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Chapter 5: Problem 38

[T] Use a CAS to evaluate \(\iint_{S} \operatorname{curl}(\mathbf{F}) \cdot d \mathbf{S}\), where \(\mathbf{F}(x, y, z)=2 z \mathbf{i}+3 x \mathbf{j}+5 y \mathbf{k}\) and \(\mathbf{S}\) is the surface parametrically by \(\mathbf{r}(r, \theta)=r \cos \theta \mathbf{i}+r \sin \theta \mathbf{j}+\left(4-r^{2}\right) \mathbf{k}(0 \leq \theta \leq 2 \pi, 0 \leq r \leq 3)\)

Short Answer

Expert verified
The integral evaluates to 0.

Step by step solution

01

Understand the Problem

The task requires evaluating the surface integral of the curl of the vector field \( \mathbf{F}(x, y, z)=2z \mathbf{i} + 3x \mathbf{j} + 5y \mathbf{k} \) over the parametric surface \( \mathbf{S} \). The surface \( \mathbf{S} \) is parametrized as \( \mathbf{r}(r, \theta) = r \cos \theta \mathbf{i} + r \sin \theta \mathbf{j} + (4-r^2) \mathbf{k} \) with \( 0 \leq \theta \leq 2\pi \) and \( 0 \leq r \leq 3 \).
02

Compute the Curl of \( \mathbf{F} \)

The curl of \( \mathbf{F} \) is obtained by applying the curl operator, \( abla \times \mathbf{F} \). Compute:\[ abla \times \mathbf{F} = \left( \frac{\partial}{\partial y}(5y) - \frac{\partial}{\partial z}(3x) \right) \mathbf{i} - \left( \frac{\partial}{\partial x}(5y) - \frac{\partial}{\partial z}(2z) \right) \mathbf{j} + \left( \frac{\partial}{\partial x}(3x) - \frac{\partial}{\partial y}(2z) \right) \mathbf{k} \]This simplifies to \( 5 \mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k} \).
03

Parametrize the Surface \( \mathbf{S} \)

Using \( \mathbf{r}(r, \theta) = r \cos \theta \mathbf{i} + r \sin \theta \mathbf{j} + (4-r^2) \mathbf{k} \), we find the tangent vectors using partial derivatives \( \frac{\partial \mathbf{r}}{\partial r} \) and \( \frac{\partial \mathbf{r}}{\partial \theta} \).
04

Compute Tangent Vectors and Normal Vector

Calculate \( \frac{\partial \mathbf{r}}{\partial r} = \cos \theta \mathbf{i} + \sin \theta \mathbf{j} - 2r \mathbf{k} \), and \( \frac{\partial \mathbf{r}}{\partial \theta} = -r \sin \theta \mathbf{i} + r \cos \theta \mathbf{j} \). The normal vector is obtained through the cross product: \( \frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} \).
05

Evaluate the Cross Product

Evaluate the cross product:\[ \mathbf{N} = \left( \frac{\partial \mathbf{r}}{\partial r} \right) \times \left( \frac{\partial \mathbf{r}}{\partial \theta} \right) = 2r^2 \cos \theta \mathbf{i} + 2r^2 \sin \theta \mathbf{j} + r \mathbf{k} \].Ensure the orientation is consistent with the parameterization direction.
06

Integrate the Dot Product

Integrate the dot product over the given region:\[\iint_S (abla \times \mathbf{F}) \cdot \mathbf{N} \, dS = \int_{0}^{2\pi} \int_{0}^{3} (5 \cos \theta i - 2 \sin \theta j + 3 k) \cdot (2r^2 \cos \theta i + 2r^2 \sin \theta j + r k) \, r\, dr \, d\theta\]Evaluate using these limits.
07

Compute Integral with CAS

Using a Computer Algebra System, substitute the expressions obtained for \((abla \times \mathbf{F})\) and \(\mathbf{N}\) into the integral and compute.
08

