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Chapter 13: Problem 5

Sketch several representative vectors in the vector field. $$ \mathbf{F}(x, y, z)=\mathbf{i}+\mathbf{j}+\mathbf{k} $$

Short Answer

Expert verified
The vector field of \( \mathbf{F}(x, y, z)=\mathbf{i}+\mathbf{j}+\mathbf{k} \) is a field of constant vectors that are the same at every point in space, with the components \(1\), \(1\), and \(1\) in the \(x\), \(y\), and \(z\) directions, respectively.

Step by step solution

01

Understanding the Vector

Vector \(\mathbf{F}(x, y, z)=\mathbf{i}+\mathbf{j}+\mathbf{k}\) is not a function of \(x\), \(y\), or \(z\). This means the vector is the same at every point in the space.
02

Identifying the Vector Components

The vector components are \(1\), \(1\), and \(1\) in \(x\), \(y\), and \(z\) directions, respectively. These figures represent the lengths of the vector in each direction.
03

Sketching the Vector

Starting from any point in 3D space, draw a vector that has equal lengths in all three directions. Repeat this process for several points. All vectors will be parallel and identical, only their starting points will be different.
04

Understanding the Vector Field

This vector field represents a constant force (assuming \(\mathbf{F}\) stands for force) in 3D space that is identical in magnitude and direction no matter where in 3D space you are.

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Most popular questions from this chapter

Find \(\operatorname{div}(\mathbf{F} \times \mathbf{G})\) $$ \begin{array}{l} \mathbf{F}(x, y, z)=x \mathbf{i}-z \mathbf{k} \\ \mathbf{G}(x, y, z)=x^{2} \mathbf{i}+y \mathbf{j}+z^{2} \mathbf{k} \end{array} $$

In Exercises \(25-30,\) evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\) where \(C\) is represented by \(\mathbf{r}(t)\) \(\mathbf{F}(x, y)=x y \mathbf{i}+y \mathbf{j}\) \(\quad C: \mathbf{r}(t)=4 t \mathbf{i}+t \mathbf{j}, \quad 0 \leq t \leq 1\)

Evaluate the line integral along the given path. \(\int_{C} 4 x y d s\) \(C: \mathbf{r}(t)=t \mathbf{i}+(2-t) \mathbf{j}\) \(0 \leq t \leq 2\)

Evaluate \(\int_{C}\left(x^{2}+y^{2}\right) d s\) \(C:\) counterclockwise around the circle \(x^{2}+y^{2}=1\) from (1,0) to (0,1)

Consider a wire of density \(\rho(x, y)\) given by the space curve \(C: \mathbf{r}(t)=x(t) \mathbf{i}+y(t) \mathbf{j}, \quad a \leq t \leq b\) The moments of inertia about the \(x\) - and \(y\) -axes are given by \(I_{x}=\int_{C} y^{2} \rho(x, y) d s\) and \(I_{y}=\int_{C} x^{2} \rho(x, y) d s\) In Exercises 63 and \(64,\) find the moments of inertia for the wire of density \(\boldsymbol{\rho}\). A wire lies along \(\mathbf{r}(t)=a \cos t \mathbf{i}+a \sin t \mathbf{j}, 0 \leq t \leq 2 \pi\) and \(a>0,\) with density \(\rho(x, y)=1\).
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