- #1
Mathman23
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Hi Guys,
Given the vector field X(x,y) = ( a + \frac{b(y^2-x^2)}{(x^2+y^2)^2}, \frac{-2bxy}{(x^2+y^2)^2}}})
Show that for a point (x,y) on the circle with radius r = \sqrt(b/a) (i.e. x^2 + y^2 = b/a), the vector X(x,y) is tangent to a circle at the point.
My strategy is that to first define a vector function r(t) and show that this function is to a point on the circle?
Cheers and Best Whishes MM23
p.s. the potential function for X(x,y) is F(x,y) = ax + (bx/x^2+y^2).
p.p.s. Do I first find the directional dereative for F(x,y) ?
And then check to see if this vector is a tangent for the circle?
my solution:
Let
r (\theta )=\sqrt{\tfrac{b}{a}}(\cos \theta , \, \sin \theta ), \qquad 0\le \theta <2\pi
be any point on the circle. Plugging into the vector field one get at such a point the vector
$a(1+\sin ^2 \theta -\cos ^2 \theta ,\, -2\cos \theta \sin \theta )=2a (\sin ^2 \theta ,\, -\cos \theta \sin \theta ).
One realizes upon a scalar multiplication by the radius vector that this vector is orthogonal to it. Thus...
If this is correct to I then conclude that r (\theta ) \cdot X = 0 ??
Given the vector field X(x,y) = ( a + \frac{b(y^2-x^2)}{(x^2+y^2)^2}, \frac{-2bxy}{(x^2+y^2)^2}}})
Show that for a point (x,y) on the circle with radius r = \sqrt(b/a) (i.e. x^2 + y^2 = b/a), the vector X(x,y) is tangent to a circle at the point.
My strategy is that to first define a vector function r(t) and show that this function is to a point on the circle?
Cheers and Best Whishes MM23
p.s. the potential function for X(x,y) is F(x,y) = ax + (bx/x^2+y^2).
p.p.s. Do I first find the directional dereative for F(x,y) ?
And then check to see if this vector is a tangent for the circle?
my solution:
Let
r (\theta )=\sqrt{\tfrac{b}{a}}(\cos \theta , \, \sin \theta ), \qquad 0\le \theta <2\pi
be any point on the circle. Plugging into the vector field one get at such a point the vector
$a(1+\sin ^2 \theta -\cos ^2 \theta ,\, -2\cos \theta \sin \theta )=2a (\sin ^2 \theta ,\, -\cos \theta \sin \theta ).
One realizes upon a scalar multiplication by the radius vector that this vector is orthogonal to it. Thus...
If this is correct to I then conclude that r (\theta ) \cdot X = 0 ??
Last edited:
