Post by Admin on Jul 23, 2018 7:11:24 GMT
In subsection 7 Needham refers to figure 13.
He refers to figure 13 on page 489 subsection 7 and writes: "Figure [13] illustrates the behaviour of $\overline{z}$ and $\overline{z}^2$ ,,,"
Notice his use of the word 'illustrates'. This is because the figure does not show an accurate picture of the vector fields of $\overline{z}$ and $\overline{z}^2$. If $R$ is the radius of the circle then the length of $\overline{z}$ is $R$ and the length of $\overline{z}^2$ is $R^2$. The vectors in the figures in the book are clearly much shorter than this. They have been scaled so that the way they rotate is easier to see.
So they represent the vector fields of $k\overline{z}$ and $k\overline{z}^2$ where $k<1$.
This is also true of figure 9 on page 485 and the last three terms of figure 11 on page 487, whereas figure 8 on page 484 is drawn without any scaling of the lengths of the vectors.
There may well be other examples where the vector lengths have been scaled to make the behaviour of the vector field clearer, The Equation Explorer software, details of which can be found in the Useful Links section of the forum, has a facility to scale the vectos. It is interesting to ask it to draw the vector field of $\overline{z}$, and observe how it chooses to scale the vectors by a factor of 0.1, and change this to 1. It will then be immediately apparent why Needham has scaled the vectors in most of his diagrams of vector fields.
Vasco/Admin
He refers to figure 13 on page 489 subsection 7 and writes: "Figure [13] illustrates the behaviour of $\overline{z}$ and $\overline{z}^2$ ,,,"
Notice his use of the word 'illustrates'. This is because the figure does not show an accurate picture of the vector fields of $\overline{z}$ and $\overline{z}^2$. If $R$ is the radius of the circle then the length of $\overline{z}$ is $R$ and the length of $\overline{z}^2$ is $R^2$. The vectors in the figures in the book are clearly much shorter than this. They have been scaled so that the way they rotate is easier to see.
So they represent the vector fields of $k\overline{z}$ and $k\overline{z}^2$ where $k<1$.
This is also true of figure 9 on page 485 and the last three terms of figure 11 on page 487, whereas figure 8 on page 484 is drawn without any scaling of the lengths of the vectors.
There may well be other examples where the vector lengths have been scaled to make the behaviour of the vector field clearer, The Equation Explorer software, details of which can be found in the Useful Links section of the forum, has a facility to scale the vectos. It is interesting to ask it to draw the vector field of $\overline{z}$, and observe how it chooses to scale the vectors by a factor of 0.1, and change this to 1. It will then be immediately apparent why Needham has scaled the vectors in most of his diagrams of vector fields.
Vasco/Admin