Hey guys!!!Could you help me at this exercise???
Draw some contour lines in the (x, y)–plane of the function h(x,y)=(a*(x+y))/(x^2+y^2+a^2).
Thanks in advance!!!
tzina, with the aid of a 3D grapher I was able to to come up with some contour lines.
let $\displaystyle \space$ $\displaystyle z = \displaystyle\frac{a(x + y)}{x^2 + a^2 + y^2}$
for each value of $\displaystyle \space$ $\displaystyle a$,$\displaystyle \space$ you get a unique set of contour lines, but each set of lines has similar features to them
.Set $\displaystyle \space$$\displaystyle z$$\displaystyle \space$ in increments of $\displaystyle \space$$\displaystyle .1$. For example$\displaystyle \space$:$\displaystyle \space$$\displaystyle z = 0, .1, .2, .3, ...$$\displaystyle \space$ etc.
For each value of $\displaystyle \space$$\displaystyle z$$\displaystyle \space$ you can create one contour line.
When$\displaystyle \space$$\displaystyle z = 0$$\displaystyle \space$ the contour line will show up as a straight line
if $\displaystyle \space$$\displaystyle z = 0$,$\displaystyle \space$$\displaystyle a(x + y) = 0$.
You will have the line $\displaystyle \space$$\displaystyle x + y = 0$
continuing with my response:
Each succeeding contour line will be a circle where the center of each circle will fall on the line $\displaystyle \space$$\displaystyle x - y = 0$
You will be able to calculate the center and radius of each circle by setting $\displaystyle \space$$\displaystyle a$$\displaystyle \space$as a specific constant and solving for $\displaystyle \space$$\displaystyle x$ and $\displaystyle y$$\displaystyle \space$ for each value of$\displaystyle \space$$\displaystyle z$.
You will end up with a series of circles whose centers fall on the line $\displaystyle \space$$\displaystyle x - y = 0$.
I hope this helps.
Hi,
Here's a graphing trick that you can use to draw contour lines. Assuming your graphing software can handle implicit functions, have it graph the function sin(s*2pi*h(x,y))=0. This is then the set of curves h(x,y)=k/s for k any integer. If your grapher has sliders, you can rapidly change the step size s. Also in your case you can rapidly change the constant a. Here's some contours for your function with s=10 (step size .1) and a =3 (the contour at height 0 is as noted above the line x+y=0, it's shown in red):