Gradient and Directional Derivatives October 2013

Transcription

1 Gradient and Directional Derivatives October 2013

2 function of one variable: makes sense to talk about the rate of change function of several variables: rate of change depends on direction slope One direction in domain

3 function of one variable: makes sense to talk about the rate of change function of several variables: rate of change depends on direction z slope One direction in domain y x

4 function of one variable: makes sense to talk about the rate of change function of several variables: rate of change depends on direction z slope One direction in domain y x

5 function of one variable: makes sense to talk about the rate of change function of several variables: rate of change depends on direction z slope One direction in domain y x

6 function of one variable: makes sense to talk about the rate of change function of several variables: rate of change depends on direction z slope One direction in domain y x Two directions in domain

7 Partial Derivatives and Directions. Partial derivatives are the rates of change in different directions: f : rate of change in î direction x f : rate of change in ĵ direction y ( f z : rate of change in ˆk direction if f (x, y, z) function of 3 variables)

8 Directional Derivatives. If v is any nonzero vector, D v f = derivative of f with respect to v, given by D v f (a) = lim t 0 f (a + tv) f (a) t

9 Directional Derivatives. If v is any nonzero vector, given by D v f = derivative of f with respect to v, D v f (a) = lim t 0 f (a + tv) f (a) t E.g., if f = f (x, y), bv = h, k, then D v f (a, b) = lim t 0 f (a + th, b + tk) f (a, b) t

10 Directional Derivatives. If v is any nonzero vector, given by D v f = derivative of f with respect to v, D v f (a) = lim t 0 f (a + tv) f (a) t E.g., if f = f (x, y), bv = h, k, then D v f (a, b) = lim t 0 f (a + th, b + tk) f (a, b) t The directional derivative of f in the direction of v is D ev f, where e v = v, unit vector. v

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12 Examples: Directional Derivative. Examples: Dîf = f / x, Dĵf = f / y, (Dˆkf = f / z) If f (x, y) = x + y 2, what is D 4,5 f?

13 Examples: Directional Derivative. Examples: Dîf = f / x, Dĵf = f / y, (Dˆkf = f / z) If f (x, y) = x + y 2, what is D 4,5 f? Answer: D 4,5 f (a, b) = lim t 0 f (a + 4t, b + 5t) f (a, b) t = lim t 0 [(a + 4t) + (b + 5t) 2 ] [a + b 2 ] t = lim t 0 4t + 10bt + 25t 2 t = b. How can we find D v f more easily?

14 Gradient. Definition: If f is a scalar-valued function then the gradient of f, denoted grad f or f, is f f = x, f f or f = y x, f y, f z or f = f, f,..., f. x 1 x 2 x n

15 Gradient. Definition: If f is a scalar-valued function then the gradient of f, denoted grad f or f, is f f = x, f f or f = y x, f y, f z or f = f, f,..., f. x 1 x 2 x n Theorem D v f (a, b) = f (a,b) v.

16 Gradient. Definition: If f is a scalar-valued function then the gradient of f, denoted grad f or f, is f f = x, f f or f = y x, f y, f z or f = f, f,..., f. x 1 x 2 x n Theorem D v f (a, b) = f (a,b) v. Example: For f (x, y) = x + y 2, f = 1, 2y.

17 Gradient. Definition: If f is a scalar-valued function then the gradient of f, denoted grad f or f, is f f = x, f f or f = y x, f y, f z or f = f, f,..., f. x 1 x 2 x n Theorem D v f (a, b) = f (a,b) v. Example: For f (x, y) = x + y 2, f = 1, 2y. = f (a,b) = 1, 2b.

18 Gradient. Definition: If f is a scalar-valued function then the gradient of f, denoted grad f or f, is f f = x, f f or f = y x, f y, f z or f = f, f,..., f. x 1 x 2 x n Theorem D v f (a, b) = f (a,b) v. Example: For f (x, y) = x + y 2, f = 1, 2y. = f (a,b) = 1, 2b. = f (a,b) 4, 5 = b.

19 Gradient. Definition: If f is a scalar-valued function then the gradient of f, denoted grad f or f, is f f = x, f f or f = y x, f y, f z or f = f, f,..., f. x 1 x 2 x n Theorem D v f (a, b) = f (a,b) v. Example: For f (x, y) = x + y 2, f = 1, 2y. = f (a,b) = 1, 2b. = f (a,b) 4, 5 = b. Therefore D 4,5 f (a, b) = b.

