# Vector Calc - Directional Derivative Question

at3rg0

## 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) = 4x^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

Gradient M = <8x, 2y>
To minimize, one must travel opposite the gradient

## The Attempt at a Solution

I got the gradient as <8x, 2y> (just the partial derivatives as a vector). Also, at (8,6), the gradient is <64, 12>, which is in the direction of the unit vector 16i + 3j/sqrt(16^2 +3^2). So, the direction he wants to travel is the negative version of that vector. However, the problem wants me to arrive at the formula (y^3 = 27x towards the origin). How do I go from the vector to the curve?

This TEX is driving me crazy. If anyone can help me by editing my post, that would be great.

$$\frac{dy}{dx}= \frac{\frac{dy}{dt}}{\frac{dx}{dt}}= \frac{1}{4}\frac{y}{x}$$.