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Chapter 14: Problem 8

How is the circulation of a vector field on a closed smooth oriented curve calculated?

Short Answer

Expert verified
Question: Calculate the circulation of the given vector field F(x, y) = (x^2 + y^2, 2xy) around the closed curve C parametrized by r(t) = (cos t, sin t) for t between 0 and 2蟺. Answer: To find the circulation of the given vector field F(x, y) around the closed curve C, follow the steps mentioned in the solution. First, write down the given vector field F(x, y) = (x^2 + y^2, 2xy) and the parametric equation r(t) = (cos t, sin t). Next, calculate the derivative r'(t) = (-sin t, cos t). Then, substitute the components of r(t) into the vector field, obtaining F(r(t)) = (cos^2 t + sin^2 t, 2cos t sin t). Compute the dot product of F(r(t)) and r'(t), which results in -2cos^2 t sin t + 2sin^2 t cos t. Finally, evaluate the line integral of this dot product over the interval [0, 2蟺]. \[ \int_{0}^{2\pi} (-2cos^2 t sin t + 2sin^2 t cos t) dt = 0 \] The circulation of the given vector field F(x, y) around the closed curve C is 0.

Step by step solution

01

Understand the concept of circulation

Circulation is a scalar quantity that measures the net "swirling" motion of a vector field around a closed curve. Intuitively, it represents the total rotation of the vector field through the curve. In a two-dimensional space, the circulation is the sum of the products of the vector field components and their respective displacements along the curve.
02

Write down the given vector field and the parametric equation of the closed curve

Suppose we have a vector field F(x, y) and a closed smooth oriented curve C given by the parametrization r(t) where t ranges from a to b. Make sure to clearly define your vector field and determine the parametric equations for your closed curve.
03

Calculate the derivative of the parametric equation for the closed curve

To find the circulation of the vector field around the closed curve, compute the derivative r'(t) with respect to the parameter t. This will give you the tangent vector to the curve at each point, which is necessary for evaluating the line integral.
04

Substitute the components of the parametric equation into the vector field

Replace the variables x and y in the vector field F(x, y) with the respective components of the parametric equation r(t) to get a new vector field F(r(t)). This transformed vector field is now expressed in terms of the parameter t.
05

Compute the dot product of F(r(t)) and r'(t)

In this step, calculate the dot product of the transformed vector field F(r(t)) and the derivative of the parametric equation r'(t). This dot product results in a scalar function that is expressed in terms of the parameter t.
06

Evaluate the line integral of the dot product along the curve

Now we need to compute the line integral of the dot product obtained in Step 5. This is done by integrating the dot product from t=a to t=b. Calculate the definite integral and find the resulting value. \[ \text{Circulation} = \int_{a}^{b} \langle F(r(t)), r'(t) \rangle dt \] The result of this integration is the circulation of the vector field on the closed smooth oriented curve.

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Most popular questions from this chapter

Find the exact points on the circle \(x^{2}+y^{2}=2\) at which the field \(\mathbf{F}=\langle f, g\rangle=\left\langle x^{2}, y\right\rangle\) switches from pointing inward to outward on the circle, or vice versa.

Let \(S\) be the paraboloid \(z=a\left(1-x^{2}-y^{2}\right),\) for \(z \geq 0,\) where \(a>0\) is a real number. Let \(\mathbf{F}=\langle x-y, y+z, z-x\rangle .\) For what value(s) of \(a\) (if any) does \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) have its maximum value?

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Prove the following identities. Assume that \(\varphi\) is \(a\) differentiable scalar-valued function and \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, all defined on a region of \(\mathbb{R}^{3}\). $$\nabla \cdot(\mathbf{F} \times \mathbf{G})=\mathbf{G} \cdot(\nabla \times \mathbf{F})-\mathbf{F} \cdot(\nabla \times \mathbf{G})$$

The rotation of a three-dimensional velocity field \(\mathbf{V}=\langle u, v, w\rangle\) is measured by the vorticity \(\omega=\nabla \times \mathbf{V} .\) If \(\omega=\mathbf{0}\) at all points in the domain, the flow is irrotational. a. Which of the following velocity fields is irrotational: \(\mathbf{V}=\langle 2,-3 y, 5 z\rangle\) or \(\mathbf{V}=\langle y, x-z,-y\rangle ?\) b. Recall that for a two-dimensional source-free flow \(\mathbf{V}=(u, v, 0),\) a stream function \(\psi(x, y)\) may be defined such that \(u=\psi_{y}\) and \(v=-\psi_{x} .\) For such a two-dimensional flow, let \(\zeta=\mathbf{k} \cdot \nabla \times \mathbf{V}\) be the \(\mathbf{k}\) -component of the vorticity. Show that \(\nabla^{2} \psi=\nabla \cdot \nabla \psi=-\zeta\). c. Consider the stream function \(\psi(x, y)=\sin x \sin y\) on the square region \(R=\\{(x, y): 0 \leq x \leq \pi, 0 \leq y \leq \pi\\}\). Find the velocity components \(u\) and \(v\); then sketch the velocity field. d. For the stream function in part (c), find the vorticity function \(\zeta\) as defined in part (b). Plot several level curves of the vorticity function. Where on \(R\) is it a maximum? A minimum?
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