1. ## Working with gradient vectors

Here's the problem:

"Let lambda = a+b+c with (a,b,c)= (2,8,5)

Captain Astro is in trouble near the sunny side of Mercury. The temperature in degrees of her ship’s hull when she is at location (x, y, z) will be given by

T(x,y,z) = e^(-ax^2-by^2-cz^2)

where x, y, and z are measured in “space units”. Unfortunately, the metal of the hull will crack if the absolute value of the instantaneous rate of change of the temperature in a given direction is greater than sqrt(6)*e^(-lambda) degrees per space unit. Describe the set of possible directions in which she may proceed to leave the point (1,1,1) to decrease the temperature without cracking the hull of the ship. It would be helpful to Captain Astro to give a description of the set of possible directions in terms of a range of angles in degrees relative to some fixed vector (for example, 'go in any direction that makes an angle with the vector v = 2.2i-3.1j-5.72k that is between 27º and 67º.')"

I have no idea how to do solve this. Help?

2. I haven't solved the problem myself, but I recommend finding the directional derivative of $\displaystyle T(x,y,z)$ at the point $\displaystyle (1,1,1)$ in the direction of some unit vector $\displaystyle \vec u = <u_1,u_2,u_3>$.

Then, use the restriction that "the metal of the hull will crack if the absolute value of the instantaneous rate of change of the temperature in a given direction is greater than $\displaystyle \sqrt{6}e^{-15}$ degrees per space unit" to construct an inequality.

Let me know how it goes.

3. So, just to make sure that I have the process right, at least:

I'm going find the direction derivative of T(x,y,z) at (1,1,1) and then, to find it in the direction of the unit vector u, I do u * T(1,1,1)?

And then, how would I construct the inequality?

4. What do you mean by "direction derivative of T(x,y,z) at (1,1,1)"? I ask because you separate the process of finding the directional derivative of T(x,y,z) at (1,1,1) and the process of finding it in the direction of the unit vector u. This doesn't make sense to me because a directional derivative isn't a directional derivative without a direction, so what exactly are you finding when you find the "direction derivative of T(x,y,z) at (1,1,1)" before you find it in the direction of the unit vector u? I just want to clarify before I say your process is correct or incorrect.

In any case, I will provide a definition of the directional derivative of the function f(x,y,z) at the point (a,b,c) in the direction of the unit vector u:

$\displaystyle \vec{D_u}f(a,b,c) = f_x(a,b,c)u_1 + f_y(a,b,c)u_2 + f_z(a,b,c)u_3$

From the above definition, to find the directional derivative 1) evaluate the partial derivatives at the point (a,b,c) and then 2) multiply the partial derivatives with the components of the unit vector. Does this agree with the process you described? If so, then your process is correct indeed.

Regarding the construction of the inequality, let's call $\displaystyle \vec{D_u}T(1,1,1)$ the directional derivative of T at the point (1,1,1) in the direction of the unit vector u. Then, $\displaystyle |\vec{D_u}T(1,1,1)| \leq \sqrt{6}e^{-15}$ else the metal of the hull will crack.

Once you find $\displaystyle \vec{D_u}T(1,1,1)$ by using the above definition, substitute it into the inequality. Any solutions are unit vectors pointing in directions in which the ship can safely travel.

5. Originally Posted by Rumor
Here's the problem:

"Let lambda = a+b+c with (a,b,c)= (2,8,5)

Captain Astro is in trouble near the sunny side of Mercury. The temperature in degrees of her ship’s hull when she is at location (x, y, z) will be given by

T(x,y,z) = e^(-ax^2-by^2-cz^2)

where x, y, and z are measured in “space units”. Unfortunately, the metal of the hull will crack if the absolute value of the instantaneous rate of change of the temperature in a given direction is greater than sqrt(6)*e^(-lambda) degrees per space unit. Describe the set of possible directions in which she may proceed to leave the point (1,1,1) to decrease the temperature without cracking the hull of the ship. It would be helpful to Captain Astro to give a description of the set of possible directions in terms of a range of angles in degrees relative to some fixed vector (for example, 'go in any direction that makes an angle with the vector v = 2.2i-3.1j-5.72k that is between 27º and 67º.')"

I have no idea how to do solve this. Help?
The directional derivative of $\displaystyle T(x,y,z)$ in direction $\displaystyle \widehat{\bf{u}}$ is $\displaystyle \nabla T.\widehat{\bf{u}}$.

You are asked to find unit vectors $\displaystyle \widehat{\bf{u}}$ such that:

$\displaystyle \left. \nabla T.\widehat{\bf{u}}\;\biggr|_{(x,y,z)=(1,1,1)}<\sqr t{6} \; e^{-\lambda}$.

CB