Use the divergence theorem to compute the flux integral P F·î dS where în is the outward pointing unit normal vector field to the surface F = (4x, 4y, 4z) and S is the surface of the unit ball x2 + y2 + z² s 1.
Arc Length
Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length.
Line Integral
A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.
Triple Integral
Examples:
![**Problem Statement:**
Use the divergence theorem to compute the flux integral
\[
\iint_S \vec{F} \cdot \hat{n} \, dS
\]
where \(\hat{n}\) is the outward pointing unit normal vector field to the surface \(S\).
\(\vec{F} = (4x, 4y, 4z)\) and \(S\) is the surface of the unit ball \(x^2 + y^2 + z^2 \leq 1\).
**Explanation:**
This problem involves calculating the flux of a vector field \(\vec{F}\) across a surface \(S\), which is the surface of a unit sphere. The vector field given is \(\vec{F} = (4x, 4y, 4z)\).
The divergence theorem, also known as Gauss's theorem, relates the flux of a vector field through a closed surface to the divergence of the field in the volume enclosed by the surface. The theorem is stated as:
\[
\iint_S \vec{F} \cdot \hat{n} \, dS = \iiint_V \nabla \cdot \vec{F} \, dV
\]
where \(V\) is the volume enclosed by the surface \(S\), and \(\nabla \cdot \vec{F}\) is the divergence of \(\vec{F}\).
In this problem:
- Surface \(S\) is the boundary of the unit ball \(x^2 + y^2 + z^2 = 1\).
- The vector field is \(\vec{F} = (4x, 4y, 4z)\).
To solve this, you will:
1. Compute the divergence \(\nabla \cdot \vec{F}\).
2. Compute the volume integral \(\iiint_V \nabla \cdot \vec{F} \, dV\) over the volume of the unit ball.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4ad20959-2831-40ff-b7b0-4d5e8b8575fb%2F70cfb877-180a-4b4a-8f7a-c36bd1f68ecd%2F0hjk2mo_processed.png&w=3840&q=75)
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