Unit 2 Multivariable Calculus PDF

BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS Course: Mathematical Foundation for Computer Science Stream-1 (23MA1BSCEM) I. PARTIAL DERIVATIVES Let u  f  x, y  be a...

BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS Course: Mathematical Foundation for Computer Science Stream-1 (23MA1BSCEM) I. PARTIAL DERIVATIVES Let u  f  x, y  be a function of two variables x and y. If we keep y as constant and vary x alone, then u is a function of x only. The derivative of u with respect to x , treating y as u constant is called the partial derivative of u w. r. t x and is denoted by one of the symbols , x u f x  x, y   f x, y  u f x, y  y   f x, y  ux , Thus  u x  lim. Similarly,  u y  lim x x 0 x y y 0 y Further ux and u y are also functions of x and y , so these can further be differentiated partially   u   2u   u   2u   u   2u w. r. t. x and y. Thus  or u    or u xy y  x   yx or u yx , x  x  x 2 , xx x y   xy     u   2u  y  y  y 2 and or u yy. Geometrically, the slope of the curve z  f  x, y0  at the point P  x0 , y0 , f  x0 , y0  in the plane y  y0 is the value of the partial derivative of f with respect to x at  x0 , y0 .  BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS In general, if f is function of n independent variables say x1 , x2 ,..., xn then the partial derivative of f is with respect to any independent variable say xi is computed by treating the remaining independent variables as constant. Examples Prove that y v y  x v x   y 2 v 3 if v  1  2 xy  y 2  1/2 1.. 2. Show that wx  wy  wz  0 if w   y  z  z  x  x  y . ex y 3. Show that u x  u y  u, if u . ex  e y 4. If w  x 2 y  y 2 z  z 2 x , prove that wx  wy  wz  ( x  y  z )2. y z 5. If u   , show that xux  yu y  zuz  0. z x u 1 v v 1 u 6. Given u  er cos cos  r sin   , v  er cos sin  r sin   , Prove that  and . r r  r r  2  z z   z z  7. If ( x  y ) z  x  y , Show that     4 1   . 2 2  x y   x y  8. Show that v x   v y 2    v  2 z 2  v 4 if v  ( x 2  y 2  z 2 ) 1 2. z z 9. If z  eax by f (ax  by ) prove that b  a  2abz. x y  y x  2u x2  y2  2u  2u 10. If u  x 2 tan 1    y 2 tan 1   , then show that (i)  2 and (ii) . x  y x y x  y 2 x y y x 11. If u  log e  x  y  x y  xy  , then show that u xx  2 u xy  u yy   3 3 2 2 4. x  y 2 2z 2  z 2 12. If z  f x  ct    x  ct  , then prove that  c. t2  x2  2u 1 u 1  2u 13. If u  ea cos  a log e r  Show that    0. r 2 r r r 2  2  2u  2u  x2  y2  , if (a) u  x y , (b) u  log e   , (c) u  sin  y / x  1 14. Prove that  xy yx  xy   2 xy   , (b) u  log e  x  y   tan  y / x  1 15. Prove that uxx  u yy  0 , if (a) u  tan 1  2 2 2 2  x y  16. Find the value of n so that the equation v  r n  3cos2   1 satisfies the relation   2 v  1   v  r   sin  0. r  r  sin      BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS 17. If Z  tan  y  ax    y  ax  2 , Show that Z xx  a 2 Z yy. 3 1  x2 4 a 2 t v  2v 18. If v  e , Prove that  a 2. t t x 2  2   r   , what value of n will make 1    r 2     .  4t 19. If   t ne  r 2  r  r   t 1 20. If x x y z z  c , show that zxy    x loge  e x  , when x  y  z. y  y  21. If z  log e e x  e , Show that rt  s 2  0 where r  z xx , s  z xy , t  z yy. 22. If v  log e ( x 2  y 2  z 2 ) , prove that ( x 2  y 2  z 2 ) vxx  v yy  vzz   2. 2     9 23. If u  log e  x3  y 3  z 3  3xyz  , show that     u .  