- #1
Jonsson
- 79
- 0
Hello there,
What is wrong with my way of finding stream lines of a vector field? Say I have this vector field:
[itex]\vec{v} = x\,y\,\vec{i} + y\,\vec{j}[/itex]
You can see a plot here: http://kevinmehall.net/p/equationexplorer/vectorfield.html#xyi+yj|[-10,10,-10,10]
It appears as if the stream lines could be [itex]y = log(x) + C[/itex].
I proceed to find out:
[itex]v_y \, \mathrm{d}x = v_x \, \mathrm{d}y\\
x\,y\,\mathrm{d}y = y\,\mathrm{d}x\\
\mathrm{d}y = \frac{1}{x}\,\mathrm{d}x\\
\int\,\mathrm{d}y = \int \frac{1}{x}\,\mathrm{d}x\\
y = log(x) + C
[/itex]
This looks about right. However there is a problem, when I look back at my vector field (http://kevinmehall.net/p/equationexplorer/vectorfield.html#xyi+yj|[-10,10,-10,10]), for values of x less than zero, it appears as if the streamlines should be a mirror-image of y = log(x) + C.
So my question, does the above streamline calculation have more solutions which I have missed? Or is there something else which is wrong, which is causing me only to find the streamlines for x values greater than 0?
Thank you for your time.
Kind regards,
Marius
What is wrong with my way of finding stream lines of a vector field? Say I have this vector field:
[itex]\vec{v} = x\,y\,\vec{i} + y\,\vec{j}[/itex]
You can see a plot here: http://kevinmehall.net/p/equationexplorer/vectorfield.html#xyi+yj|[-10,10,-10,10]
It appears as if the stream lines could be [itex]y = log(x) + C[/itex].
I proceed to find out:
[itex]v_y \, \mathrm{d}x = v_x \, \mathrm{d}y\\
x\,y\,\mathrm{d}y = y\,\mathrm{d}x\\
\mathrm{d}y = \frac{1}{x}\,\mathrm{d}x\\
\int\,\mathrm{d}y = \int \frac{1}{x}\,\mathrm{d}x\\
y = log(x) + C
[/itex]
This looks about right. However there is a problem, when I look back at my vector field (http://kevinmehall.net/p/equationexplorer/vectorfield.html#xyi+yj|[-10,10,-10,10]), for values of x less than zero, it appears as if the streamlines should be a mirror-image of y = log(x) + C.
So my question, does the above streamline calculation have more solutions which I have missed? Or is there something else which is wrong, which is causing me only to find the streamlines for x values greater than 0?
Thank you for your time.
Kind regards,
Marius
Last edited: