Engineering Mathematics (4)

Transcription

1 Engineering Mathematics (4) Zhang, Xinyu Department of Computer Science and Engineering, Ewha Womans University, Seoul, Korea

2 Example With respect to parameter: s (arc length) r( t) acos t, asin t, ct r * s s s ( s) acos, asin, c a c a c a c

3 Example With respect to parameter: s (arc length) r( t) acos t, asin t, ct t 0 0 t r s dt a c dt t a c t a s c

4 Example With respect to parameter: s (arc length) s t a 2 c 2 r( t) acos t, asin t, ct r * s s s ( s) acos, asin, c a c a c a c

5 Gradient Directional Derivative

6 Gradient Definition f ( x, y, z) grad f f f f i j k x y z grad f f f f i j k x y z 6

7 Gradient Definition: the gradient of a given scalar function f (x,y,z) is the vector function defined by grad f f f f i j k x y z f (x,y,z) is differentiable 7

8 nabla or del Operator i j k x y z f f f f grad f i j k x y z 8

9 Example f ( x, y, z) 2x yz 3y 2 f 2, f z 6 y, f y x y z grad f 2i z 6y j yk f 2i z 6y j yk 9

10 Directional Derivative Definition f ( x, y, z) df ds lim f Q f P s 0 s DPQ f 10

11 Directional Derivative Direction: straight line r ( s) x( s) i y( s) j z( s) k p sb 0 Function f ( x, y, z) = f x( s), y( s), z( s) 11

12 Directional Derivative f ( x, y, z)= f x( s), y( s), z( s) Chain Rule df f dx f dy f dz ds x ds y ds z ds 12

13 Directional Derivative f dx f dy f dz x ds y ds z ds f f f i j k x y z dx dy dz i j k ds ds ds f f b r r( s) x( s) i y( s) j z( s) k p sb b 0 13

14 Directional Derivative D f f b b b f b is a unit vector 14

15 Directional Derivative If b is not a unit vector b D f b b f If the given vector is a a D f a a f 15

16 Example Find the directional derivative of f ( x, y, z) 2x 3y z at the point P: (2,1,3) in the direction of the vector a i 2k 16

17 Solution b D f b b f 17

18 Solution The gradient f ( x, y, z) 2x 3y z f 4xi 6y j 2z k at the point P: (2,1,3) f (2,1,3) 8i 6 j 6k i 2k Df(2,1,3) 8i 6 j 6k a

19 Direction Directional Derivative Point P 19

20 Surface Normal Vector A surface in space f ( x, y, z) C A curve on the surface r x( t) i y( t) j z( t) k f x( t), y( t), z( t) C 20

21 Surface Normal Vector At a point P, tangent vector r x( t) i y( t) j z( t) k Differentiating f dx f dy f dz f x( t), y( t), z( t) x dt y dt z dt f f f x y z x y z 21

22 Surface Normal Vector f f f x y z x y z f r 0 f ( x, y, z) C0 22

23 Surface Normal Vector Let f be a differentiable scalar function that represents a surface S: f(x,y,z)=c. Then if the gradient of f at a point P is not the zero vector, it is a normal vector of S at P f 23

24 Example Find the unit normal vector of z 2 4 x 2 y 2 at the point P: (1,0,2) 24

25 Solution The given surface is Let f ( x, y, z) 4 x y z x y z 0 25

26 Solution Then the normal vector is f 8xi 8y j 2z k at the point P: (1,0,2) f 8i 4k The unit normal vector is f 2 1 n i k f

27 Assignment Find the directional derivative of f ( x, y, z) x 3y 4z at the point P: (1,0,1) in the direction of the vector a i j k 27

28 Assignment Find the directional derivative of f ( x, y, z) e x cos y at the point P: (2,, 0) in the direction of the vector a 2i 3j 28

29 Assignment Find the unit normal vector for the surface z x 2 y 2 at the point P: (6,8,10) 29

30 Divergence and Curl of a Vector Field