Problem 304 - Calculus Volume 3

Question

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

Step-by-Step Solution

Answer

The surface integral is 0.

Step 1: Understand the Problem

We are asked to compute the surface integral \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \, dS \), where \( \mathbf{F} = 2yz \mathbf{i} + (\tan^{-1} xz) \mathbf{j} + e^{xy} \mathbf{k} \) and \( \mathbf{N} \) is the outward normal vector to a sphere with radius 1, centered at the origin.

Step 2: Use the Divergence Theorem

According to the Divergence Theorem, \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \, dS = \iiint_{V} abla \cdot \mathbf{F} \, dV \), where \( V \) is the volume enclosed by \( S \). Here, \( S \) is the surface of the sphere \( x^2 + y^2 + z^2 = 1 \), so \( V \) is the interior of this sphere.

Step 3: Compute the Divergence of \( \mathbf{F} \)

Calculate the divergence \( abla \cdot \mathbf{F} = \frac{\partial}{\partial x}(2yz) + \frac{\partial}{\partial y}(\tan^{-1} xz) + \frac{\partial}{\partial z}(e^{xy}) \). The partial derivatives are: \( \frac{\partial}{\partial x}(2yz) = 0 \), \( \frac{\partial}{\partial y}(\tan^{-1} xz) = 0 \), and \( \frac{\partial}{\partial z}(e^{xy}) = 0 \), since these expressions do not depend on \( x \), \( y \), or \( z \) respectively. Therefore, \( abla \cdot \mathbf{F} = 0 \).

Step 4: Evaluate the Volume Integral

Since \( abla \cdot \mathbf{F} = 0 \), the volume integral \( \iiint_{V} abla \cdot \mathbf{F} \, dV = \iiint_{V} 0 \, dV = 0 \). This means that the surface integral \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \, dS = 0 \) as well.

Key Concepts

Divergence Theorem

The Divergence Theorem is a powerful tool in vector calculus that connects the flux of a vector field across a closed surface to the divergence of the vector field within the volume it encloses. This theorem is especially useful when dealing with complex three-dimensional shapes. It states:

• \( \iint_{S} \mathbf{F} \cdot \mathbf{N} \, dS = \iiint_{V} abla \cdot \mathbf{F} \, dV \)

Here, \( S \) represents a closed surface, \( \mathbf{F} \) a vector field, \( \mathbf{N} \) the outward normal vector to the surface, and \( V \) the volume enclosed by \( S \).

This theorem allows us to transform a difficult surface integral into a potentially simpler volume integral over the region enclosed by the surface. It highlights the relationship between outward flux and local flow trends indicated by the divergence.

The transformation is particularly advantageous when the divergence is relatively simpler to compute or when simplifying assumptions about the symmetry of the field can be applied.

Sphere

A sphere is a three-dimensional, perfectly symmetrical shape, where every point on the surface is equidistant from its center. The sphere used in our exercise is defined by the equation:

• \( x^2 + y^2 + z^2 = 1 \)

This sphere has a radius of 1 and is centered at the origin of the coordinate system.

Due to its inherent symmetry, a sphere often simplifies the calculation of surface integrals, as different components cancel out due to symmetry across different axes. When considering a vector field over such a surface, like in our exercise, each perpendicular normal vector points directly outward from the center.

Understanding the properties of spheres is crucial when applying the Divergence Theorem. This symmetrical shape simplifies not only the calculation of surface integrals but also the volume integrals due to a consistent relationship between surface and volume measurements.

Vector Fields

Vector fields represent complex data where each point in space is associated with a vector, describing a range of phenomena such as electromagnetic fields or fluid flow. In our scenario, the vector field \( \mathbf{F}(x, y, z)=2yz \mathbf{i}+(\tan^{-1} xz) \mathbf{j}+e^{xy} \mathbf{k} \) is defined in three dimensions.

Each component of the vector field corresponds to a force or velocity in a particular direction:

• \( 2yz \mathbf{i} \)
• \( (\tan^{-1} xz) \mathbf{j} \)
• \( e^{xy} \mathbf{k} \)

These influences are the driving factor in determining the behavior of the field over any surface or volume.

Analyzing vector fields necessitates considering the interaction between different components, how they behave under transformations such as divergence, and how they contribute to an integral, especially over symmetrical surfaces like spheres.

Understanding vector fields leads to a better grasp of how the motion or field strength changes through space and how these changes can be leveraged using the Divergence Theorem.

Divergence of a Vector Field

Divergence is a measure of the "outflowing-ness" of a vector field at a particular point and is integral in applying the Divergence Theorem. Mathematically, it's represented as \( abla \cdot \mathbf{F} \), where \( \mathbf{F} \) is the vector field.

Divergence can help determine whether a vector field is a source or sink at a particular location, or if it's perfectly balanced (zero divergence). In our exercise, computing the divergence of \( \mathbf{F} \) involves calculating partial derivatives for each component of the vector:

• The partial derivative with respect to \( x \) for \( 2yz \) equals 0.
• The partial derivative with respect to \( y \) for \( \tan^{-1} xz \) equals 0.
• The partial derivative with respect to \( z \) for \( e^{xy} \) equals 0.

Since each derivative is zero, the overall divergence of \( \mathbf{F} \) is zero.

A zero divergence indicates that there is no net flow out of or into the volume enclosed by the sphere, simplifying the surface integral to zero. Understanding this property helps streamline evaluations using the Divergence Theorem in vector calculus.