Chapter 13: Q4E (page 760)
Sketch the vector field by drawing a diagram like Figure 4 or Figure 8.
\(F(x,y) = yi{\rm{ + }}\left( {x{\rm{ + }}y} \right)j\)
Short Answer
Step by step solution
Concept
To sketch a vector field, draw the arrow representing the vector\({\rm{F}}\left( {{\rm{x,y}}} \right){\rm{ = yi + }}\left( {{\rm{x + y}}} \right){\rm{j}}\)starting at point\(\left( {{\rm{x, y}}} \right)\).
It is impossible to do this for all points\(\left( {{\rm{x, y}}} \right)\)but we can gain a reasonable impression of\({\rm{F}}\)by doing it for a few representative points in the domain.
Given Information
The given vector field is\(F\left( {x,y} \right) = yi + \left( {x + y} \right)j\).
Calculation
We will first calculate several representative values of \(F\left( {x,y} \right)\)in the table form, and then, by using the concept, draw the corresponding vectors to represent the vector field.
Several representative values of \(F\left( {x,y} \right)\) are:
\(\left( {x,y} \right)\) | \(F\left( {x,y} \right)\) |
\((1,0)\) | \(\left( {0,1} \right)\) |
\((0,1)\) | \(\left( {1,1} \right)\) |
\(( - 1,0)\) | \(\left( {0, - 1} \right)\) |
\((0, - 1)\) | \(\left( { - 1,1} \right)\) |
\((2,2)\) | \(\left( {2,4} \right)\) |
\((2, - 2)\) | \(\left( { - 2,0} \right)\) |
\(( - 2,2)\) | \(\left( {2,0} \right)\) |
\((1,1)\) | \(\left( {1,2} \right)\) |
\(( - 1, - 1)\) | \(\left( { - 1, - 2} \right)\) |
Sketch the vector field.
The sketch of the vector field will be:
Over 30 million students worldwide already upgrade their learning with Vaia!
