^{2}+z^{2}=1\).

Short Answer

Expert verified
The surface integral is 0.

Step by step solution

01

Understand the Surface

The surface S is defined as the sphere given by the equation \( x^2 + y^2 + z^2 = 1 \). We identify this as a closed surface, specifically the unit sphere.
02

Vector Field Analysis

Given vector field is \( \mathbf{F}(x, y, z) = 2yz \mathbf{i} + \tan^{-1}(xz) \mathbf{j} + e^{xy} \mathbf{k} \). This vector field needs to be integrated over the surface of the sphere.
03

Apply Divergence Theorem

According to the Divergence Theorem, for a closed surface \( S \), \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \, d S = \iiint_{V} abla \cdot \mathbf{F} \, dV \) where \( V \) is the volume inside \( S \). Calculate \( abla \cdot \mathbf{F} \).
04

Compute the Divergence

The divergence \( abla \cdot \mathbf{F} \) is the dot product of the nabla operator and \( \mathbf{F} \): \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(2yz) + \frac{\partial}{\partial y}(\tan^{-1}(xz)) + \frac{\partial}{\partial z}(e^{xy}) \).
05

Evaluate Divergence Terms Individually

Calculate each term separately:- \( \frac{\partial}{\partial x}(2yz) = 0 \).- \( \frac{\partial}{\partial y}(\tan^{-1}(xz)) = 0 \).- \( \frac{\partial}{\partial z}(e^{xy}) = 0 \).Thus, \( abla \cdot \mathbf{F} = 0 \).
06

Integrate Divergence Over Volume

Since \( abla \cdot \mathbf{F} = 0 \), the volume integral \( \iiint_{V} 0 \, dV = 0 \). This implies \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \, d S = 0 \).

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Key Concepts

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

Closed Surface
A closed surface is a geometric shell that completely encloses a volume. It has no openings or boundaries. Think of it as similar to a balloon, which neatly encloses the air inside it.
One important characteristic of a closed surface is that what goes in must come out, influencing field calculations and flow analysis.
  • Closed surfaces include spheres, cubes, and any shape you can imagine that fully encapsulates a space.
  • When dealing with a closed surface, we can make use of the Divergence Theorem, which connects the surface integral of a vector field over a closed surface with a corresponding volume integral inside the surface.
In our specific case, the surface in question is the unit sphere. When we calculate integrals over a closed surface, the normal vectors point in the outward direction, which helps ensure consistency with the Divergence Theorem.
Vector Field
A vector field assigns a vector to every point in space, much like assigning a direction and strength of a flowing river at each point along its path.
In mathematical terms, a vector field is represented by a function like \( \mathbf{F}(x, y, z) \). This function assigns vectors based on the values of \( x, y, \) and \( z \).
  • For instance, in our exercise, we have the vector field \( \mathbf{F}(x, y, z) = 2yz \mathbf{i} + \tan^{-1}(xz) \mathbf{j} + e^{xy} \mathbf{k} \).
  • This equation specifies a different vector for every point \( (x, y, z) \) in three-dimensional space.
Using vector fields, we can understand and visualize factors like forces or flows. In calculations, knowing the vector field allows us to perform integrations over surfaces, helping to determine how much of the vector field passes through the surface.
Unit Sphere
A unit sphere is a sphere with a radius of 1. It's a simple, perfectly round shape fundamental in geometry and is used frequently in mathematical exercises as an ideal closed surface.
The equation \( x^2 + y^2 + z^2 = 1 \) defines this unit sphere. Any point \((x, y, z)\) that satisfies this equation is on the surface of the unit sphere.
  • The unit sphere is particularly convenient for calculations due to its symmetry and simplicity.
  • It is utilized in problems involving integration because it makes applying the Divergence Theorem straightforward.
For exercises like the one at hand, the unit sphere provides the surface over which the vector field is integrated, allowing you to determine various properties of the vector field with respect to this spherical surface.

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Most popular questions from this chapter

[T Evaluate \(\iint_{S}(x z / y) d S\), where \(S\) is the portion of cylinder \(x=y^{2}\) that lies in the first octant between planes \(z=0, z=5, y=1\), and \(y=4\)

Use the divergence theorem to evaluate \(\iint_{S} \mathbf{F} \cdot d S .\)Use the divergence theorem to compute the value of the flux integral over the unit sphere with \(\mathbf{F}(x, y, z)=3 z \mathbf{i}+2 y \mathbf{j}+2 x \mathbf{k}\)

Find the divergence of \(\mathrm{F}\) at the given point. $$ \mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k} \text { at }(1,2,3) $$

For the following exercises, express the surface integral as an iterated double integral by using a projection on \(S\) on the \(x z\) -plane $$ \iint_{S} x y^{2} z^{3} d \mathrm{~S} ; \mathrm{S} \text { is the first-octant portion of plane } 2 x+3 y+4 z=12 \text { . } $$

For the following exercises, find the gradient vector field of each function f. $$ f(x, y, z)=z e^{-x y} $$
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