Question 04 (b)
The temperature at a point (x,y,z) is given by
where T is measured in centigrade and x,y,z in metres.
In which direction does the temperature decrease most rapidly?
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Recall that for two vectors
How would we choose to make this dot product the smallest? How can we relate this idea to the directional derivative?
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
For any given unit vector , the rate of change of temperature at P in direction is the scalar
where θ is the angle between the given vectors. (Recall that .) For optimum decrease, we should make the directional derivative as small as possible. Since our only degree of freedom is the angle above and we know that
we should arrange by choosing θ=π, i.e., by making parallel to :
This is the direction of most rapid temperature decrease.