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Chapter 17: Problem 28

For the vector field \(\mathbf{F}\) and curve \(C\), complete the following: a. Determine the points (if any) along the curve C at which the vector field \(\mathbf{F}\) is tangent to \(C\). b. Determine the points (if any) along the curve C at which the vector field \(\mathbf{F}\) is normal to \(C\) c. Sketch \(C\) and a few representative vectors of \(\mathbf{F}\) on \(C\). $$\mathbf{F}=\langle y,-x\rangle ; C=\left\\{(x, y): x^{2}+y^{2}=1\right\\}$$

Short Answer

Expert verified
Question: Find the points on the curve \(C: x^2 + y^2 = 1\) where the vector field \(\mathbf{F} = \langle y, -x \rangle\) is either tangent or normal to the curve. Sketch the curve \(C\) and include a few representative vectors for \(\mathbf{F}\). Answer: There are no points where \(\mathbf{F}\) is tangent to \(C\). However, there are two pairs of points where \(\mathbf{F}\) is normal to \(C\): \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\), \((-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\), \((-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\), and \((\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2})\). These points correspond to the values of \(t\) in \(\{(\pi/4, \pi/4+2m\pi)\), \((3\pi/4, 3\pi/4+2n\pi)\}\).

Step by step solution

01

Parameterize the curve \(C\)

As \(C\) is the unit circle, we can parameterize it using trigonometric substitution. Let \(x = \cos{t}\) and \(y = \sin{t}\), then we have $$C(t) = (\cos{t}, \sin{t})$$ for \(0 \leq t \leq 2\pi\).
02

Calculate the tangent and normal vectors of \(C\)

In order to find the tangent and normal vectors of the curve \(C\), we need to compute the derivative \(C'(t)\). $$C'(t) = (-\sin{t}, \cos{t})$$ The tangent vector of \(C\) at \(t\) is just \(C'(t)\), and the normal vector of \(C\) at \(t\) is orthogonal to the tangent vector, which can be represented by \(\mathbf{N}=\langle -\cos(t),-\sin(t)\rangle\).
03

Find the points where \(\mathbf{F}\) is tangent or normal to \(C\)

To find where the vector field \(\mathbf{F}\) is tangent to \(C\), we need to find points where \(\mathbf{F}\) is parallel to the tangent vector of \(C\). To do this, we will calculate \(\mathbf{F}(C(t))\) and compare it to \(C'(t)\). $$\mathbf{F}(C(t)) = \langle \sin{t}, -\cos{t} \rangle$$ \(\mathbf{F}(C(t))\) is parallel to \(C'(t)\) if their cross product is zero; therefore, their determinant should be zero as well. $$\det( \mathbf{F}(C(t)), C'(t) ) = \sin{t} \cdot \cos{t} - (-\cos{t})\cdot (-\sin{t}) = 0$$ This equation yields no solutions for \(t\), so there is no point where \(\mathbf{F}\) is tangent to \(C\). To find where \(\mathbf{F}\) is normal to \(C\), we need to find points where \(\mathbf{F}\) is parallel to the normal vector of \(C\). We will compare \(\mathbf{F}(C(t))\) and \(\mathbf{N}(t)\). $$\mathbf{N}(t) = \langle -\cos{t}, -\sin{t} \rangle$$ \(\mathbf{F}(C(t))\) is parallel to \(\mathbf{N}(t)\) if their cross product is zero; therefore, their determinant should be zero as well. $$\det( \mathbf{F}(C(t)), \mathbf{N}(t)) = \sin{t} \cdot (-\sin{t}) - (-\cos{t})\cdot (-\cos{t}) = 0$$ This equation simplifies to \(\sin^2{t} - \cos^2{t} = 0\), which has solutions \(\{(\pi/4, \pi/4+2m\pi)\), \((3\pi/4, 3\pi/4+2n\pi)\}\). These solutions correspond to points on the curve \(C\) where the vector field \(\mathbf{F}\) is normal to \(C\).
04

Sketch the curve \(C\) and the representative vectors of \(\mathbf{F}\)

Sketch the unit circle, which represents \(C\), as well as the vector field \(\mathbf{F}\) using the points where \(\mathbf{F}\) is normal to \(C\). Include a few representative vectors of \(\mathbf{F}\) along the curve \(C\) at points such as \((1,0)\), \((0,1)\), \((-1,0)\), and \((0,-1)\).

