Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 16: Problem 14
Verify that Stokes' Theorem is true for the given vector field \(\mathbf{F}\) and surface \(S .\) \(\mathbf{F}(x, y, z)=-2 y z \mathbf{i}+y \mathbf{j}+3 x \mathbf{k}\) \(S\) is the part of the paraboloid \(z=5-x^{2}-y^{2}\) that lies above the plane \(z=1,\) oriented upward
Short Answer
Expert verified
Stokes' Theorem holds for \(\mathbf{F}\) and \(S\); both integrals equal 6\pi.
Step by step solution
01
Understand Stokes' Theorem
Stokes' Theorem relates the surface integral of the curl of a vector field over a surface \( S \) to the line integral of the vector field along the boundary \( C \) of the surface. Mathematically, it is expressed as: \( \int_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r} \). Here, \( \mathbf{n} \) is the outward-pointing unit normal vector to the surface \( S \). We need to verify this theorem for the given vector field \( \mathbf{F} \) and surface \( S \).
02
Compute the Surface Description
The surface \( S \) is a part of the paraboloid \( z = 5 - x^2 - y^2 \) which is above the plane \( z = 1 \). This means \( 1 \leq z \leq 5 - x^2 - y^2 \), so we have \( x^2 + y^2 \leq 4 \). The boundary \( C \) of \( S \) is the circle \( x^2 + y^2 = 4 \) in the plane \( z = 1 \). The surface is oriented upward.
03
Compute \( \nabla \times \mathbf{F} \)
First, compute the curl of \( \mathbf{F} \). Given \( \mathbf{F} = -2yz \mathbf{i} + y \mathbf{j} + 3x \mathbf{k} \), we find \( abla \times \mathbf{F} = \left( \frac{\partial}{\partial y}(3x) - \frac{\partial}{\partial z}(y) \right) \mathbf{i} - \left( \frac{\partial}{\partial x}(3x) - \frac{\partial}{\partial z}(-2yz) \right) \mathbf{j} + \left( \frac{\partial}{\partial x}(y) - \frac{\partial}{\partial y}(-2yz) \right) \mathbf{k} \). This simplifies to \( abla \times \mathbf{F} = 0 \mathbf{i} + 2z \mathbf{j} + 1 \mathbf{k} \).
04
Set up the Surface Integral
The surface integral is \( \int_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \int_{S} (0 \mathbf{i} + 2z \mathbf{j} + 1 \mathbf{k}) \cdot \mathbf{n} \, dS \). Since \( S \) is oriented upward, \( \mathbf{n} \) is the unit vector pointing upward. We set up \( \int_{S} (2z \cdot n_j + n_k) \ dS \). The unit normal to the surface is derived from the gradient of the surface equation. As the surface is \( z = 5 - x^2 - y^2 \), \( \mathbf{n} \) pointing upward is indeed \( \langle -2x, -2y, 1 \rangle/\sqrt{4x^2 + 4y^2 + 1} \). The upward unit normal simplifies to approximately align with \( \mathbf{k} \).
05
Parameterize and Compute the Surface Integral
Parameterize \( S \) using polar coordinates, \( x = r \cos \theta \), \( y = r \sin \theta \), with \( 0 \leq r \leq 2 \) and \( 0 \leq \theta < 2\pi \). Substitute into the surface integral: \( \int_{0}^{2\pi} \int_{0}^{2} (2 - r^2) \, r \, dr \, d\theta \). Calculate the inner integral: \( \int_{0}^{2} (2r - r^3) \, dr = [r^2 - \frac{r^4}{4}]_{0}^{2} = 4 - 4/4 = 3 \). The outer integral becomes \( 3 \int_{0}^{2\pi} d\theta = 6\pi \).
06
Compute the Line Integral Around C
The boundary \( C \) is the circle \( x^2 + y^2 = 4 \) at \( z = 1 \). In polar coordinates, \( x = 2\cos \theta \), \( y = 2\sin \theta \), \( z = 1 \). \( \mathbf{F} = -4\sin \theta \mathbf{i} + 2 \sin \theta \mathbf{j} + 6 \cos \theta \mathbf{k} \). On \( C \), \( d\mathbf{r} = -2\sin \theta \mathbf{i} \, d\theta + 2\cos \theta \mathbf{j} \, d\theta \). Evaluate the line integral \( \oint_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{0}^{2\pi} (-4\sin \theta)(-2\sin \theta) + 2\sin \theta (2\cos \theta) \, d\theta \). Simplifying, integrate \( \int_{0}^{2\pi} (8\sin^2 \theta + 4\sin \theta \cos \theta) \, d\theta = 6\pi \).
