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Chapter 16: Problem 8

In Exercises \(7-12,\) use the surface integral in Stokes' Theorem to calculate the circulation of the field \(\mathbf{F}\) around the curve \(C\) in the indicated direction. \begin{equation} \begin{array}{l}{\mathbf{F}=2 y \mathbf{i}+3 x \mathbf{j}-z^{2} \mathbf{k}} \\\ {\text { C: The circle } x^{2}+y^{2}=9 \text { in the } x y \text { -plane, counterclockwise }} \\ {\text { when viewed from above }}\end{array} \end{equation}

Short Answer

Expert verified
The circulation of the field is \(9\pi\).

Step by step solution

01

Understand Stokes' Theorem

Stokes' Theorem relates a surface integral over a surface \(S\) to a line integral over its boundary curve \(C\). The theorem states that \( \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S}\), where \( abla \times \mathbf{F} \) is the curl of \( \mathbf{F} \) and \( d\mathbf{S} \) is the vector area element of \(S\).
02

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

Given \(\mathbf{F} = 2y \mathbf{i} + 3x \mathbf{j} - z^2 \mathbf{k}\), we calculate its curl: \( abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \partial / \partial x & \partial / \partial y & \partial / \partial z \ 2y & 3x & -z^2 \end{vmatrix} = (0)\mathbf{i} + (0)\mathbf{j} + (3 - 2)\mathbf{k} = \mathbf{k}\). So, \( abla \times \mathbf{F} = \mathbf{k}\).
03

Determine the Surface \(S\)

The curve \(C\) is the circle \(x^2 + y^2 = 9\) in the \(xy\)-plane. We choose \(S\) to be the disk bounded by this circle in the \(xy\)-plane. Thus, \(S\) is the surface \(z = 0\) and \(r\leq 3\).
04

Set Up the Surface Integral

The vector area element \(d\mathbf{S}\) for the plane at \(z=0\) pointing upwards is \(d\mathbf{S} = dA \mathbf{k} = dx\,dy\, \mathbf{k}\). Using Stokes' Theorem, the integration becomes \( \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S} = \iint_S (\mathbf{k}) \cdot (\mathbf{k})\, dx\, dy = \iint_S dx\, dy \).
05

Compute the Surface Integral

The integral becomes \( \iint_S dx\, dy \), which computes the area of the disk of radius 3. The area is \( \pi \times 3^2 = 9\pi \).
06

Conclusion

The surface integral \( \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S}\) equals \(9\pi\). Therefore, by Stokes' Theorem, the circulation of the field \(\mathbf{F}\) around the curve \(C\) is also \(9\pi\).

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Key Concepts

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

Surface Integral
The concept of a surface integral is crucial when working with Stokes' Theorem. It involves integrating a function over a surface in three-dimensional space. Imagine you have a sheet of paper and you want to find the sum of a variable, like temperature, over its entire surface. This is essentially what a surface integral does.
  • Defining the Surface: The surface is the geometric shape over which the integration is performed. In this problem, our surface is the area enclosed by the circle in the xy-plane.
  • Surface Orientation: The direction of integration is important too. Here, the surface is oriented upward as given by the vector \(d\mathbf{S} = dx \,dy \, \mathbf{k}\).
The act of performing the integration over this surface measures something like total flow across that area, and it brings us closer to solving for the circulation of the field.
Curl of a Vector Field
The curl of a vector field is a measure of how much a vector field tends to rotate or twist. It tells you the infinitesimal circulation at each point of the field.
  • Mathematical Definition: Mathematically, the curl of a vector field \( \mathbf{F}\) is expressed as \( abla \times \mathbf{F}\). It involves computing the determinant of a matrix-style arrangement using \( \mathbf{i}\), \( \mathbf{j}\), and \( \mathbf{k}\) unit vectors, along with partial derivatives.
  • Physical Interpretation: The curl can be thought of as the amount of "twisting" or "whirl" of the vectors around a point. In our exercise, \( abla \times \mathbf{F} = \mathbf{k}\), indicating a uniform slight rotation in the z-direction.
Understanding curl is key for applying Stokes' Theorem because it relates directly to the surface integral calculation.
Line Integral
A line integral takes into account the values of a vector field along a curve, summing the field's projections along the line. This is particularly useful in finding work done by forces along paths or circuits.
  • Curve Path: The curve, noted as \(C\), in our problem is a circle defined by \(x^2 + y^2 = 9\), walked counterclockwise when viewed from above.
  • Integration Process: The line integral, symbolized as \( \oint_C \mathbf{F} \cdot d\mathbf{r}\), is a summation over this curve where \(\mathbf{F}\) values are evaluated and dotted with small path elements \(d\mathbf{r}\).
In physical terms, it represents the "total circulation" along the curve鈥攁 direct measure of rotational effect captured by the path.
Circulation of a Field
The circulation of a vector field around a closed loop is essentially the line integral of the field over that loop. It's a measurement of how the field "circles" around a given path.
  • Connection to Physical Concepts: Picture a closed racetrack where the field might represent wind direction and speed. The circulation calculates how much wind tends to push along the track.
  • Using Stokes' Theorem: In our solution, we use Stokes' Theorem to equate the circulation (line integral) to the surface integral of the curl: \(\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (abla \times \mathbf{F}) \cdot d\mathbf{S}\).
By converting the problem of calculating around a closed curve to one on a flat surface, the task becomes simpler and harnesses powerful calculus techniques.

