Directional Derivative, Gradient and Level Set
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1 Directional Derivative, Gradient and Level Set Liming Pang 1 Directional Derivative Te partial derivatives of a multi-variable function f(x, y), f f and, tell us te rate of cange of te function along te x-axis and y-axis respectively. But in general wat about te rate of cange in oter directions? On te xy-plane, eac direction can be represented by a unit vector u. We are going to define te directional derivative of a function z = f(x, y) at (x 0, y 0 ) in te direction u: On te xy-plane, consider te line l passing troug (x 0, y 0 ) and parallel to te unit vector u. Passing troug te line l, tere is a unique vertical plane α, and α intersects te grap of z = f(x, y) along a curve C, so C projects to l on te xy-plane. If we start at (x 0, y 0 ) and travel along u direction for a distance, arriving at (x, y). Ten te vector wit initial point (x 0, y 0 ) and terminal point (x, y) is (x x 0, y y 0 ) = u. Since u is a unit vector, let u = (a, b), were a 2 + b 2 = 1. So: (x x 0, y y 0 ) = u = (a, b) wic implies x = x 0 + a, y = y 0 + b so te rate of cange of f along u at (x 0, y 0 ) is D u f(x 0, y 0 ) = lim 0 f(x 0 + a, y 0 + b) f(x 0, y 0 ) We call it te directional derivative of f at (x 0, y 0 ) in te direction of u. Indeed tere is a faster way to evaluate te directional derivative. 1
2 Teorem 1. If f is a differentiable function of x and y, ten f as a directional derivative in te direction of any unit vector u = (a, b) and D u f(x 0, y 0 ) = a f Proof. Define g() = f(x 0 + a, y 0 + b). We get g (0) = lim 0 g() g(0) On te oter and, by te cain rule, = lim 0 f(x 0 + a, y 0 + b) f(x 0, y 0 ) = D u f(x 0, y 0 ) g () = f x (x 0 + a, y 0 + b) d(x 0 + a) + f d y (x 0 + a, y 0 + b) d(y 0 + b) d = a f x (x 0 + a, y 0 + b) + b f y (x 0 + a, y 0 + b) wen = 0, we get g (0) = a f Combining te above results, we conclude D u f(x 0, y 0 ) = a f 2
3 Remark 2. If te unit vector u forms an angle θ wit te positive x-axis, ten u = (cos θ, sin θ). We can compute te directional derivative by D u f(x 0, y 0 ) = f x (x 0, y 0 ) cos θ + f sin θ Example 3. Find te directional derivative D u if f(x, y) = x 3 3xy + 4y 2, and u is te unit vector given by angle θ = π 6. Wat is D uf(1, 2)? D u f(x, y) = f x (x, y) cos π 6 + f y (x, y) sin π 3 6 = (3x2 3y) 2 + ( 3x + 8y)1 2 and D u f(1, 2) = Gradient In te previous section, we ave seen tat if a unit vector u = (a, b), te directional derivative of f along u is given by D u f(x, y) = a f (x, y) + b f (x, y) We can rewrite it in te following form as a dot product: D u f(x, y) = ( f f (x, y), (x, y)).(a, b) Definition 4. Te gradient of a function f(x, y) is f(x, y) = ( f f (x, y), (x, y)) By tis definition, we can write D u f(x, y) = f(x, y). u Example 5. Find te gradient of te function f(x, y) = x 2 y 3 4y at (2, 1), and find te directional derivative in te direction of te vector v = (2, 5). f(x, y) = ( f f (x, y), (x, y)) = (2xy3, 3x 2 y 2 4), so f(2, 1) = ( 4, 8). 3
4 Note tat v is not a unit vector, so we first compute te unit vector in te direction of v, wic is v = ( 2 5 v 29, 29 ). So D v f(2, 1) = f(2, 1). v v v 2 5 = ( 4, 8).(, ) = Teorem 6. If f(x, y) is a differentiable function, ten te maximum value of D u f(x 0, y 0 ) is f(x 0, y 0 ), and it is acieved wen u = f(x 0,y 0 ) f(x 0,y 0 ) Proof. D u f(x 0, y 0 ) = f(x 0, y 0 ). u = f(x 0, y 0 ) u cos θ = f(x 0, y 0 ) cos θ, so it acieves maximum wen θ = 0. Example 7. f(x, y) = xe y. In wic direction does f ave te maximum rate of cange at (2, 0)? Wat is te maximum rate of cange? f(x, y) = (e y, xe y ), so f(2, 0) = (1, 2). Te maximum rate of cange is along te direction of (1, 2) and te maximum rate of cange is (1, 2) = 5 3 Level Set Definition 8. f(x, y) is a function, and c is a real number. Define te set {(x, y) R 2 f(x, y) = c} to be te level set of f corresponding to te value c. In oter words, te level set of f(x, y) corresponding to te value c is te set of points (x, y) at wic te value of f is c. Example 9. f(x, y) = x + y. Te level set f(x, y) = 0 is te set of points satisfying x + y = 0, i.e. te straigt line y = x. Example 10. Let f(x, y) = x 2 + y 2, ten te level set f(x, y) = 1 is te unit circle centred at origin. Exercise 11. Sketc te level set of te function f(x, y) = xy = c. (You may need to discuss te cases c > 0, c = 0, c < 0 separately.) Example 12. In geology, te altitude is a function of te location on eart. People often use a topograpic map to describe altitude by sketcing some level sets, in wic case is often a curve. Wen te curves are denser, it means te area is steeper. If we travel along a level curve, te altitude doesn t cange. 4
5 It turns out tere is a close relation between level sets and gradient of a function: Teorem 13. f(x, y) is a differentiable function. If f(x, y) = c is a level set and f(x, y) (0, 0), ten f(x, y) is perpendicular to te tangent line of f(x, y) = c at (x, y). Coming back to te previous example regarding topograpic maps, te above teorem indicates tat if we want to climb onto a mountain in a sortest pat, we sould always go in te direction perpendicular to te level curve, since tis is te direction of te gradient. Remark 14. Te concept of level sets also applies to functions of more variables. For example, f(x, y, z) = x 2 + y 2 + z 2, te level set f(x, y, z) = 1 is te unit spere centred at origin. 5
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