Describe curve to reduce field intensity in fastest time

[toforfiltum]

Homework Statement

Igor, the inchworm, is crawling along graph paper in a magnetic field. The intensity of the field at the point $(x,y)$ is given by $M(x,y)=3x^2+y^2+5000$. If Igor is at the point $(8,6)$, describe the curve along which he should travel if he wishes to reduce the field intensity as rapidly as possible.

Homework Equations

The Attempt at a Solution

The problem I have with this question is that I'm asked to describe a curve instead of direction.
Here are my steps so far: $$\nabla M(x,y)=(6x,2y)$$
$$\nabla M(8,6)=(48,12)$$
Since they want the most rapid reduction, I must go in the direction opposite to $(48,12)$, which if given in unit vector is $(\frac{-4}{\sqrt 17},\frac{-1}{\sqrt 17})$.

I have no idea how to come up with an equation of a curve based on this.

Any hints?

Thanks.

 

[Mark44]

Homework Statement

Igor, the inchworm, is crawling along graph paper in a magnetic field. The intensity of the field at the point $(x,y)$ is given by $M(x,y)=3x^2+y^2+5000$. If Igor is at the point $(8,6)$, describe the curve along which he should travel if he wishes to reduce the field intensity as rapidly as possible.

Homework Equations

The Attempt at a Solution

The problem I have with this question is that I'm asked to describe a curve instead of direction.
Here are my steps so far: $$\nabla M(x,y)=(6x,2y)$$
$$\nabla M(8,6)=(48,12)$$
Since they want the most rapid reduction, I must go in the direction opposite to $(48,12)$, which if given in unit vector is $(\frac{-4}{\sqrt 17},\frac{-1}{\sqrt 17})$.

I have no idea how to come up with an equation of a curve based on this.

Any hints?

Thanks.

Igor is crawling on graph paper, which is flat. The curves of constant magnetic field intensity are all in the shape of ellipses. If he maintains the direction you calculated, he should be able to get to a location with a lower field intensity (although not by much, since the minimum intensity is 5000 whatevers).

 

[LCKurtz]

Igor is crawling on graph paper, which is flat. The curves of constant magnetic field intensity are all in the shape of ellipses. If he maintains the direction you calculated, he should be able to get to a location with a lower field intensity (although not by much, since the minimum intensity is 5000 whatevers).

But that direction is only correct at that point. To get the curve on which it must travel you need the orthogonal trajectory through that point. Find the necessary slope $y'$ in terms of $y$ and $x$ and solve the resulting differential equation.

 

[Mark44]

But that direction is only correct at that point. To get the curve on which it must travel you need the orthogonal trajectory through that point. Find the necessary slope $y'$ in terms of $y$ and $x$ and solve the resulting differential equation.

I stand corrected. What I said would be applicable if the level curves were circles, but that isn't the case here.
Thanks for the correction.