20 Example: Directional Derivative. Example: f (x, y) = e xy+x2, v = 12, 5, P = ( 2, 2). What is the directional derivative of f at P in the direction of v? Answer:

21 Example: Directional Derivative. Example: f (x, y) = e xy+x2, v = 12, 5, P = ( 2, 2). What is the directional derivative of f at P in the direction of v? Answer: v = = 13

22 Example: Directional Derivative. Example: f (x, y) = e xy+x2, v = 12, 5, P = ( 2, 2). What is the directional derivative of f at P in the direction of v? Answer: v = = 13 Let u = v v = 12 13, 5 13

23 Example: Directional Derivative. Example: f (x, y) = e xy+x2, v = 12, 5, P = ( 2, 2). What is the directional derivative of f at P in the direction of v? Answer: v = = 13 Let u = v v = 12 13, 5 13 We want to find D u f (P).

24 Example: Directional Derivative. Example: f (x, y) = e xy+x2, v = 12, 5, P = ( 2, 2). What is the directional derivative of f at P in the direction of v? Answer: v = = 13 Let u = v v = 12 13, 5 13 We want to find D u f (P). This is f P u

25 Example: Directional Derivative. Example: f (x, y) = e xy+x2, v = 12, 5, P = ( 2, 2). What is the directional derivative of f at P in the direction of v? Answer: v = = 13 Let u = v v = 12 13, 5 13 We want to find D u f (P). This is f P u f = e xy+x2 (y + 2x), e xy+x2 x

26 Example: Directional Derivative. Example: f (x, y) = e xy+x2, v = 12, 5, P = ( 2, 2). What is the directional derivative of f at P in the direction of v? Answer: v = = 13 Let u = v v = 12 13, 5 13 We want to find D u f (P). This is f P u f = e xy+x2 (y + 2x), e xy+x2 x so f P = e 4+4 (2 4), e 4+4 ( 2)

27 Example: Directional Derivative. Example: f (x, y) = e xy+x2, v = 12, 5, P = ( 2, 2). What is the directional derivative of f at P in the direction of v? Answer: v = = 13 Let u = v v = 12 13, 5 13 We want to find D u f (P). This is f P u f = e xy+x2 (y + 2x), e xy+x2 x so f P = e 4+4 (2 4), e 4+4 ( 2) = 2, 2

28 Example: Directional Derivative. Example: f (x, y) = e xy+x2, v = 12, 5, P = ( 2, 2). What is the directional derivative of f at P in the direction of v? Answer: v = = 13 Let u = v v = 12 13, 5 13 We want to find D u f (P). This is f P u f = e xy+x2 (y + 2x), e xy+x2 x so f P = e 4+4 (2 4), e 4+4 ( 2) = 2, 2

29 Example: Directional Derivative. Example: f (x, y) = e xy+x2, v = 12, 5, P = ( 2, 2). What is the directional derivative of f at P in the direction of v? Answer: v = = 13 Let u = v v = 12 13, 5 13 We want to find D u f (P). This is f P u f = e xy+x2 (y + 2x), e xy+x2 x so f P = e 4+4 (2 4), e 4+4 ( 2) = 2, 2 Hence D u f (P) = f P u = 2, , 5 13 = =

30 Worksheet. Worksheet #2 (skip #1 for now)

31 Clicker Question: Let f P = 3, 5. What is the sign of the directional derivative of f in the direction of the vector and in the direction of the vector? A. positive and positive B. positive and negative C. negative and positive D. negative and negative receiver channel: 41 session ID: bsumath275

32 Rate of change along path. Say r(t) is a path and f is a scalar-valued function. = f (r(t)) is a scalar-valued function of a single parameter. What is its rate of change? What is d dt f (r(t))? Function notation: r : R R 2, f : R 2 R, so f r : R R

33 Rate of change along path. Say r(t) is a path and f is a scalar-valued function. = f (r(t)) is a scalar-valued function of a single parameter. What is its rate of change? What is d dt f (r(t))? Function notation: r : R R 2, f : R 2 R, so f r : R R Intuitive idea: At a given value of t, r(t) is going in the direction of its tangent vector, r (t). So the rate of change is just the rate of change of f, in the direction of r (t).