07
Verify Stokes' Theorem
The surface integral \( \int_{S} (abla \times \mathbf{F}) \cdot d\mathbf{S} = 6\pi \). The line integral \( \oint_{C} \mathbf{F} \cdot d\mathbf{r} = 6\pi \). Both integrals yield the same value, verifying Stokes' Theorem for the given vector field and surface.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector field
A vector field is a function that assigns a vector to every point in space. Imagine a vector, which has both direction and magnitude, attached to each point within a region. This way, you can visualize how vectors behave at different locations.
In the context of the given problem, the vector field is described by the expression \( \mathbf{F}(x, y, z) = -2yz \mathbf{i} + y \mathbf{j} + 3x \mathbf{k} \). Here are a few points about this vector field:
In the context of the given problem, the vector field is described by the expression \( \mathbf{F}(x, y, z) = -2yz \mathbf{i} + y \mathbf{j} + 3x \mathbf{k} \). Here are a few points about this vector field:
- Each variable \( x, y, \) and \( z \) contributes differently to the vector's components.
- As \( y \) and \( z \) change, they affect the \( i \)-component by \( -2yz \).
- The \( j \)-component is purely from \( y \), showing a simpler relationship.
- The \( k \)-component depends only on \( x \), depicted by \( 3x \).
Surface integral
A surface integral helps to generalize the concept of integrating over a surface in three-dimensional space. It is especially useful in physics and engineering to measure quantities, such as flux, over a surface.
For our problem, Stokes' Theorem involves the surface integral of the curl of the vector field \( \mathbf{F} \) over the surface \( S \). Here, the surface \( S \) is part of a paraboloid and is defined by \( z = 5 - x^2 - y^2 \) above the plane \( z = 1 \).
Key points regarding surface integral:
For our problem, Stokes' Theorem involves the surface integral of the curl of the vector field \( \mathbf{F} \) over the surface \( S \). Here, the surface \( S \) is part of a paraboloid and is defined by \( z = 5 - x^2 - y^2 \) above the plane \( z = 1 \).
Key points regarding surface integral:
- It consists of adding up contributions from tiny surface patches for the whole surface.
- The vector field is evaluated on these surface patches.
- Orientation is crucial: the surface is oriented with its normal vector pointing upward.
Line integral
A line integral calculates the work done by a vector field along a curve. Imagine it as summing the field's influence along a specific path or boundary.
In this exercise, we look at the boundary \( C \) of the surface \( S \), which is a circle given by \( x^2 + y^2 = 4 \) in the plane \( z = 1 \). We compute how \( \mathbf{F} \) behaves along this boundary.
Here are important aspects concerning line integrals:
In this exercise, we look at the boundary \( C \) of the surface \( S \), which is a circle given by \( x^2 + y^2 = 4 \) in the plane \( z = 1 \). We compute how \( \mathbf{F} \) behaves along this boundary.
Here are important aspects concerning line integrals:
- Determine the parametric equations for \( C \); here \( x = 2\cos \theta \) and \( y = 2\sin \theta \).
- The differential path length element is \( d\mathbf{r} = -2\sin \theta \mathbf{i} \, d\theta + 2\cos \theta \mathbf{j} \, d\theta \).
- Calculate the influence by evaluating the integral of \( \mathbf{F} \cdot d\mathbf{r} \).
Curl of a vector field
The curl of a vector field is a measure of its tendency to rotate around a point. Think of it as capturing the rotational effect of the vectors around a point in space.
The curl is fundamental to Stokes' Theorem as it links line integrals to surface integrals. For \( \mathbf{F} \), we compute the curl as \( abla \times \mathbf{F} = 0 \mathbf{i} + 2z \mathbf{j} + 1 \mathbf{k} \).
Important points about the curl of a vector field:
The curl is fundamental to Stokes' Theorem as it links line integrals to surface integrals. For \( \mathbf{F} \), we compute the curl as \( abla \times \mathbf{F} = 0 \mathbf{i} + 2z \mathbf{j} + 1 \mathbf{k} \).
Important points about the curl of a vector field:
- It's a vector resulting from the cross-product of the del operator \( abla \) and \( \mathbf{F} \).
- Indicates how much and in which way vectors rotate around a given point.
- A non-zero curl implies a more pronounced rotational effect.
