Gradient and finding the direction of maximum rate of change

1. Mar 28, 2017

Taylor_1989

1. The problem statement, all variables and given/known data
Hi guys, it a very simple question, but it causing me a great deal of confusion. The questions are as follows:

So I worked out the ans for one which I have displayed below. But what I don't understand is what they want from the second question. Because the way I see it, the direction of maximum rate of change is the gradient itself. Am I missing something here because I am really lost.

2. Relevant equations

3. The attempt at a solution
1: $\nabla f=\frac{-xi}{(x^2+y^2+z^2)^3} -\frac{yj}{(x^2+y^2+z^2)^3}-\frac{zk}{(x^2+y^2+z^2)^3}$

$\nabla f = le^{lx+my+nz}i+me^{lx+my+nz}j+ne^{lx+my+nz}k$

2: I make the direction as follows: $(-xi-yj-zk)$ & $li+mj+nk$

Is this correct or have I miss understood the question?

2. Mar 28, 2017

BvU

Check the exponent in the denominator in the first f in part 1).
For part 2: I agree with you : $-\bf\hat r$ and $(l,m,n)$.

 perhaps for 'direction' the exercise composer wants to see $\bf \hat r$ ?

3. Mar 28, 2017

PeroK

The direction of maximum rate of change is the direction of the gradient. That generally means a unit vector,

4. Mar 28, 2017

Taylor_1989

Thank you I did not put the 3/2 in thanks.

5. Mar 28, 2017

Taylor_1989

when u say unit vector you mean $\frac{\nabla f}{|\nabla f|}$?

6. Mar 28, 2017

PeroK

In this case, yes. If you look at the difference between the gradient and the direction if the gradient for the second function - the exponential - you'll see the point.