The question is:
Find a unit vector in the direction in which f(r,θ)=e^(-r)sinθ decreases most rapidly at P(0, π/3) and determine the rate of range of f at P in that direction.
I know usually it would be as simple as finding the gradient. Taking the negative gradient. Normalize it (divide by its length to reduce its length to 1). But now obviously the function is given in cylindrical coordinates. I have tried converting the function to rectangular coordinates, but the function is not defined at that point in rectangular coordinates (as r=0 causes both x and y to be zero). I have found the equation for the gradient in cylindrical coordinates on Wikipedia, but I end up with 1/r in the partial derivative with respect to θ, which makes the gradient undefined. Any ideas?
Answers & Comments
Verified answer
The problem you are running into is that there is really no such point as (0, pi/3)
in cylindrical coordinates. The origin cannot be associated with any particular angle.
If I think of a very small circle around the origin,
the "e^(-r)" factor will have a constant value quite close to 1.
The sin(theta) factor will have a value changing from 0 along the x-axis
to 1 along the y-axis to 0 along the negative x-axis and -1 along the negative y-axis.
At any point on such a circle, other than on the y-axis,
the direction of most rapid change will be tangent to the circle.
If one were to consider a point (epsilon, pi/3),
which is 60 degrees "above" the x-axis,
the direction of most rapid decrease would be clockwise around the circle,
and 30 degrees "below" the x-direction.
In rectangular coordinates, the unit vector would be i sqrt(3)/2 - j/2.
That's probably the answer your teacher wants...at least, I hope so.
Note that d(e^(-r))/dr is zero at the origin,
so that only the variation with respect to theta is important.
devide the vector by way of its importance. importance of ai +bj +ck is [ (a)^2 + (b)^2 + (c)^2 ]^a million/2. so which you may desire to devide the vector by way of [ (2)^2 + (-4)^2 + (a million)^2 ]^a million/2 = (21)^a million/2