/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 48 Let \(\mathbf{F}=\langle z, 0,-y... [FREE SOLUTION] | 91影视

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Chapter 17: Problem 48

Let \(\mathbf{F}=\langle z, 0,-y\rangle\) a. Find the scalar component of curl \(\mathbf{F}\) in the direction of the unit vector \(\mathbf{n}=\langle 1,0,0\rangle\). b. Find the scalar component of curl \(\mathbf{F}\) in the direction of the unit vector \(\mathbf{n}=\left\langle\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right\rangle\). c. Find the unit vector \(\mathbf{n}\) that maximizes \(\operatorname{scal}_{\mathbf{n}}\langle-1,1,0\rangle\) and state the value of \(\operatorname{scal}_{\mathbf{n}}\langle-1,1,0\rangle\) in this direction.

Short Answer

Expert verified
##Part (b)## #tag_title#Compute the scalar component of the curl of \(\mathbf{F}\) in the direction of the unit vector \(\mathbf{u}\) and \(\mathbf{v}\)# #tag_content#We need to find the scalar component of the curl of \(\mathbf{F}\) in the direction of the given unit vectors \(\mathbf{u}\) and \(\mathbf{v}\). To do this, we will use the formula for the scalar projection: $$ \text{Scalar Projection} = \frac{\mathbf{A} \cdot \mathbf{B}}{\|\mathbf{B}\|}, $$ where \(\mathbf{A}\) is the curl of \(\mathbf{F}\), and \(\mathbf{B}\) will be the unit vectors \(\mathbf{u}\) and \(\mathbf{v}\) respectively. For \(\mathbf{u} = \left\langle\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \right\rangle\), $$ \text{Scalar Component} = \frac{\left\langle0, 1,0\right\rangle \cdot \left\langle\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0 \right\rangle}{1} = \frac{1}{\sqrt{2}}. $$ For \(\mathbf{v} = \left\langle0, 1, 0 \right\rangle\), $$ \text{Scalar Component} = \frac{\left\langle0, 1,0\right\rangle \cdot \left\langle0, 1, 0 \right\rangle}{1} = 1. $$ So, the scalar component of the curl of \(\mathbf{F}\) in the direction of the unit vector \(\mathbf{u}\) is \(\frac{1}{\sqrt{2}}\), and in the direction of \(\mathbf{v}\) is \(1\).

Step by step solution

01

Find the curl of the vector field \(\mathbf{F}\)#

To find the curl of the vector field \(\mathbf{F}\), we use the formula: $$ \nabla \times \mathbf{F} = \left\langle\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle \times \left\langle z, 0,-y\right\rangle = \left\langle\frac{\partial(-y)}{\partial z} - \frac{\partial(0)}{\partial y}, \frac{\partial(z)}{\partial x} - \frac{\partial(-y)}{\partial z}, \frac{\partial(0)}{\partial x} - \frac{\partial(z)}{\partial y}\right\rangle = \left\langle0, 1,0\right\rangle. $$ Now we have \(\nabla \times \mathbf{F} = \left\langle0, 1,0\right\rangle\).

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Key Concepts

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

Scalar Component
Understanding the scalar component is crucial in vector calculus, especially when dealing with vector fields like \(\mathbf{F} = \langle z, 0, -y \rangle\). The scalar component essentially tells us the magnitude of a vector in the direction of another vector. In this case, it focuses on finding the component of the curl of \(\mathbf{F}\) along a specific direction. This can be seen in parts (a) and (b) of the original exercise, where different unit vectors (such as \(\langle 1,0,0 \rangle\) and \(\langle \frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \rangle\)) are mentioned.

To find the scalar component of a vector \(\mathbf{A}\) in the direction of a vector \(\mathbf{n}\), we use the formula: \( ext{scal}_\mathbf{n} (\mathbf{A}) = \mathbf{A} \cdot \mathbf{n}\). The formula uses the dot product, which effectively projects \(\mathbf{A}\) along the direction of \(\mathbf{n}\). For example, for the curl \(abla \times \mathbf{F} = \langle 0, 1, 0 \rangle\), this would evaluate the influence of \(\mathbf{n}\) on this vector.

