# Equation of a curve in 3 dimensions

1. Feb 6, 2012

### sdevoe

1. The problem statement, all variables and given/known data
A heat-seeking missile is located at (2,-3) on a plane. The temperature function is
given by T(x; y) = 20-4x^2-y^2. Find the equation of the curve along which the
missile travels, if it continuously moves in the direction of maximum temperature
increase. Express your answer in the form x = f(y). Show the calculations.

2. Relevant equations

T(x; y) = 20-4x^2-y^2

3. The attempt at a solution
I know the missile will travel along the direction of the gradient. The gradient with respect to x is -8x and the gradient with respect to y is -2y. The problem I'm having is getting the equation in terms of x. My only idea is to take δx(2,-3)(x-2)+δy(2,-3)(y+3)=0 and solve for x where x and δy are the gradients with respect to x and y. Is that correct?

2. Feb 6, 2012

### ehild

The missile moves in the direction of the temperature gradient at any point of its path. That means that its velocity points in the direction of the gradient vector. But the velocity is tangent to the path. How do you get the tangent of a curve y=f(x)?

ehild