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Magazine Chapter 5: Problem 9
For the following exercises, use Stokes' theorem to evaluate \(\iint_{S}(\operatorname{cur} l \mathbf{F} \cdot \mathbf{N}) d S\) for the vector fields and surface.\(\mathbf{F}(x, y, z)=4 y \mathbf{i}+z \mathbf{j}+2 y \mathbf{k}\) and \(C\) is the intersection of sphere \(x^{2}+y^{2}+z^{2}=4\) with plane \(z=0\), and using the outward normal vector
Short Answer
Expert verified
The integral evaluates to \(-16\pi\).
Step by step solution
01
Understanding Stokes' Theorem
Stokes' Theorem relates the surface integral of a curl of a vector field over a surface, \( S \), to a line integral of the vector field over the boundary \( C \) of the surface: \( \iint_{S}(abla \times \mathbf{F}) \cdot \mathbf{N} \, dS = \oint_{C}\mathbf{F} \cdot \mathbf{T} \, ds \), where \( \mathbf{N} \) is the outward normal to \( S \) and \( \mathbf{T} \) is the tangent vector along \( C \).
02
Compute the Curl of \( \mathbf{F} \)
Determine the curl of the vector field \( \mathbf{F}(x, y, z) = 4y \mathbf{i} + z \mathbf{j} + 2y \mathbf{k} \). Use the definition of the curl: \( abla \times \mathbf{F} = \left( \frac{\partial}{\partial y} 2y - \frac{\partial}{\partial z} z \right)\mathbf{i} - \left( \frac{\partial}{\partial x} 2y - \frac{\partial}{\partial z} 4y \right)\mathbf{j} + \left( \frac{\partial}{\partial x} z - \frac{\partial}{\partial y} 4y \right)\mathbf{k} \). The calculation yields \( -1 \mathbf{i} + 0 \mathbf{j} - 4 \mathbf{k} \).
03
Identify the Surface and Its Boundary
The surface \( S \) is part of the sphere \( x^2 + y^2 + z^2 = 4 \) lying above the plane \( z = 0 \). Its boundary \( C \) is a circle in the \( xy \)-plane with \( z = 0 \) at the radius \( r = 2 \). The path of \( C \) is a circle of radius 2 centered at the origin in the plane.
04
Parameterize the Circle \( C \)
Parameterize \( C \) by \( \mathbf{r}(t) = 2\cos(t)\mathbf{i} + 2\sin(t)\mathbf{j} \) where \( t \) ranges from 0 to \( 2\pi \). Calculate \( \mathbf{T} = \frac{d\mathbf{r}}{dt} = -2\sin(t)\mathbf{i} + 2\cos(t)\mathbf{j} \).
05
Set up the Line Integral
Using Stokes' Theorem, evaluate the line integral \( \oint_{C}\mathbf{F} \cdot \mathbf{T} \, ds \). Integrate from 0 to \( 2\pi \): \( \int_{0}^{2\pi} \mathbf{F}(\mathbf{r}(t)) \cdot \mathbf{T}(t) \, dt \).
06
Evaluate \( \mathbf{F} \cdot \mathbf{T} \)
Substitute \( \mathbf{r}(t) \) into \( \mathbf{F} \): \( \mathbf{F}(\mathbf{r}(t)) = 8\sin(t)\mathbf{i} \).\( \mathbf{F} \cdot \mathbf{T} = (8\sin(t)\mathbf{i}) \cdot (-2\sin(t)\mathbf{i} + 2\cos(t)\mathbf{j}) = -16\sin^2(t) + 0 \).
07
Compute the Integral
Evaluate the integral \( \int_{0}^{2\pi} -16\sin^2(t) \, dt \). Use the identity \( \sin^2(t) = \frac{1 - \cos(2t)}{2} \) to rewrite the integral: \( -16 \int_{0}^{2\pi} \frac{1 - \cos(2t)}{2} \, dt = -16 \left( \pi - 0 \right) = -16\pi \). Thus, the integral evaluates to \( -16\pi \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Fields
A vector field is a function that assigns a vector to each point in space. Imagine it as a set of arrows pointing in various directions, and each has a magnitude (or length) associated with it. These vectors can represent many things, like the direction of water flow in a river or the strength of a magnetic field in space.
The vector field given in the problem is \(\mathbf{F}(x, y, z)=4y \mathbf{i} + z \mathbf{j} + 2y \mathbf{k}\). This field assigns a vector with components that depend on the coordinates \(x\), \(y\), and \(z\). In simple terms, at each point in the space \((x,y,z)\), you can determine a specific direction and intensity of the vector based on this equation.