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Most popular questions from this chapter

Conservation of mass Let \(\mathbf{v}(t, x, y, z)\) be a continuously differentiable vector field over the region \(D\) in space and let \(p(t, x)\) \(y, z )\) be a continuously differentiable scalar function. The variable \(t\) represents the time domain. The Law of Conservation of Mass asserts that $$\frac{d}{d t} \iiint_{D} p(t, x, y, z) d V=-\iint_{S} p \mathbf{v} \cdot \mathbf{n} d \sigma$$ where \(S\) is the surface enclosing \(D\) a. Give a physical interpretation of the conservation of mass law if \(\mathbf{v}\) is a velocity flow field and \(p\) represents the density of the fluid at point \((x, y, z)\) at time \(t\) b. Use the Divergence Theorem and Leibniz's Rule, $$\frac{d}{d t} \iiint_{D} p(t, x, y, z) d V=\iiint_{D} \frac{\partial p}{\partial t} d V$$ to show that the Law of Conservation of Mass is equivalent to the continuity equation, $$\nabla \cdot p \mathbf{v}+\frac{\partial p}{\partial t}=0$$ (In the first term \(\nabla \cdot p \mathbf{v},\) the variable \(t\) is held fixed, and in the second term \(\partial p / \partial t,\) it is assumed that the point \((x, y, z)\) in \(D\) is held fixed.)

Harmonic functions A function \(f(x, y, z)\) is said to be harmonic in a region \(D\) in space if it satisfies the Laplace equation $$\nabla^{2} f=\nabla \cdot \nabla f=\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}+\frac{\partial^{2} f}{\partial z^{2}}=0$$ throughout \(D\) a. Suppose that \(f\) is harmonic throughout a bounded region \(D\) enclosed by a smooth surface \(S\) and that \(n\) is the chosen unit normal vector on \(S .\) Show that the integral over \(S\) of \(\nabla f \cdot \mathbf{n},\) the derivative of \(f\) in the direction of \(\mathbf{n},\) is zero. b. Show that if \(f\) is harmonic on \(D,\) then $$\iint_{S} f \nabla f \cdot \mathbf{n} d \sigma=\iiint_{D}|\nabla f|^{2} d V$$

In Exercises \(19-24,\) use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(F\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\) . \begin{equation} \begin{array}{l}{\mathbf{F}=3 y \mathbf{i}+(5-2 x) \mathbf{j}+\left(z^{2}-2\right) \mathbf{k}} \\ {S : \quad \mathbf{r}(\phi, \theta)=(\sqrt{3} \sin \phi \cos \theta) \mathbf{i}+(\sqrt{3} \sin \phi \sin \theta) \mathbf{j}+} \\ {(\sqrt{3} \cos \phi) \mathbf{k}, \quad 0 \leq \phi \leq \pi / 2, \quad 0 \leq \theta \leq 2 \pi}\end{array} \end{equation}

Evaluating a work integral two ways Let \(F=\nabla\left(x^{3} y^{2}\right)\) and let \(C\) be the path in the \(x y\) -plane from \((-1,1)\) to \((1,1)\) that consists of the line segment from \((-1,1)\) to \((0,0)\) followed by the line segment from \((0,0)\) to \((1,1)\) . Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) in two ways. \begin{equation}\begin{array}{l}{\text { a. Find parametrizations for the segments that make up } C \text { and }} \\ {\text { evaluate the integral. }} \\ {\text { b. Use } f(x, y)=x^{3} y^{2} \text { as a potential function for } \mathbf{F} \text { . }}\end{array}\end{equation}

Green's first formula Suppose that \(f\) and \(g\) are scalar functions with continuous first- and second-order partial derivatives throughout a region \(D\) that is bounded by a closed piecewise smooth surface \(S .\) Show that $$\iint_{S} f \nabla g \cdot \mathbf{n} d \sigma=\iiint_{D}\left(f \nabla^{2} g+\nabla f \cdot \nabla g\right) d V \quad \quad\quad(10)$$ Equation \((10)\) is Green's first formula. (Hint: Apply the Divergence Theorem to the field \(\mathbf{F}=f \nabla g . )\)
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