x y z   x  y  z 2 24. If w  r m , prove that w  w  w  mm  1r m2 where r 2  x 2  y 2  z 2. xx yy zz 25. Verify that v satisfies Laplace’s equation vxx  v yy  vzz  0 if 3x4 y a) ve cos5z 1 b) v  ( x2  y 2  z 2 ) 2 c) v  cos 3 x cos 4 y sinh 5 z II. COMPOSITE FUNCTIONS Total Derivative: du u dx u dy If u  f  x, y  where x   t  and y    t  then  . dt x dt y dt Examples 1. Find the total differential of the following functions: xyz a) f ( x, y)  x cos y  y cos x (b) f ( x, y )  e 2. Find du for the following functions: dt a) u  x 2  y 2 , x  et cos t , y  et sin t at t  0. b) u  x y , when y  tan 1 t , x  sin t c) u  sin(e x  y ) , x  f (t ) , y  g (t ) d) u  tan 1  y / x  , x  et  et , y  et  et  BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS e) u  xy  yz  zx, x  1/ t , y  et and z  et f) u  x3 ye z where x  t , y  t 2 and z  log e t at t = 2. x g) u  sin   and x  et , y  t 2.  y h) u  x2  y2  z 2 , x  e2t , y  e2t cos3t and z  e2t sin 3t 3. The altitude of a right circular cone is 15cm and is increasing at 0.2cm s. The radius of the base is 10cm and is decreasing at 0.3cm s. How fast is the volume changing? 4. Find the rate at which the area of a rectangle is increasing at a given instant when the sides of the rectangle are 4 ft and 3 ft and are increasing at the rate of 1.5 fts-1 and 0.5 fts-1 respectively. 5. In order that the function u  2 xy  3x2 y remains constant, what should be the rate of change of y w.r.t t , given 𝑥 increases at the rate of 2cm/sec at the instant when 𝑥 = 3𝑐𝑚 and 𝑦 = 1𝑐𝑚. 6. The temperature function for a bird in flight is given by T  x, y, z   0.9x2  1.4xy  95z 2. Find approximately the change in temperature when head wind x increases from 1 meter per second to 2 meters per second, bird heart rate y increases from 50 beats per minute to 55 beats per minute and flapping rate z increases from 3 flaps per second to 4 flaps per second. 7. Elevation of land above sea level, H , depends on two map coordinates  x, y  in the following way: H  x, y   e  0.01 x 2  y 2 . A car travels thorough this terrain, so its coordinates depend on time in the following way: x t   7  10cos(10t ) , y  t   4  10sin 10t . Find the speed with which the altitude of the car increases or decreases at t  0. 8. If w  xy  z is the temperature at any point  x, y, z  along a curve C with parametric equations x  cos t, y  sin t, z  t. Find the instantaneous rate of change of temperature due to the motion along the curve. 9. The voltage V in a circuit that satisfies the law V  IR is slowly dropping as the battery wears out. At the same time, the resistance R is increasing as the resistor heats up. Find how the is current changing dR dV at the instant when R  600 ohms, I  0.04amp,  0.5ohm / sec, and  0.01volt / sec. dt dt Partial differentiation of Composite functions: If u  f  x, y  where x  g  s, t  and y  h  s, t  then u u x u y u u x u y   and   ds x s y s dt x t y t Examples u u 1. If u  x 2  y 2 , x  2r  3s  4 , y  r  8s  5 find and. r s  BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS W W 2. If W  u 2v and u  e x  y2 , v  sin  xy 2  find 2 and. x y  y  x z  x u u u 3. If u  u  ,  ,show that x 2  y2  z2 0.  