Question 04 (a)
The temperature at a point (x,y,z) is given by
where T is measured in centigrade and x,y,z in metres.
Find the rate of change of temperature at the point P(1,2,-1) in the direction toward the point (1,1,0).
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
Recall that the directional derivative of a scalar function f in the direction is
where is a unit vector.
What is the direction vector that we need?
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
For , the gradient is
At the point , we have Each component has units of .
The vector from to is
A unit vector in this direction is
Each component is dimensionless because the units of the vector and its magnitude are the same.
The rate of change of T at P in direction is
Its units are .