Each choice of a unit vector represents a different direction in space, and hence changes the scalar component result, offering valuable insight into the vector field's behavior in that direction.
Unit Vector
A unit vector is a vector of length one, denoted usually by \(\mathbf{n}\). Think of it like a pointer which gives only the direction, without any magnitude of its own. In vector fields and physics, unit vectors are critical because they provide a standard way to express direction no matter the scale of the real-world quantities involved.

In the original exercise, the unit vectors used are \(\langle 1,0,0 \rangle\) and \(\langle \frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \rangle\), each indicating a distinct direction in three-dimensional space. Here鈥檚 why their lengths are crucial:
  • A vector's length is calculated using \(\sqrt{x^2+y^2+z^2}\). A unit vector will always have a length of 1.
  • Unit vectors are used to scale other vectors or compare directions without altering the length of the vector being analyzed.
This property is very useful when computing the curl, as it's necessary to get equal weighing along each axis. Using unit vectors helps us determine just the direction-specific component of another vector.
Maximize Scalar Component
Maximizing the scalar component involves finding the direction where the magnitude of a vector along that direction is the greatest. In vector calculus, this concept is crucial for determining where a vector field exerts its maximum influence in space.

For part (c) of the exercise, the goal is to find the unit vector \(\mathbf{n}\) that maximizes the scalar component of \(\langle -1, 1, 0 \rangle\). We do this by ensuring \(\mathbf{n}\) aligns with \(\langle -1, 1, 0 \rangle\). When this alignment happens, the dot product is maximized because \(\cos(\theta)\) of 0 degrees is 1, enhancing magnitude expressions.

To achieve the maximum scalar component:
  • Identify the existing vector, such as \(\langle -1, 1, 0 \rangle\).
  • Find the direction vector \(\mathbf{n}\) that points in the same direction.
  • The resulting unit vector should mirror \(\langle -1, 1, 0 \rangle\)鈥檚 orientation, yielding a maximum scalar component equal to the magnitude of \(\langle -1, 1, 0 \rangle\).
This optimization is significant in engineering and physics to find output forces, fluxes, or other field dimensions.

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Most popular questions from this chapter

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. $$T(x, y, z)=100(1+\sqrt{x^{2}+y^{2}+z^{2}})$$

Let \(S\) be the cylinder \(x^{2}+y^{2}=a^{2},\) for \(-L \leq z \leq L\) a. Find the outward flux of the field \(\mathbf{F}=\langle x, y, 0\rangle\) across \(S\) b. Find the outward flux of the field \(\mathbf{F}=\frac{\langle x, y, 0\rangle}{\left(x^{2}+y^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}\) across \(S\), where \(|\mathbf{r}|\) is the distance from the \(z\) -axis and \(p\) is a real number. c. In part (b), for what values of \(p\) is the outward flux finite as \(a \rightarrow \infty(\text { with } L\) fixed)? d. In part (b), for what values of \(p\) is the outward flux finite as \(L \rightarrow \infty\) (with \(a\) fixed)?

Heat flux The heat flow vector field for conducting objects is \(\mathbf{F}=-k \nabla T,\) where \(T(x, y, z)\) is the temperature in the object and \(k > 0\) is a constant that depends on the material. Compute the outward flux of \(\mathbf{F}\) across the following surfaces S for the given temperature distributions. Assume \(k=1\) $$T(x, y, z)=100 e^{-x^{2}-y^{2}-z^{2}} ; S \text { is the sphere } x^{2}+y^{2}+z^{2}=a^{2}$$

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 \(x^{2}+y^{2}+z^{2}=a^{2}\) \(z \geq 0\)

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