Problem 17
Question
Assume that \(S\) forms the boundary of a closed and bounded region \(D\). If a is a constant vector, show that \(\iint_{S}(\mathbf{a} \cdot \mathbf{n}) d S=0\).
Step-by-Step Solution
Verified Answer
The integral \( \iint_{S} (\mathbf{a} \cdot \mathbf{n}) \, dS = 0 \) by the divergence theorem, since the divergence of a constant vector field is zero.
1Step 1: Define the Problem
We are given a surface integral \( \iint_{S}(\mathbf{a} \cdot \mathbf{n}) \, dS = 0 \) where \( \mathbf{a} \) is a constant vector and \( \mathbf{n} \) is the outward unit normal to the surface \( S \). Our task is to show that this integral results in zero.
2Step 2: Apply the Divergence Theorem
The Divergence Theorem relates a surface integral over a closed surface \( S \) to a volume integral over the region \( D \) it encloses. For a vector field \( \mathbf{F} \), the theorem states: \[ \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{D} abla \cdot \mathbf{F} \, dV \]. In our case, set \( \mathbf{F} = \mathbf{a} \), a constant vector.
3Step 3: Compute the Divergence
The divergence of a constant vector field \( \mathbf{a} \) is zero, i.e., \( abla \cdot \mathbf{a} = 0 \). This is because the components of \( \mathbf{a} \) are constant and do not change with respect to position.
4Step 4: Evaluate the Volume Integral
Since \( abla \cdot \mathbf{a} = 0 \), the volume integral over the region \( D \) is \[ \iiint_{D} 0 \, dV = 0 \].
5Step 5: Conclude Using the Divergence Theorem
According to the divergence theorem, the integral over the surface \( S \) is equal to the integral over the volume \( D \): \( \iint_{S}(\mathbf{a} \cdot \mathbf{n}) \, dS = \iiint_{D} 0 \, dV \). Therefore, \( \iint_{S}(\mathbf{a} \cdot \mathbf{n}) \, dS = 0 \).
Key Concepts
Understanding Surface IntegralsExploring the Divergence TheoremUnderstanding Constant Vectors
Understanding Surface Integrals
Surface integrals are a type of integral used in vector calculus. They help calculate the flow of a vector field across a surface. Imagine water pouring over the top of a hill; a surface integral can tell us how much water actually crosses over the "surface" of that hill.
A surface integral of a vector field is defined over a surface, and it incorporates two main components:
This integral calculates the total "flow" of the field across the surface. When \(\mathbf{n}\) and \(\mathbf{F}\) are perpendicular, their dot product is zero, indicating no flow across that portion of the surface.
In our original exercise, the constant vector \(\mathbf{a}\) plays the role of the vector field \(\mathbf{F}\), and \(\mathbf{n}\) is the normal vector that varies over the surface \(S\).
A surface integral of a vector field is defined over a surface, and it incorporates two main components:
- The vector field, which gives a magnitude and direction at each point
- The unit normal vector, which is perpendicular to the surface at each point
This integral calculates the total "flow" of the field across the surface. When \(\mathbf{n}\) and \(\mathbf{F}\) are perpendicular, their dot product is zero, indicating no flow across that portion of the surface.
In our original exercise, the constant vector \(\mathbf{a}\) plays the role of the vector field \(\mathbf{F}\), and \(\mathbf{n}\) is the normal vector that varies over the surface \(S\).
Exploring the Divergence Theorem
The Divergence Theorem is a key principle in vector calculus that connects the flow of a vector field through a closed surface to behavior within the volume it encloses.
It is often stated as:
\[ \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{D} abla \cdot \mathbf{F} \, dV \]
Where:
The right side is a volume integral over \(D\), highlighting how the field expands or contracts within the region.
In the case of our exercise, by substituting \(\mathbf{F}\) with the constant vector \(\mathbf{a}\), we find the divergence (\(abla \cdot \mathbf{a}\)) is zero. This means there is no expansion or contraction within volume \(D\), leading to a zero result for the surface integral.
It is often stated as:
\[ \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{D} abla \cdot \mathbf{F} \, dV \]
Where:
- \(S\) is a closed surface
- \(D\) is the volume enclosed by \(S\)
- \(\mathbf{F}\) is a vector field
- \(abla \cdot \mathbf{F}\) is the divergence of \(\mathbf{F}\)
The right side is a volume integral over \(D\), highlighting how the field expands or contracts within the region.
In the case of our exercise, by substituting \(\mathbf{F}\) with the constant vector \(\mathbf{a}\), we find the divergence (\(abla \cdot \mathbf{a}\)) is zero. This means there is no expansion or contraction within volume \(D\), leading to a zero result for the surface integral.
Understanding Constant Vectors
A constant vector is a vector that has a fixed magnitude and direction, meaning its components do not change based on location or other variables. In simpler terms, it's like a steady wind that blows in the same direction with the same strength everywhere.
Constant vectors simplify many calculations in vector calculus because their derivatives are zero. This property is why the divergence of a constant vector, \(\mathbf{a}\), is zero. Since divergence measures the extent to which a vector field spreads out from a certain point, a constant vector has no spreading or curving behavior.
In the original exercise, the constant vector \(\mathbf{a}\) plays a crucial role. Because its divergence is \(0\), we can apply the Divergence Theorem to reach the solution quickly. It's as if the vector field \(\mathbf{a}\) is "neutral" in the region it encloses, leading to the surface integral resolving to zero.
Constant vectors simplify many calculations in vector calculus because their derivatives are zero. This property is why the divergence of a constant vector, \(\mathbf{a}\), is zero. Since divergence measures the extent to which a vector field spreads out from a certain point, a constant vector has no spreading or curving behavior.
In the original exercise, the constant vector \(\mathbf{a}\) plays a crucial role. Because its divergence is \(0\), we can apply the Divergence Theorem to reach the solution quickly. It's as if the vector field \(\mathbf{a}\) is "neutral" in the region it encloses, leading to the surface integral resolving to zero.
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