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Key Concepts

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

Tangency
In vector calculus, tangency refers to the condition where a vector field is tangential to a curve at a certain point. This means that the vector field does not pass through the curve but rather lies along it at that particular point. To check for tangency of a vector field \( \mathbf{F} \) with a curve \( C \), we need to ensure that the vector is parallel to the tangent vector of the curve.
For the curve \( C \), which is a unit circle, the tangent vector can be determined via derivative of the parametrization. If \( \mathbf{F}(C(t)) \) and the tangent vector \( C'(t) \) are parallel, the cross product should be zero. In our example, the calculated determinant (akin to the 2D cross product) resulted in no solutions, indicating there are no points where \( \mathbf{F} \) is tangent to \( C \). This implies \( \mathbf{F} \) never aligns exactly with the trajectory of \( C \) at any point, remaining purely perpendicular.
Unit Circle
The unit circle is a fundamental concept in mathematics and particularly in trigonometry. It is defined as the set of points equidistant from the origin in a plane, specifically at a distance of 1 unit. Mathematically, it is expressed by the equation \( x^2 + y^2 = 1 \).
This equation represents a circle centered at the origin with a radius of 1. When working with the unit circle, we often use trigonometric parametrization, substituting \( x = \cos{t} \) and \( y = \sin{t} \) to describe points on the circle as \( C(t) = (\cos{t}, \sin{t}) \) over the interval \( 0 \leq t \leq 2\pi \).
This parametrization covers the entire circumference of the unit circle, rotating counterclockwise from the positive x-axis starting at \( t = 0 \). For any given angle \( t \), the point on the unit circle is directly related to the angle in the trigonometric sense, making it an essential tool for understanding circular motion and waveforms.
Parametrization
Parametrization is the process of expressing a curve or geometric object in terms of a parameter, usually \( t \). For a circle, particularly the unit circle, parametrization allows us to describe every point on the circle in terms of trigonometric functions. This is useful because it simplifies the calculation of other properties, such as tangents and normals.
For the unit circle in the xy-plane described by \( x^2 + y^2 = 1 \), we use the parametrization \( x = \cos{t} \) and \( y = \sin{t} \). Hence, every point \( P \) on this circle can be represented as \( (\cos{t}, \sin{t}) \). This representation is smooth and continuous because \( \cos{t} \) and \( \sin{t} \) are periodic functions with a period of \( 2\pi \).
Parametrization efficiently maps the real number domain onto the unit circle, turning an algebraic setup into a trigonometric one, which can be more convenient for certain analytical procedures, such as determining points of tangency and normalcy against vector fields.
Normal Vectors
Normal vectors play an important role in vector calculus as they are orthogonal to tangent vectors at a given point on a curve or surface. They provide geometrical information about the curve's directional tendencies. In the context of the unit circle, a normal vector is perpendicular to the tangent vector derived through derivation of the parameterized curve function \( C(t) \).
For a parameterization \( C(t) = ( \cos{t}, \sin{t}) \), its derivative \( C'(t) = (-\sin{t}, \cos{t}) \) is the tangent vector. A corresponding normal vector \( \mathbf{N}(t) \) at any point on the unit circle can be represented as \( \mathbf{N}(t) = \langle -\cos{t}, -\sin{t} \rangle \).
In our example, finding where \( \mathbf{F}(C(t)) \) is normal involves comparing it with \( \mathbf{N}(t) \) to check for parallelism. The determinant condition \( \det( \mathbf{F}(C(t)), \mathbf{N}(t)) = 0 \) helps in identifying points like \( t = \pi/4 \) and \( t = 3\pi/4 \) where the field is indeed normal to the circle. Identifying such points is crucial for applications like understanding field interactions or optimizing force trajectories.

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Most popular questions from this chapter

Outward normal to a sphere Show that \(\left|\mathbf{t}_{u} \times \mathbf{t}_{\mathbf{v}}\right|=a^{2} \sin u\) for a sphere of radius \(a\) defined parametrically by \(\mathbf{r}(u, v)=\langle a \sin u \cos v, a \sin u \sin v, a \cos u\rangle,\) where \(0 \leq u \leq \pi\) and \(0 \leq v \leq 2 \pi\)

Mass and center of mass Let \(S\) be a surface that represents a thin shell with density \(\rho .\) The moments about the coordinate planes (see Section 16.6 ) are \(M_{y z}=\iint_{S} x \rho(x, y, z) d S, M_{x z}=\iint_{S} y \rho(x, y, z) d S\) and \(M_{x y}=\iint_{S} z \rho(x, y, z) d S .\) The coordinates of the center of mass of the shell are \(\bar{x}=\frac{M_{y z}}{m}, \bar{y}=\frac{M_{x z}}{m},\) and \(\bar{z}=\frac{M_{x y}}{m},\) where \(m\) is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. The constant-density hemispherical shell \(x^{2}+y^{2}+z^{2}=a^{2}\) \(z \geq 0\)

Consider the radial field \(\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}=\frac{\langle x, y, z\rangle}{|\mathbf{r}|^{p}},\) where \(p>1\) (the inverse square law corresponds to \(p=3\) ). Let \(C\) be the line segment from (1,1,1) to \((a, a, a),\) where \(a>1,\) given by \(\mathbf{r}(t)=\langle t, t, t\rangle,\) for \(1 \leq t \leq a\) a. Find the work done in moving an object along \(C\) with \(p=2\) b. If \(a \rightarrow \infty\) in part (a), is the work finite? c. Find the work done in moving an object along \(C\) with \(p=4\) d. If \(a \rightarrow \infty\) in part (c), is the work finite? e. Find the work done in moving an object along \(C\) for any \(p>1\) f. If \(a \rightarrow \infty\) in part (e), for what values of \(p\) is the work finite?

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 the conductivity \(k>0\) is a property of the material. Note that 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 \(\nabla \cdot \mathbf{F}=-k \nabla \cdot \nabla T=-k \nabla^{2} T(\text {the Laplacian of } T) .\) Compute the heat flow vector field and its divergence for the following temperature distributions. $$T(x, y, z)=100(1+\sqrt{x^{2}+y^{2}+z^{2}})$$

Conditions for Green's Theorem Consider the radial field \(\mathbf{F}=\langle f, g\rangle=\frac{\langle x, y\rangle}{\sqrt{x^{2}+y^{2}}}=\frac{\mathbf{r}}{|\mathbf{r}|}\) a. Explain why the conditions of Green's Theorem do not apply to F on a region that includes the origin. b. Let \(R\) be the unit disk centered at the origin and compute \(\iint_{R}\left(\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}\right) d A\) c. Evaluate the line integral in the flux form of Green's Theorem on the boundary of \(R\) d. Do the results of parts (b) and (c) agree? Explain.
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