xy xz  x y z u u u 4. If u  F x  y, y  z, z  x  , prove that   0. x y z z z z z 5. Prove that  x y if z is a function of x and y and x  eu  e v , y  eu  ev. u v x y x y z V V V 6. If V  f (r , s, t ) and r  , s  , t  show that x y z  0. y z x x y z 7. If x  u  v  w, y  vw  wu  uv, z  uvw and F is a function of x, y, z show that F F F F F F u v w x  2y  3z u v w x y z 8. If z is a function of x and y and x  e cos v, y  e u sin v. u   z   2 z z z  z   z  2 u  z  2 2 2 Prove that (i) x  y  e2u (ii )       e       v u y  x   y   u   v   III. JACOBIAN If u and v are functions of two independent variable x and y then the Jacobian of u, v w.r.t   u, v   u, v  u x u y x , y is denoted by or J     x, y   x, y  vx v y Properties of Jacobian 1. If u, v are functions of x, y and x, y are functions of u, v.   u, v    x, y  J and J   then JJ   1   x, y    u, v    u, v    u, v    r, s  2. If u, v are functions of r , s and r , s are functions of x, y then  .   x, y    r , s    x, y   u, v  3. Two functions u and v are functionally dependent if their Jacobian J  0  x, y  Examples (u, v) 1 2 1. If u  a cosh x cos y, v  a sinh xsin y Show that  a (cosh 2 x  cos 2 y ). ( x, y) 2 (u, v, w) 2. Find for the following:  ( x, y , z ) a) u  x2 , v  sin y, w  e3z.  BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS b) u  x 2  y 2  z 2 , v  xy  yz  zx , w  x  y  z. c) u  x  3 y 2  z 3 , v  4 x 2 yz , w  2 z 2  xy at (1,-1,0). 2 yz 3zx 4 xy d) u  , v , w x y z  ( x, y , z ) 3. Find if  (u, v, w) u 2  v2 a) x  , y  uv, z  w. 2 b) u  x  y  z , uv  y  z , uvw  z , 4. Verify that JJ '  1 a) x  e u cos v, y  e u sin v. b) x  u 1  v  , y  uv. y2 y2 c) u  x  ,v. x x d) x  u, y  u tan v, z  w x x xx xx e) y1  2 3 , y2  3 1 , y3  1 2 x1 x2 x3  (u, v) 5. Find when,  (r ,  ) a) u  x 2  2 y 2 , v  2 x 2  y 2 and x  r cos , y  r sin  b) u  x 2  y 2 , v  2 xy and x  r cos , y  r sin  c) u  2 xy, v  x 2  y 2 and x  r cos , y  r sin  d) u  2axy & v  a  x 2  y 2  and x  r cos , y  r sin  ( X , Y ) where X  u 2v , Y  uv 2 and u  x  y , v  yx 2 2 6. Find  ( x, y ) 7. If x  vw , y  wu , z  uv and u  r sin  cos , v  r sin  sin , w  r cos  , calculate  ( x, y , z ).  (r ,  ,  ) 8. Show that the following u & v are functionally dependent and hence find their functional relationship. (i) u  x 1  y 2  y 1  x 2 , v  sin 1 x  sin 1 y x y (ii) u , v  tan 1 x  tan 1 y 1  xy (iii) u  e x sin y, v  x  loge sin y . (iv) u  x  y, y  uv  BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS IV. TAYLOR’S SERIES FOR FUNCTION OF TWO VARIABLES 2 f  a, b  3 f  a, b    f  x, y   f  a, b   f  a, b     where    x  a    y  b 2! 3! x y. Examples Expand the following functions up to second-degree terms. 1. f ( x, y )  x 2  xy  y 2 in powers of ( x  1) and ( y  2). 2. f ( x, y)  (1  x  y)1 in powers of ( x  1) and ( y  1). 3. f ( x, y)  e x cos y in powers of ( x 1) and  y  4. 4. f ( x, y)  x y in powers of ( x 1) and  y 1 and also find (1.1)1.1. 5. f ( x, y )  tan 1 y x  in powers of ( x 1) and  y 1. Hence compute f (1.1,0.9). 6. f ( x, y)  x3  xy 2  y 3 in powers of  x  1 and  y  2. 7. f ( x, y)  cot 1  xy  in powers of ( x  0.5) and ( y  2) and hence compute f (0.4, 2.2). 8. f ( x, y )  x 2 y  3 y  2 in powers of  x  1 and  y  2. 9. f ( x, y )  sin  xy  in powers of ( x  1) and y   2   10. f ( x, y)  xy 2  cos xy in powers of ( x  1) and y   2.   11. f ( x, y )  x 2 y  sin y  e x about 1,  . 12. f ( x, y )  e x sin y about  1,  / 4. V. MACLAURIN’S SERIES FOR FUNCTION OF TWO VARIABLES  2 f  0,0  3 f  0,0    f  x, y   f  0,0   f  0,0     where   x y. 2! 3! x y Examples 1. cos x cos y in powers of x and y up to second-degree terms. 2. Expand e y loge 1  x  in powers of x and y up to third-degree terms. 3. Expand eax sin by in the ascending powers of x and y up to third-degree terms. 4. Find the Maclaurin’s expansion of ex loge 1  y  up to third-degree terms. 