LectureDerivatives of Vector Function

Derivatives of Vector Function Instructor by Assist.Prof.Siriwan Wasukree LOGO

Derivatives of Vector Function List of Topics in this class 1. Vector function of one variable 2. Limit of vector function 3. Continuity of vector function 4. Derivative of vector function 5. Partial derivatives of vector function 6. Operator del , gradient , divergence and curl www.themegallery.com Company Logo

1. Vector function of one variable FLedt eff1i(nt)e,dfb2(yt) and f3(t) are functions of one variable t F ( t )  f1 (t) i  f2 (t)j  f3 (t)k F or F(t) is called that a vector function of one variable Let 3421F....(FFFt)((((tttt)))a)AnGdGGi(((sGttt)))v(iiietsss)cvsvtaoceerracceltftaoouvrrrnefcfcfuuuttnoinnocrccntfttiiiuooonnnnctions and (t) is scalar function www.themegallery.com Company Logo

1. Vector function of one variable Ex 1 Give (t)  t2  2t  3 , F(t)  ti  t2 j  3k and G (t)  t( i  j k) Find 1. (t)F(t) 2. F(t)  G(t) www.themegallery.com Company Logo

1. Vector function of one variable Ex 1 Give (t)  t2  2t  3 , F(t)  ti  t2 j  3k and G (t)  t( i  j k) Find 3. F(t)  G(t) www.themegallery.com Company Logo

2. Limit of vector function Let F(t) is vector function Here is the limit of a vector function. F(t)  f1(t)i  f2 (t)j  f3 (t)k [ ] [ ] [ ]tlima F(t)  tlima f1(t) i  tlima f2 (t) j  tlima f3(t) k So, all that we do is take the limit of each of the components functions and leave it as a vector. www.themegallery.com Company Logo

2. Limit of vector function Ex 2 Compute tlim1F(t) when F(t)  t3i  e2t j  3cos(3t  3)k www.themegallery.com Company Logo

2. Limit of vector function Ex 3 Let A (t)  2cos t i  2sin t j  3t k and B (t)  t2i  t j  5 k Find tlim0[A(t)  B(t)] www.themegallery.com Company Logo

3. Continuity of vector function A vector function F(t) is continuous at the point t = c if and only if 1. F(c) is defined at t = c 2. tlimc F(t) exists, 3. tlimc F(t)  F(c) www.themegallery.com Company Logo

3. Continuity of vector function Ex 4 Consider that F(t)  sin t i  cos t j  t k is continuous at t =  www.themegallery.com Company Logo

4. Derivative of vector function Derivative of vector function F(t) denoted by ddt F(t) or F /(t) ddt F(t)  ltim0 F(tt)t F(t) exists Here is the derivative of a vector function. F(t)  f1(t) i  f2 (t)j  f3 (t)k d F ( t )  d f1 ( t )i  d f2 (t)j  d f3 (t)k dt dt dt dt So, all that we do is take the derivative of each of the components functions and leave it as a vector. www.themegallery.com Company Logo

4. Derivative of vector function Ex 5 Giving that x(t)  e2t , y(t)  sin3t and z(t)  4  t2 Find a vector function F(t) and derivative of vector function at t = 0 www.themegallery.com Company Logo

4. Derivative of vector function Most of the basic facts that we know about derivatives still hold however, just to make it clear here are some facts about derivatives of vector functions. 1. ddt c  0 2. ddt [F(t)  G(t)]  ddt F(t)  ddt G(t) 3. ddt [F(t)  G(t)]  F(t)  ddt G(t)  G(t)  ddt F(t) 4. ddt [F(t) G(t)]  F(t) ddt G(t)  ddt F(t) G(t) 5. ddt [(t)F(t)]  (t) ddt F(t)  F(t) ddt (t) 6. ddt [F(t)  G(t)  H(t)]  ddt F(t)  G(t)  H(t)  F(t)  ddt G(t)  H(t)  F(t)  G(t)  ddt H(t) 7. ddt [F(t)[G(t) H(t)]]  ddt F(t)[G(t) H(t)]  F(t)[ddt G(t)  H(t)]  F(t)[G(t) ddt H(t)] www.themegallery.com Company Logo

4. Derivative of vector function Ex 6 Giving that (t)  t2  5 , F(t)  ti  t2j  3k and G (t)  ti  sin tj  et k Find 1. ddt [(t)F(t)] 2. ddt [F(t)  G(t)] www.themegallery.com Company Logo

4. Derivative of vector function Ex 6 Giving that (t)  t2  5 , F(t)  ti  t2j  3k and G (t)  ti  sin tj  et k Find 3. ddt [F(t)  G(t)] www.themegallery.com Company Logo

4. Derivative of vector function Ex 6 Giving that (t)  t2  5 , F(t)  ti  t2j  3k and G (t)  ti  sin tj  et k Find 4. ddt [F(t) G(t)] www.themegallery.com Company Logo

4. Derivative of vector function Ex 7 Let A (t)  e2ti  e3t j Show that A // (t)  6A(t)  5A /(t) www.themegallery.com Company Logo

Google Jamboard • Click the google link Jamboard in Line group. • Find your name on the Jamboard page and complete the action assigned to it. • You can write responses on the Jamboard, or from somewhere else and insert pictures. • When finished sharing on Jamboard, we will do one more activity. www.themegallery.com Company Logo