The vector field given in the problem is \(\mathbf{F}(x, y, z)=4y \mathbf{i} + z \mathbf{j} + 2y \mathbf{k}\). This field assigns a vector with components that depend on the coordinates \(x\), \(y\), and \(z\). In simple terms, at each point in the space \((x,y,z)\), you can determine a specific direction and intensity of the vector based on this equation.
- The \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) components refer to the standard unit vectors along the x, y, and z-axes respectively.
- \(4y \mathbf{i}\) implies that in the x-direction, the strength is proportional to \(y\).
- The \(z \mathbf{j}\) component affects only the y-direction based on the height above the xy-plane.
- \(2y \mathbf{k}\) illustrates influence along the z-axis, which increases by twice the value of \(y\).
Line Integral
A line integral allows us to integrate vector field functions along a curve or path. Unlike regular integration, which is over an interval, a line integral is evaluated along a specific path in space.
In our exercise, the boundary of the surface \( C \) is a circle, which makes it convenient to compute a line integral. We use parameterization to represent this path. By changing the parameter \( t \), we move along the path of the circle.
In our exercise, the boundary of the surface \( C \) is a circle, which makes it convenient to compute a line integral. We use parameterization to represent this path. By changing the parameter \( t \), we move along the path of the circle.
- The parameterization \( \mathbf{r}(t) = 2\cos(t) \mathbf{i} + 2\sin(t) \mathbf{j} \) is used, where \( t \) varies from 0 to \( 2\pi \).
- The result of this parameterization is the path traversing the boundary of our surface, allowing the vector field to act upon it.
- \( \mathbf{T} = \frac{d\mathbf{r}}{dt} = -2\sin(t)\mathbf{i} + 2\cos(t)\mathbf{j} \) is the tangent vector needed for the integral calculation.
- By evaluating the line integral \( \oint_{C}\mathbf{F} \cdot \mathbf{T} \, ds \), we measure the total impact of the vector field along the circle.
Curl of a Vector Field
The curl of a vector field gives us an idea of the rotation of the field at a point. It measures how much and in what direction the vector field rotates around a point.
The curl \( abla \times \mathbf{F} \) is a new vector field that describes the rotational tendency or swirling strength of the original field \( \mathbf{F} \). For our exercise, the vector field \( \mathbf{F}(x, y, z) = 4y \mathbf{i} + z \mathbf{j} + 2y \mathbf{k} \) has a curl calculated as:
The curl \( abla \times \mathbf{F} \) is a new vector field that describes the rotational tendency or swirling strength of the original field \( \mathbf{F} \). For our exercise, the vector field \( \mathbf{F}(x, y, z) = 4y \mathbf{i} + z \mathbf{j} + 2y \mathbf{k} \) has a curl calculated as:
- \( abla \times \mathbf{F} = \left( \frac{\partial}{\partial y} 2y - \frac{\partial}{\partial z} z \right)\mathbf{i} - \left( \frac{\partial}{\partial x} 2y - \frac{\partial}{\partial z} 4y \right)\mathbf{j} + \left( \frac{\partial}{\partial x} z - \frac{\partial}{\partial y} 4y \right)\mathbf{k} \)
- After performing calculations, we find the curl is \(-1 \mathbf{i} + 0 \mathbf{j} - 4 \mathbf{k} \).
- This means there is a rotation in the negative x-direction and negative z-direction in the space where the field is applied.
Surface Integral
A surface integral allows us to calculate a two-dimensional integral over a surface in space. It's an extension of line integrals into two dimensions.
In the context of the exercise, the surface integral \( \iint_{S}(abla \times \mathbf{F}) \cdot \mathbf{N} \, dS \) evaluates how the curl of a vector field interacts over the specific surface. The surface \( S \) highlighted in the problem is part of a sphere's cap intersecting with the plane \( z=0 \).
In the context of the exercise, the surface integral \( \iint_{S}(abla \times \mathbf{F}) \cdot \mathbf{N} \, dS \) evaluates how the curl of a vector field interacts over the specific surface. The surface \( S \) highlighted in the problem is part of a sphere's cap intersecting with the plane \( z=0 \).
- \( \mathbf{N} \) is the outward normal vector to each point on the surface \( S \).
- Stokes' Theorem states that this surface integral will equal the line integral over the boundary curve \( C \).
- Thus, computing the surface integral tells us the net rotation through the surface \( S \), while the line integral captures circulation along the surface's boundary.