5. Expand loge 1  x  y  in the neighbourhood of (0,0) up to second degree terms.  BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS VI. MAXIMA AND MINIMA OF FUNCTIONS OF TWO VARIABLES  A function z  f  x, y  is said to have a maximum or minimum at x  a, y  b as f  a  h, b  k   f  a, b  or f  a  h, b  k   f  a, b  for all values of h and k.  Necessary conditions for f  x, y  to have a maximum or a minimum at  a, b  are f x  a, b   0 and f y  a, b   0.  Given the function z  f  x, y  , the point  x , y , f  x , y  0 0 0 0 is a saddle point if f x  x0 , y0   0 , f y  x0 , y0   0 but f does not have a local extremum at  x0 , y0 . Working rule to find the maxima and minima of z  f  x, y   z z 1) Find , and equate them to zero, solve these as simultaneous equations in x and y.  x y Let  a, b , c, d ........... be the roots of the simultaneous equations. 2 z 2 f 2 f 2) Calculate the values of r  , s  , t  for each of points. x 2 xy y 2 3) If rt  s2  0 and r  0 at  a, b , f  x, y  has a maximum value. 4) If rt  s2  0 and r  0 at  a, b , f  x, y  has a minimum value. 5) If rt  s2  0 at  a, b  ,  a, b  is a saddlepoint. 6) If rt  s2  0 at  a, b  then the case is doubtful and needs further investigation. Examples 1. Find the extreme values of the following functions: a) f ( x, y )  x3  3xy 2  15 x 2  15 y 2  72 x.   b) f ( x, y)  2 x 2  y 2  x 4  y 4. a3 a3 c) f ( x, y )  xy   x y d) f  x, y   x4  y 4  x2  y 2  1  BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS e) f  x, y   x2  2 y 2  3z 2  2xy  2 yz  2 f) f ( x, y )  x3 y 2 (1  x  y ) g) f  x, y   x3  y3  3 y  12x  20 h) f  x, y   cos x cos y cos  x  y  i) x 2  y 2  6 x  12  0 j) f ( x, y)  x 4  y 4  2 x 2  4 xy  2 y 2 2. Find the shortest distance from origin to the plane x  2 y  2 z  3. 3. Find the points on the surface z 2  xy  1 nearest to the origin. 4. Find the shortest distance from origin to the surface xyz 2  2. 5. Sum of three numbers is a constant. Prove that their product is maximum when they are equal. 6. Divide 120 into three parts so that the sum of their products taken two at a time shall be maximum. 7. Examine the function f ( x, y)  sin x  sin y  sin( x  y) , x, y   0,   for extreme values. 8. In a plane triangle find the maximum value of cos A cos B cos C where A, B and C are the angles of the triangle. 9. Find the minimum value of x 2  y 2  z 2 given ax  by  cz  p 10. A flat circular plate is heated so that the temperature at any point  x, y  is u  x, y   x2  2 y 2  x. Find the coldest point on the plate. 11. A rectangular box open at the top is to have a volume 108 cubic meters. Find its dimensions if its total surface area is minimum. 12. The temperature T at any point  x, y, z  in space is T  x, y, z   kxyz 2 where k is a constant. Find the highest temperature on the surface of the sphere 13. Assume that you are in charge of erecting a radio telescope on a newly discovered planet. To minimize interference, you want to place it where the magnetic field of the planet is weakest. The planet is spherical with a radius of 6 units. Based on a coordinate system whose origin is at the center of the planet, the strength of the magnetic field is given by M  x, y, z   6x  y 2  xz  60. Where should you locate the radio telescope? VII. ERRORS AND APPROXIMATIONS Let z  f  x, y . If  x and  y be the small errors occur in observing the values of x and y respectively, then the corresponding error in z is  z. z z (i) Absolute error =  z  z  z0 , (ii) Relative error = , (iii) Percentage error =  100 z z 1. The diameter and altitude of a can in the shape of a right circular cylinder are measured as 4 𝑐𝑚 and 6 𝑐𝑚 respectively. The possible error in each measurement is 0.1 𝑐𝑚. Find approximately the maximum possible error in the values computed for the volume and the lateral surface.  