Today’activities • Do activity1 in Classroom. • Submit work at Classroom Week 8 • Click the google form link Activity1 • When you have successfully submitted the activity, you can leave the meet classroom. • See you next Thursday. www.themegallery.com Company Logo

5. Partial derivatives of vector function F(x, y,z) are vector function of x , y and z can be written as F(x, y, z)  f1(x, y, z)i  f2 (x, y, z)j  f3 (x, y, z)k or, write short F  f1i  f2 j  f3 k F x Partial derivatives of F with respect to x is  f1 i  f 2 j  f3 k x x x Partial derivatives of F with respect to y is F  f1 i  f 2 j  f3 k y y y y Partial derivatives of F with respect to z is F  f1 i  f 2 j  f3 k z z z z www.themegallery.com Company Logo

5. Partial derivatives of vector function Higher Partial Derivatives  2 F  x ( Fx )  Fxx  2 F  y ( Fy )  Fyy x 2 y 2 y2Fx  y ( Fx )  Fxy x2Fy  x ( Fy )  Fyx  3 F  y [x ( Fx )]  Fxxy yx 2 www.themegallery.com Company Logo

5. Partial derivative of vector function Ex 8 If   e xy iˆ  x sin ( y) kˆ Find F(x, y) 1. Fx 2. Fy 3.  2 F x 2 www.themegallery.com Company Logo

5. Partial derivative of vector function Ex 8 If   e xy iˆ  x sin ( y) kˆ Find F(x, y) 4. Fx  Fy www.themegallery.com Company Logo

5. Partial derivative of vector function Ex 9 If (x, y, z)  xyz , F(x,y,z)  x2i  y2j  z2 k and G (x,y,z)  xyi  yzj  xzk Find 1. x (F) at point (1,1,2) www.themegallery.com Company Logo

5. Partial derivative of vector function Ex 9 If (x, y, z)  xyz , F(x,y,z)  x2i  y2j  z2 k and G (x,y,z)  xyi  yzj  xzk Find 2. y (F  G ) www.themegallery.com Company Logo

5. Partial derivative of vector function Ex 9 If (x, y, z)  xyz , F(x,y,z)  x2i  y2j  z2 k and G (x,y,z)  xyi  yzj  xzk Find 3. x2y (F  G ) www.themegallery.com Company Logo

5. Partial derivative of vector function Ex 9 If (x, y, z)  xyz , F(x,y,z)  x2i  y2j  z2 k and G (x,y,z)  xyi  yzj  xzk Find 4. x2z (F G ) at point (1,2,0) www.themegallery.com Company Logo

6. Operator del , gradient , divergence and curl Operator del Think of the symbol  as an “operator” on a function that produces a vector is given by   i   j   k x y z Gradient If the partial derivatives of (x,y,z) are defined, then the Gradient of  , denoted by grad  or  (pronounced “ del phi ”) grad     (  i   j   k) x y z   i   j   k x y z www.themegallery.com Company Logo

6. Operator del , gradient , divergence and curl Ex 10 Find  for (x, y , z )  x 2  xz  y 2  yz at point (1,1,3) www.themegallery.com Company Logo

6. Operator del , gradient , divergence and curl Ex 11 Find f for f (x, y)  arc tan(2xy) at point (1,3) www.themegallery.com Company Logo

6. Operator del , gradient , divergence and curl divergence If F(x, y, z)  f1i  f2 j  f3 k is a vector function dthievepragretniacledoefrivFatdivenesotaetdpboyint (x,y,z) are defined, then the div F or   F div F    F  (  i   j   k)  (f1i  f2 j  f3 k) x y z  f1  f2  f3 x y z div F is a scalar function Note that www.themegallery.com Company Logo

6. Operator del , gradient , divergence and curl Ex 12 Find div F for F(x, y, z)  x2zi  2y3zj  x2 cos(yz)k www.themegallery.com Company Logo

6. Operator del , gradient , divergence and curl Ex 13 Giving that (x, y , z )  3x 2 y 2  yz 2 Find div(grad ) www.themegallery.com Company Logo

6. Operator del , gradient , divergence and curl Ex 14 Giving that (x, y,z )  2x3z  12x2y  4y 3  2xz 3 Find 2  www.themegallery.com Company Logo

6. Operator del , gradient , divergence and curl curl If F(x, y, z)  f1i  f2 j  f3 k is a vector function dthievepragretniacledoefrivFatdivenesotaetdpcbouyirnltF(x,yo,rz) are defined, then the   F curl F    F  i j k  x y z f1 f2 f3  ( f3  f2 )i  ( f3  f1 )j  ( f2  f1 )k y z x z x y www.themegallery.com Company Logo

6. Operator del , gradient , divergence and curl Ex 15 GFiinvdingcuthrlaFt F(x, y, z)  x3yz 2 i  x 2 y 2 z 2 j  x 2 yz3 k at point (1,3,1) www.themegallery.com Company Logo

Homework Today • Do activity2 • Submit work at Classroom Week 9 • Click the google form link Activity2 • Deadline for submission, Monday March 15. • See you next Thursday. www.themegallery.com Company Logo