34 Chain Rule For Paths. d dt f (r(t)) = (D r (t)f ) }{{} direction of r (t) (r(t)) }{{} at the point r (t) = f r(t) r (t)

35 Chain Rule For Paths. d dt f (r(t)) = (D r (t)f ) }{{} direction of r (t) (r(t)) }{{} at the point r (t) Theorem (Chain Rule for Paths) = f r(t) r (t) If r is a differentiable path and f is a differentiable scalar-valued function then d dt f (r(t)) = f r(t) r (t).

36 Chain Rule For Paths. d dt f (r(t)) = (D r (t)f ) }{{} direction of r (t) (r(t)) }{{} at the point r (t) Theorem (Chain Rule for Paths) = f r(t) r (t) If r is a differentiable path and f is a differentiable scalar-valued function then d dt f (r(t)) = f r(t) r (t). Example: f (x) = 2x 7y + 5z, r(t) = t 2, t 3, t 4, at t = 2: d dt f (r(t)) t=2 = f (4,8,16) 2t, 3t 2, 4t 3 t=2 = 2, 7, 5 4, 12, 32 = 84.

37 Worksheet. Worksheet #1

38 Steepest Ascent. Which direction u has the maximum rate of change of f?

39 Steepest Ascent. Which direction u has the maximum rate of change of f?

40 Steepest Ascent, continued. Which direction u has the maximum rate of change of f? Out of all unit vectors u, which one maximizes D u f (P)?

41 Steepest Ascent, continued. Which direction u has the maximum rate of change of f? Out of all unit vectors u, which one maximizes D u f (P)? D u f (P) = f P u = f P u cos θ f P : constant (doesn t depend on u) u = 1 θ = angle between f P and u

42 Steepest Ascent, continued. Which direction u has the maximum rate of change of f? Out of all unit vectors u, which one maximizes D u f (P)? D u f (P) = f P u = f P u cos θ f P : constant (doesn t depend on u) u = 1 θ = angle between f P and u Therefore Rate of change maximized when cos θ = 1 = θ = 0 = u = direction of f P Minimum: cos θ = 1, u = opposite direction to f P Rate of change is zero if θ = π/2, i.e., u f P The gradient f P points uphill.

43 Clicker Question: The table below gives values of the function f (x, y) which is smoothly varying around the point (3, 5). Estimate the vector f (3,5). If the gradient vector is placed with its tail at the origin, which quadrant does the vector point into? x y A. I B. II C. III D. IV E. Need more information. receiver channel: 41 session ID: bsumath275

44 Level Surfaces. Surface defined by F (x, y, z) = k is a level surface of F :

45 Level Surfaces. Surface defined by F (x, y, z) = k is a level surface of F :

46 Tangent Planes to Level Surfaces. S = level surface defined by F (x, y, z) = k, P = point on S such that F is differentiable at P and F P 0. What is the tangent plane to S at P?

47 Tangent Planes to Level Surfaces. S = level surface defined by F (x, y, z) = k, P = point on S such that F is differentiable at P and F P 0. What is the tangent plane to S at P? If S is a graph, z f (x, y) = 0, equation is local linearization.

48 Tangent Planes to Level Surfaces. S = level surface defined by F (x, y, z) = k, P = point on S such that F is differentiable at P and F P 0. What is the tangent plane to S at P? If S is a graph, z f (x, y) = 0, equation is local linearization. We know tangent plane passes through P. Need normal vector.

49 Tangent Planes to Level Surfaces. S = level surface defined by F (x, y, z) = k, P = point on S such that F is differentiable at P and F P 0. What is the tangent plane to S at P? If S is a graph, z f (x, y) = 0, equation is local linearization. We know tangent plane passes through P. Need normal vector. Answer: F P is a normal vector to the tangent plane. Why?

50 Geometrical Explanation. Say r(t) is any curve on S, through P. F (r(t)) = k, constant so d F (r(t)) = 0, dt so r (t) F P = 0 so F P r (t). That applies to all the tangent vectors so it covers the whole tangent plane.

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53 Worksheet. Worksheet #3

54 Clicker Question: The figure below shows the contour diagram of f (x, y). Which of the vectors shown could be f at the point where the tail is attached? 3 1 A B 4 2 C D receiver channel: 41 session ID: bsumath275