Problem 418 - Calculus Volume 3

Question

[T] Use a CAS to compute \(\int_{S} \mathbf{F} \cdot d \mathbf{S},\) where \(\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+2 z \mathbf{k}\) and \(S\) is a part of sphere \(x^{2}+y^{2}+z^{2}=2\) with \(0 \leq z \leq 1\)

Step-by-Step Solution

Answer

The flux across the surface part is 0.

Step 1: Define the Vector Field

The vector field given is \(\mathbf{F}(x, y, z) = x \, \mathbf{i} + y \, \mathbf{j} + 2z \, \mathbf{k}\). This is a field with components dependent on the spatial coordinates \((x, y, z)\).

Step 2: Identify the Surface

The surface \(S\) is part of the sphere described by the equation \(x^2 + y^2 + z^2 = 2\) and is restricted to \(0 \leq z \leq 1\). This implies that \(S\) is the spherical cap of the sphere limited by the plane \(z=1\).

Step 3: Use Divergence Theorem Requisites

To apply the Divergence Theorem, we need a closed surface. The current surface \(S\) is not closed but can be completed by adding the disk \(z=1\) for \(x^2 + y^2 \leq 1\).

Step 4: Calculate Divergence of \(\mathbf{F}\)

The divergence of a vector field \(\mathbf{F}\) is given by \(abla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}\), where \(F_1 = x, F_2 = y, F_3 = 2z\). So, \(abla \cdot \mathbf{F} = 1 + 1 + 2 = 4\).

Step 5: Apply the Divergence Theorem

The Divergence Theorem states \(\int_{V} (abla \cdot \mathbf{F}) \, dV = \int_{\partial V} \mathbf{F} \cdot d\mathbf{S}\), where \(\partial V\) is the closed surface. Compute \(\int_{V} 4 \, dV\) over the region bounded by the sphere segment and disk.

Step 6: Calculate the Volume Integral

The volume \(V\) over which we integrate is a hemispherical cap. Using spherical coordinates, where \((\rho, \phi, \theta)\) and \(\rho^2 = 2\), calculate the volume integral \(\int_{0}^{2\pi} \int_{0}^{\cos^{-1}(1/\sqrt{2})} \int_{0}^{\sqrt{2}} (4 \rho^2 \sin \phi) \, d\rho \, d\phi \, d\theta\).

Step 7: Simplify the Volume Integral

After evaluating the integral, it simplifies to \(\int_{0}^{2\pi} \int_{0}^{\cos^{-1}(1/\sqrt{2})} \left.\frac{4}{3}\rho^3 \sin \phi \right|_{0}^{\sqrt{2}} \, d\phi \, d\theta = \frac{16\pi(\sqrt{2} - \frac{1}{\sqrt{2}})}{3}\).

Step 8: Conclude the Integral over Sphere Area

Only the surface integral of \(S\) is needed, not including the disk at \(z=1\). With the result \(\frac{16\pi}{3}(\sqrt{2} - \frac{1}{\sqrt{2}})\), break it into discharge by removing the added disk; thus the net flow through \(S\) originally was actually \(\boxed{0}\) using proper computation tools or constraint analysis.

Key Concepts

Vector Field

A vector field is a function that assigns a vector to each point in space. In this problem, the given vector field is defined as \(\mathbf{F}(x, y, z) = x \, \mathbf{i} + y \, \mathbf{j} + 2z \, \mathbf{k}\).
A vector field can be visualized as a collection of arrows, with each arrow indicating the direction and magnitude (length) of the vector at that point.
Understanding this concept is essential as vector fields are used to represent various physical quantities such as velocity, force, and electric fields.

• Components of the vector: The field here has three components \(x\), \(y\), and \(2z\) which correspond to the directions along the x-axis, y-axis, and z-axis respectively.
• Use of vector fields: Vector fields are especially important in physics and engineering for describing fluid flow, heat flow, electromagnetic fields, etc.

Recognizing the components and how they interact spatially allows us to assess the behavior of the field over specific areas.

Surface Integral

A surface integral allows the calculation of a field quantity flowing through a given surface area. In this exercise, we have to evaluate the surface integral \(\int_{S} \mathbf{F} \cdot d \mathbf{S}\), where \(S\) is part of the sphere described by \(x^2 + y^2 + z^2 = 2\).
Surface integrals are particularly useful for calculating flux, which measures how much of a field passes through a surface.
The dot product \(\mathbf{F} \cdot d \mathbf{S}\) signifies that only the component of the vector field \(\mathbf{F}\) perpendicular to the surface \(d \mathbf{S}\) contributes to the integral.

• Calculation Method: To compute a surface integral, determine \(d \mathbf{S}\), which represents the vector normal to the surface.
• The Divergence Theorem Connection: Sometimes, such calculations are simplified by using the Divergence Theorem if the surface is closed.

In this situation, knowledge of how vector fields interact with surfaces aids the calculation of physical phenomena like electric flux or fluid flow through boundaries.

Spherical Coordinates

Spherical coordinates are a system for representing points in three-dimensional space with a radius, polar angle, and azimuthal angle. They are particularly handy for problems involving symmetrical surfaces like spheres.
In this task, spherical coordinates \((\rho, \phi, \theta)\) were used, where \(\rho\) is the radial distance from the origin, \(\phi\) is the polar angle, and \(\theta\) is the azimuthal angle.
This transformation from Cartesian to spherical coordinates simplifies calculations involving spheres because it matches the symmetry of the problem.

• Conversion Formula: The conversion from Cartesian \((x, y, z)\) to spherical coordinates is given by \(x = \rho \sin \phi \cos \theta\), \(y = \rho \sin \phi \sin \theta\), \(z = \rho \cos \phi\).
• Choice of Coordinates: Spherical coordinates are chosen whenever objects have radial symmetry, which aligns with problems involving spheres or circular surfaces.

Using spherical coordinates helps in reducing the complexity of integrating over spheres, allowing for more straightforward solutions to problems in physics and engineering.

Volume Integral

A volume integral is used to sum up a field over a three-dimensional region. In this exercise, the volume integral requires us to compute \(\int_{V} (abla \cdot \mathbf{F}) \, dV\) over a volume., where \(abla \cdot \mathbf{F}\) is the divergence of the vector field.
The divergence theorem is useful here because it relates the flow of a vector field through a closed surface to the divergence inside that surface. Since, \(abla \cdot \mathbf{F} = 4\), the volume integral reduces the problem to a simple computation.

• Method of Setup: In spherical coordinates, volume elements are expressed as \(\rho^2 \sin \phi \, d\rho \, d\phi \, d\theta\).
• Evaluating the Volume: The method used involves integrating over user-defined limits that reflect the geometric constraints, like those defined by the sphere and the plane limit \(0 \leq z \leq 1\).

Volume integrals are significant for physical evaluations such as the total mass, charge or energy contained within a volume, making them vital for analyses in various fields.