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Multivariate Calculus - Bruce E. Shapiro

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78 LECTURE 11. GRADIENTS AND THE DIRECTIONAL DERIVATIVE<br />

Example 11.5 Simplify the expression<br />

∂f<br />

∂u = D uf(p)<br />

when u = k, i.e., a unit vector in the z-direction.<br />

Solution.<br />

(<br />

D u f(p) = u · ∇f = k · ∇f = k · i ∂f<br />

)<br />

∂x + j∂f ∂y + k∂f ∂z<br />

= k · i ∂f<br />

∂x + k · j∂f ∂y + k · k∂f ∂z<br />

= (0) ∂f<br />

∂x + (0)∂f ∂y + (1)∂f ∂z<br />

= ∂f<br />

∂z = D zf <br />

Theorem 11.3 The directional derivatives along a coordinate axis is the partial<br />

derivative with respect to that axis:<br />

D i f = ∂f<br />

∂x = f x<br />

D j f = ∂f<br />

∂y = f y<br />

D k f = ∂f<br />

∂z = f z<br />

Theorem 11.4 A function f(x, y) increases the most rapidly (as you move away<br />

from p=(x,y)) in the direction of the gradient. The magnitude of the rate of change is<br />

given by ‖∇f(p)‖. The function decreases most rapidly in the direction of −∇f(p),<br />

with magnitude −‖∇f(p)‖.<br />

Proof. The directional derivative, which gives the rate of change in any direction u<br />

at p is<br />

D u f = u · ∇f = |u| |∇f| cos θ<br />

Consider the set of all possible unit vectors u emanating from p. Then the maximum<br />

of the directional derivative occurs when cos θ = 1,i.e., when u is parallel to ∇f,<br />

and<br />

max(D u f(p)) = ‖∇f(p)‖<br />

Similarly the rate of maximum decrease occurs when cos θ = −1 <br />

Example 11.6 Find a unit vector in the direction in which f(x, y) = e y sin x increases<br />

most rapidly at p = (5π/6, 0)<br />

Revised December 6, 2006. Math 250, Fall 2006

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