Need Help With What SHOULD Be a Simple Directional Derivative?

Question

I've done questions harder than this with no issues, but I simply can't seem to get this one right. Can anyone lend a hand?

The temperature W is given by:

W = 50+xyz

Find the rate of change of temperature with respect to distance at point P(3,4,1) in the direction v<1,2,2>

This seems like it should be a fairly straightforward question. I take the partial derivatives with respect to xyz to get the vector <4,3,12>. I then get the unit vector of v, which is <1/3, 2/3, 2/3>. I take the length of the dot product of those to get 8.4. But the answer the book gives is 34/3, which I can't figure out. I'd be very grateful if someone could help me out here.

Johan · Accepted Answer

Well, your calculations are quite good and you were very close. Here's where it went wrong.

We can't take the "length of the dot product" of (4, 3, 12) and (1/3, 2/3, 2/3). The dot product is a scalar number, it's not a vector; so it doesn't have a length. The answer is much simpler: just calculate the dot product, and the result (whether positive or negative) is the requested solution.

(4, 3, 12) * (1/3, 2/3, 2/3) = 
= 4*1/3 + 3*2/3 + 12*2/3 
= 4/3 + 6/3 + 24/3
= 34/3

(If v had been (-1, -2, -2), i.e. in the opposite direction, then the correct answer would have been -34/3. See where you went wrong? If you had calculated the length of some vector, then your resulting answer would always have been a non-negative number; but there's no reason why the rate of change of temperature can't be negative.)

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Need Help With What SHOULD Be a Simple Directional Derivative?

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