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Magazine Chapter 16: Problem 15
Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\).$$\begin{aligned} &\mathbf{F}=x^{2} y \mathbf{i}+2 y^{3} z \mathbf{j}+3 z \mathbf{k}\\\ &S: \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+r \mathbf{k}\\\ &0 \leq r \leq 1, \quad 0 \leq \theta \leq 2 \pi \end{aligned}$$
Short Answer
Expert verified
The flux of the curl across the surface \( S \) is zero.
Step by step solution
01
Understand Stokes' Theorem
Stokes' Theorem relates a surface integral of the curl of a vector field \( abla \times \mathbf{F} \) over a surface \( S \) to a line integral of \( \mathbf{F} \) over the boundary curve \( C \) of the surface: \[ \iint_S (abla \times \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_C \mathbf{F} \cdot d\mathbf{r} \]In this exercise, the task is to compute the surface integral on the left side since the boundary curve isn't explicitly defined.
02
Compute the Curl of \(\mathbf{F}\)
First, we need to find the curl of the vector field \( \mathbf{F} = x^2 y \, \mathbf{i} + 2y^3 z \, \mathbf{j} + 3z \, \mathbf{k} \). The curl is given by:\[ abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ x^2 y & 2y^3 z & 3z \end{vmatrix} \]After computing the determinant, we get:\[ abla \times \mathbf{F} = 6y^2 z \, \mathbf{i} - 2z \, \mathbf{j} + 0 \, \mathbf{k} \] So, the curl of \( \mathbf{F} \) is \( 6y^2 z \, \mathbf{i} - 2z \, \mathbf{j} \).
03
Parametrize the Surface \(S\)
The surface \( S \) is given parametrically as \( \mathbf{r}(r, \theta) = (r\cos\theta) \, \mathbf{i} + (r\sin\theta) \, \mathbf{j} + r \, \mathbf{k} \). Identify limits \( 0 \leq r \leq 1 \) and \( 0 \leq \theta \leq 2\pi \) which define a disk surface parallel to the \( xy \)-plane and rise in the \( z \)-direction.
04
Find the Normal Vector \(\mathbf{n}\)
To find the outward normal vector \( \mathbf{n} \), compute the cross product of the partial derivatives:\[ \frac{\partial \mathbf{r}}{\partial r} = \cos\theta \, \mathbf{i} + \sin\theta \, \mathbf{j} + \mathbf{k}, \quad \frac{\partial \mathbf{r}}{\partial \theta} = -r\sin\theta \, \mathbf{i} + r\cos\theta \, \mathbf{j} \]\[ \mathbf{n} = \frac{\partial \mathbf{r}}{\partial r} \times \frac{\partial \mathbf{r}}{\partial \theta} = -r \, \mathbf{i} - r \, \mathbf{j} + r^2 \, \mathbf{k} \]The outward normal vector is simplified to \( \mathbf{n} = - \mathbf{i} - \mathbf{j} + r \, \mathbf{k} \).
05
Integrate the Scalar Product over the Surface
The surface integral requires calculating the dot product \( (abla \times \mathbf{F}) \cdot \mathbf{n} \). Substituting, we get:\[ (6y^2 z \, \mathbf{i} - 2z \, \mathbf{j}) \cdot (- \mathbf{i} - \mathbf{j} + r \, \mathbf{k}) = -6y^2 z + 2z \]Using \( y = r\sin\theta, \, z = r \), the expression becomes:\[ -6(r\sin\theta)^2 r + 2r = -6r^3 \sin^2 \theta + 2r \]Integrate over \( 0 \leq r \leq 1 \) and \( 0 \leq \theta \leq 2\pi \):\[ \iint_S -6r^3 \sin^2 \theta + 2r \, dr \, d\theta \] After integration the result is \( 0 \).
06
Conclude the Flux Calculation
The result \( 0 \) implies that the net flux of the curl of the field across the surface \( S \) in the direction of the outward unit normal is zero. This balances the circulation due to symmetry based on the geometry and field properties.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Surface Integrals
Surface integrals are a powerful tool in calculus used for finding the flux of a vector field through a given surface. You can think of it as measuring how much of the field penetrates the surface. In our exercise, we're using Stokes' Theorem to evaluate the flux of the curl of a vector field across a given surface. This theorem helps simplify calculations by converting a surface integral into a line integral over the boundary. However, in this case, we directly compute the surface integral because the boundary isn鈥檛 explicitly defined.
- Understanding the Surface: The surface is parametrized, describing it using parameters like radius and angle. Here, it forms a disk parallel to the xy-plane.
- Integration Domain: The integration is over the defined parameter limits, precisely where the surface exists.
- Flux Calculation: The dot product of the curl of the field and the surface's normal vector is integrated over the surface.
Vector Fields
Vector fields are mathematical constructs where each point has a vector representing a direction and magnitude. They help describe various physical phenomena such as fluid flow, magnetic fields, and more. In our exercise, vector field \( \mathbf{F} = x^2 y \mathbf{i} + 2y^3 z \mathbf{j} + 3z \mathbf{k} \) is the entity whose behavior across a surface we analyze through surface integrals.
- Components of a Vector Field: Notice the vector \( \mathbf{F} \) consists of three parts: \( x^2 y \) in the i-direction, \( 2y^3 z \) in the j-direction, and \( 3z \) in the k-direction. These components vary with space, providing a field across three dimensions.
- Application: Understanding vector fields is crucial as they model real-world systems and allow analyses like determining flux and circulation.
- Calculation Techniques: Computations involve determining derivatives and performing integrations to quantify field behavior over surfaces and lines.
Curl of a Field
The curl of a vector field is a vector measure indicating how the field 'circulates' around a point. Stokes' Theorem involves the curl because it translates the closed-loop circulation of the field into a surface integral. In the exercise, we compute the curl of the field \( \mathbf{F} \), finding it as \( abla \times \mathbf{F} = 6y^2 z \mathbf{i} - 2z \mathbf{j} \). This indicates how the field rotates around axes.
- Defining Curl: Mathematically, the curl is derived using a determinant of a matrix formed with unit vectors and partial derivatives.
- Physical Meaning: The curl captures the tendency of the vector field to induce a rotational motion at any given point.
- Second Nature: Calculating curls is routine for solving many physics problems, revealing valuable information about rotational tendencies within a field.
Flux Calculation
Flux calculation is essential for understanding how a vector field behaves when passing through a specific surface. This is where surface integrals shine, helping compute how much of the field penetrates the surface.
- Why Compute Flux? It quantifies the field's effect, identifying whether it's strong or weak across a surface.
- Integration Process: The integral, after parameterizing the surface and finding normal vectors, involves evaluating \((abla \times \mathbf{F}) \cdot \mathbf{n}\).
- Interpreting Results: In our exercise, the result is zero, meaning the net penetration is balanced. This suggests symmetry in the set-up or that the field and surface interact without a net directional penetration.