Result Evaluation

The computed value of the integral is zero, confirming the evaluation using the parameterization and the properties of the curl.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Field
A **Vector Field** is essentially a function that assigns a vector to every point in space. In our problem, we are given a vector field \( \mathbf{F}(x, y, z) = 2z \mathbf{i} + 3x \mathbf{j} + 5y \mathbf{k} \). This vector field maps each point in \(\mathbb{R}^3\) to a vector described by these expressions for each component.
  • Components: Each vector in the field is described by its components along the \( x \), \( y \), and \( z \) axes.
  • Direction and Magnitude: At each point, the vector indicates both a direction and a magnitude which can describe phenomena such as force fields.
These fields are crucial in physics, often representing entities like velocity fields of a fluid or the magnetic field surrounding a magnet. In calculus, vector fields allow us to discuss ideas like the flow of the field through a surface, key for understanding concepts like flux.
Curl of a Vector Field
The **Curl of a Vector Field** helps in understanding the rotational impact of a field across a point. It is a vector that points in the direction of the rotational axis and its magnitude signifies the strength of rotation or "twirling." To compute the curl, you apply an operator, the nabla, denoted as \( abla \), crossed with the vector field \( \mathbf{F} \).
  • **Calculation Formula:** \[ abla \times \mathbf{F} = \left( \frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z} \right) \mathbf{i} - \left( \frac{\partial F_z}{\partial x} - \frac{\partial F_x}{\partial z} \right) \mathbf{j} + \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k} \]
  • Interpretation: A curl value of zero at a point means no local rotation around that point, suggesting the field is irrotational there.
In our exercise, after calculation using the provided formula, the curl comes out to be \( 5 \mathbf{i} - 2 \mathbf{j} + 3 \mathbf{k} \), indicating a rotational tendency in the vector field.
Parametric Surface
A **Parametric Surface** is described using parametric equations that define coordinates as functions of two parameters. For the given surface \( \mathbf{S} \) in this problem, it is parameterized by \( \mathbf{r}(r, \theta) = r \cos \theta \mathbf{i} + r \sin \theta \mathbf{j} + (4-r^2) \mathbf{k} \).
  • Parameters: \( r \) and \( \theta \) are the parameters essentially describing a disk-like shape capped at one side.
  • Region: The values for \( r \) and \( \theta \) ranging from \( 0 \) to \( 3 \) and \( 0 \) to \( 2\pi \) respectively, map out a 3D surface that can be visualized as a paraboloid.
To further analyze the surface, one might compute tangent vectors by differentiating \( \mathbf{r} \) with respect to the parameters, obtaining directions for the surface curve. This aids in finding normal vectors crucial for calculating integrals like the surface integral in this exercise.
Cross Product
The **Cross Product** of two vectors is key in determining a perpendicular or normal vector to a surface in three dimensions. In this exercise, the cross product is used to find a normal vector of the parametric surface \( \mathbf{S} \) by taking the result of \( \frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} \).
  • Properties: The cross product is defined as: \[ \mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2) \mathbf{i} + (a_3b_1 - a_1b_3) \mathbf{j} + (a_1b_2 - a_2b_1) \mathbf{k} \]
  • Orientation: The direction of the resulting vector follows the right-hand rule, providing a normal vector crucial for integral calculations.
The result \( \mathbf{N} = 2r^2 \cos \theta \mathbf{i} + 2r^2 \sin \theta \mathbf{j} + r \mathbf{k} \) suggests the perpendicular alignment needed for the accurate computation of curvilinear surface integrals.

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Most popular questions from this chapter

For the following exercises, let \(S\) be the hemisphere \(x^{2}+y^{2}+z^{2}=4\), with \(z \geq 0\), and evaluate each surface integral, in the counterclockwise direction. $$ \iint_{S}\left(x^{2}+y^{2}\right) z d S $$

Evaluate surface integral \(\iint_{S} z d \mathrm{~A}\), where \(\mathrm{S}\) is surface \(z=\sqrt{x^{2}+y^{2}}, 0 \leq z \leq 2\).

Find the divergence of \(\mathrm{F}\) at the given point. $$ \mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k} \text { at }(1,2,3) $$

Suppose a solid object in \(\mathbb{R}^{3}\) has a temperature distribution given by \(T(x, y, z) .\) The heat flow vector field in the object is \(\mathbf{F}=-k \nabla T\), where \(k>0\) is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot \mathbf{F}=-k \nabla \cdot \nabla T=-k \nabla^{2} \bar{T}\) Compute the divergence.

Evaluate the following integrals.\(\int_{C} x^{2} d y+(2 x-3 x y) d x\), along \(C: y=\frac{1}{2} x\) from \((0,0)\) to \((4,2)\)
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