BMS COLLEGE OF ENGINEERING, BENGALURU-19 Autonomous Institute, Affiliated to VTU DEPARTMENT OF MATHEMATICS l 2. The period of a simple pendulum is T  2 , find the maximum error in 𝑇 due to the possible g error up to 1% in 𝑙 and 2.5% in 𝑔. 3. A balloon is in the form of a right circular cylinder of radius 1.5 𝑚 and length 4 𝑚 and is surmounted by hemispherical ends. If the radius is increased by 0.01 𝑚 and the length by 0.05 𝑚, find the percentage change in the volume of the balloon. 4. In estimating the cost of a pile of bricks measured as 2m  15m 1.2m , the tape stretched 1% beyond the standard length. If the count is 450 bricks to 1 cu. m. and bricks cost 𝑅𝑠530 per 1000, find the approximate error in the cost. 5. The height ℎ and semi-vertical angle 𝛼of a cone are measured and from them 𝐴, the total area of the surface of the cone including the base is calculated. If ℎ and 𝛼are in error by small quantities 𝜋 𝛿ℎ and 𝛿𝛼respectively, find the corresponding error in the area. Show further that if 𝛼 = 6 , an error of 1% in ℎ will be approximately compensated by an error of −0.33 degrees in 𝛼. 6. If the H.P. required to propel a steamer varies as the cube of the velocity and square of the length. Prove that a 3% increase in velocity and 4% > increase in length will require an increase of about 17% in H.P. 7. The range 𝑅 of a projectile which starts with a velocity 𝑣 at an elevation 𝛼is given by v 2 sin  2  R. Find the percentage error in 𝑅 due to an error of 1% in 𝑣 and an error of 0.5 % g in 𝛼. 8 Il 8. The torsional rigidity of a length of wire is obtained from the formula N  2 4. If 𝑙 is decreased t r by 2%, 𝑟 is increased by 2%, 𝑡 is increased by 1.5%, show that the value of 𝑁 is diminished by 13% approximately. 9. The volume V of a right circular cylinder is to be calculated from measured values of r and h. Suppose that r is measured with an error of no more than 2% and h with an error of no more than 0.5%. Estimate the resulting possible percentage error in the calculation of V. 10. The resistance R produced by wiring resistors of R1 and R2 ohms in parallel can be calculated from 1 1 1 the formula  . R R1 R2 (i) Now, you have to design a two-resistor circuit have resistances of R1  100ohms and R2  400 ohms , but there is always some variation in manufacturing and the resistors received by your firm will probably not have these exact values. Will the value of R be more sensitive to variation in R1 or to variation in R2 ? (ii) In another circuit like the one shown, you plan to change R1 from 20 to 20.1 ohms and R2 from 25 to 24.9 ohms. By about what percentage will this change R ? 11. The area of a triangle is A   ab  sin C , where a and b are the length of two sides of the triangle 2 and C is the measure of the included angle. In surveying a triangular plot, you have measured a , b , and C to be 150ft, 200ft, and 60 , respectively. By abut how much could your area calculation be in error if your values of a and b are off by half a foot each and your measurement of C is off by 2 ? Use in radians.