Suppose the temperature at point (x,y,z) in space is given by the following function:

f(x,y,z) = 1/(1 + x^2 + y^2 + z^2)

A bee is located at the point P = (1,1,1) in space.

Find the direction (as a three-dimensional vector) in which the initial rate of change in the temperature is 0.
Doesn't this show a sphere centered at the origin with radius sqrt3?

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Yahoo! Answers

Find the gradient, (del(f)/delz(x),del(f)/delz(y),del(f)/delz(z))
Then you want some vector displacement to result in no change, e.g.

ds = (dx, dy, dz)

Then you want df = ds(dot)gradient = 0

You can then get a function of dx, dy, dz = 0

You will be moving in a linear direction, so you can safely assume that one parameter, x say, can describe this motion, then dy = adx, and dz = bdx.

You then have a function of a and b. Without further constraints, this is unsolvable, so let us take the vector as having a unit length, or (1 + a^2 +b^2)=1. You now have enough equations to solve for a and b, and therefore the unit vector in which direction df is initially zero.

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