Directional Derivatives
Directional Derivatives The Question Suppose that you leave the point (a, b) moving with velocity ~v = v , v . Suppose further h 1 2i that the temperature at (x, y) is f(x, y). Then what rate of change of temperature do you feel? The Answers Let’s set the beginning of time, t = 0, to the time at which you leave (a, b). Then at time t you are at (a + v1t, b + v2t) and feel the temperature f(a + v1t, b + v2t). So the change in temperature between time 0 and time t is f(a + v1t, b + v2t) f(a, b), the average rate of change of temperature, −f(a+v1t,b+v2t) f(a,b) per unit time, between time 0 and time t is t − and the instantaneous rate of f(a+v1t,b+v2t) f(a,b) change of temperature per unit time as you leave (a, b) is limt 0 − . We apply → t the approximation f(a + ∆x, b + ∆y) f(a, b) fx(a, b) ∆x + fy(a, b) ∆y − ≈ with ∆x = v t and ∆y = v t. In the limit as t 0, the approximation becomes exact and we have 1 2 → f(a+v1t,b+v2t) f(a,b) fx(a,b) v1t+fy (a,b) v2t lim − = lim t 0 t t 0 t → → = fx(a, b) v1 + fy(a, b) v2 = fx(a, b),fy(a, b) v , v h i·h 1 2i The vector fx(a, b),fy(a, b) is denoted ~ f(a, b) and is called “the gradient of the function h i ∇ f at the point (a, b)”.
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