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Magazine Chapter 15: Problem 1
Use Stokes' theorem for the evaluation of$$\iint_{S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S.$$ \(\mathbf{F}=3 y \mathbf{i}-2 x \mathbf{j}+x y z \mathbf{k} ; \quad S\) is the hemispherical surface \(z=\sqrt{4-x^{2}-y^{2}}\) with upper unit normal vector.
Short Answer
Expert verified
The surface integral is \(-8\pi\).
Step by step solution
01
Understanding Stokes' Theorem
Stokes' Theorem states that \( \iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r} \), where \(C\) is the boundary of the surface \(S\), and \(\mathbf{n}\) is the unit normal vector to \(S\).
02
Determine the Boundary of Surface S
The given surface \(S\) is the hemisphere \(z = \sqrt{4 - x^2 - y^2}\), which is part of a sphere with radius 2. Its boundary is the circle \(x^2 + y^2 = 4, z = 0\) in the \(xy\)-plane.
03
Finding the Curl of \(\mathbf{F}\)
Calculate the curl of \(\mathbf{F} = 3y\mathbf{i} - 2x\mathbf{j} + xyz\mathbf{k}\). Use the formula \(abla \times \mathbf{F} = ( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z})\mathbf{i} + (\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x})\mathbf{j} + (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\mathbf{k}\), where \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\).
04
Calculate \(\nabla\times\mathbf{F}\)
For \(\mathbf{F} = 3y\mathbf{i} - 2x\mathbf{j} + xyz\mathbf{k}\), we have: \(P = 3y, Q = -2x, R = xyz\).Calculate each component:- \(\frac{\partial R}{\partial y} = xz, \frac{\partial Q}{\partial z} = 0\Rightarrow xz - 0 = xz\mathbf{i}\)- \(\frac{\partial P}{\partial z} = 0, \frac{\partial R}{\partial x} = yz\Rightarrow 0 - yz = -yz\mathbf{j}\)- \(\frac{\partial Q}{\partial x} = -2, \frac{\partial P}{\partial y} = 3\Rightarrow -2 - 3 = -5\mathbf{k}\)Therefore, \(abla \times \mathbf{F} = xz\mathbf{i} - yz\mathbf{j} - 5\mathbf{k}\).
05
Parameterize the Boundary C
The boundary \(C\) is the circle \(x^2 + y^2 = 4\) located in the plane \(z = 0\). Parameterize \(C\) using \(x = 2\cos\theta, y = 2\sin\theta, z=0\) with \(\theta\) from \(0\) to \(2\pi\).
06
Evaluate the Line Integral \(\oint_{C} \mathbf{F} \cdot d\mathbf{r} \)
Substitute the parameterization into \(\mathbf{F}\) and calculate \(\mathbf{F} = 3y\mathbf{i} - 2x\mathbf{j} + xyz\mathbf{k} = 6\sin\theta \mathbf{i} - 4\cos\theta \mathbf{j}\).Compute \(d\mathbf{r} = (\frac{dx}{d\theta}\mathbf{i} + \frac{dy}{d\theta}\mathbf{j} + \frac{dz}{d\theta}\mathbf{k}) d\theta = (-2\sin\theta\mathbf{i} + 2\cos\theta\mathbf{j})d\theta\).Evaluate the line integral:\[\oint_{C} \mathbf{F} \cdot d\mathbf{r} = \int_{0}^{2\pi}(6 \sin\theta (-2 \sin\theta) + (-4 \cos\theta )(2 \cos\theta)) d\theta \].Simplify and solve: \[\int_{0}^{2\pi}(-12 \sin^2\theta - 8 \cos^2\theta) d\theta = \int_{0}^{2\pi}(-12 \sin^2\theta - 8(1-\sin^2\theta)) d\theta\].This simplifies to \[\int_{0}^{2\pi}(-4) d\theta = -8\pi\].
07
Result of the Surface Integral
Using Stokes' Theorem, the surface integral \(\iint_{S} (abla \times \mathbf{F}) \cdot \mathbf{n} dS\) over the surface \(S\) is equal to the line integral we calculated, which is \(-8\pi\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Calculus
Vector Calculus is a field of mathematics that studies how multivariable functions behave in space. These functions can take in multiple variables and output vectors in different dimensions. In this exercise, Stokes' Theorem, a powerful tool in vector calculus, is utilized to convert a complex surface integral into a simpler line integral that is easier to evaluate.
The main components we deal with in Vector Calculus include:
The main components we deal with in Vector Calculus include:
- Vectors: These are entities that have both a magnitude and a direction, represented usually as bold letters or with an arrow above, such as \( \mathbf{F} \).
- Fields: These are functions that associate a vector to each point in space. For example, the vector field \( \mathbf{F} = 3 y \mathbf{i} - 2 x \mathbf{j} + x y z \mathbf{k} \) assigns a vector to every point \((x, y, z)\).
- Operations on Vectors: Includes operations like gradient, divergence, and curl, which help in understanding changes in the vector field.
Surface Integrals
Surface integrals are used to find the integral of a scalar or vector field over a surface in three-dimensional space. In simple terms, they measure the total value of a field over a particular surface, whether it be a plane, cylinder, or sphere.
When working with surface integrals:
When working with surface integrals:
- Identify the surface \( S \) over which the integral is to be calculated. In this problem, the surface is the hemisphere defined by \( z = \sqrt{4 - x^2 - y^2} \).
- The integral involves the dot product of the curl of a vector field, \( abla \times \mathbf{F} \), with the unit normal vector \( \mathbf{n} \) to the surface.
Line Integrals
Line integrals help calculate the interaction of vector fields along curves or paths. They measure, for instance, the work done by a force field in moving an object along a path.
In our example, Stokes' Theorem simplifies the surface integral to a line integral around the boundary \( C \) of the surface \( S \):
In our example, Stokes' Theorem simplifies the surface integral to a line integral around the boundary \( C \) of the surface \( S \):
- Parameterize the boundary by expressing its coordinates in terms of a single parameter, such as \( \theta \). For the circular boundary here, we use \( x = 2 \cos \theta, y = 2 \sin \theta, \text{ and } z = 0 \).
- The line integral then involves integrating the dot product of the field \( \mathbf{F} \) with the differential path element \( d\mathbf{r} \).
Curl of a Vector Field
Curl provides an insight into the rotational characteristics of a vector field. It is a vector that describes the infinitesimal rotation at any point in the field and is denoted by \( abla \times \mathbf{F} \).
In this exercise, we calculated the curl of the vector field \( \mathbf{F} = 3y\mathbf{i} - 2x\mathbf{j} + xyz\mathbf{k} \) using the formula:
In this exercise, we calculated the curl of the vector field \( \mathbf{F} = 3y\mathbf{i} - 2x\mathbf{j} + xyz\mathbf{k} \) using the formula:
- \( abla \times \mathbf{F} = ( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z})\mathbf{i} + (\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x})\mathbf{j} + (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\mathbf{k} \)
