Chapter 15, Problem 1RE

Sketching a Vector Field In Exercises 1 and 2, find $\Vert F\Vert$ and sketch several representative vectors in the vector field.

$F(x,y,z)=x\hat{i}+\hat{j}+2\hat{k}$

To determine

To calculate:

$F(x,y,z)=xi+j+2k$ and sketch several representative vectors in the vector field.

Answer to Problem 1RE

Solution:

$\Vert F\Vert =\sqrt{5+x^2}$

Explanation of Solution

Given:

$F(x,y,z)=xi+j+2k$

Formula used:

$\Vert F(x,y,z)\Vert =\sqrt{M^2+N^2+P^2}$, where $F(x,y,z)=M(x,y,z)i+N(x,y,z)j+P(x,y,z)k$ ----- (1)

Calculation:

Using (1), the magnitude of a vector function $F(x,y,z)=xi+j+2k$, is given by

$\Vert F(x,y,z)\Vert =\sqrt{x^2+1^2+2^2}=\sqrt{5+x^2}$

The divergence of a vector field $F(x,y,z)=xi+j+2k$ is given by

$\begin{array}{ccc}\hfill \nabla .F(x,y)& =& \nabla x\left(xi+j+2k\right)\hfill \\ \hfill & =& \left(\frac{\partial }{\partial x}i+\frac{\partial }{\partial y}j+\frac{\partial }{\partial z}k\right).\left(xi+j+2k\right)\hfill \\ \hfill & =& \frac{\partial }{\partial x}(x)+\frac{\partial }{\partial y}(1)+\frac{\partial }{\partial z}(2)\hfill \\ \hfill & =& 1+0+0\hfill \end{array}$

$\nabla .F(x,y,z)=1$

We start by making a table showing the vector field at several points. The table shown is a small sample. The vector field lines at many other points should be calculated to get a representative vector field, which can be obtained by using a computer algebra system.

x	-2	-1	0	0	1	2
y	1	-1	-1	1	-1	-1
z	1	1	1	1	1	1
$F=xi+j+2k$	$-2i+j+2k$	$-i+j+2k$	$j+2k$	$j+2k$	$i+j+2k$	$2i+j+2k$

Next, we draw line segments at the points to represent the vector field

The sketch of the representative vectors, is shown below

Interpretation: The vector field is irrotational in nature and has a positive divergence $\nabla .F(x,y,z)=1$, suggesting the presence of a source.