Multivariable Calculus Elementary Vector Analysis

%< . « Multivariable Calculus: Review Session 1 1. Elementary Vector Analysis ( from Stewart , Calculus , Chapters 12 and 13) Vectors, Points, and Position Vectors: The position vector of a point, P is the vector from the origin to P . For example, in Ra , a vector v { x , y, z ) is the position vector of the point P ( x , y , z ) In this way, we often identify points as vectors and vice versa. — = . Parametrized Curves Parametrized Curves: Given a continuous vector function f { t ) ** { x( t ) , y( t ) ) or r (< ) = ( x( l ) , y ( t ) , z( t ) ) , the set of points with position vector r( t ) for some t defines a curve C The components x( t ) , j/ (£ ) , and z ( t ) are called parametric equations for the parametrized curve C The variable t is called the parameter . . . The tangent vector to a curve at the point P f ( t ) is r' ( t ) ~ (x'( t ), j/'(< ) ) or r'( l ) = { x' ( t ) , y' ( t ), r'( t )). The line through the point P = f ( t ) parallel to the vector r' ( t ) is called the tangent line to C at P. - Ex. X. Sketch the curve C with parametric equations x ( t ) = t + 1, y( t ) 3 y 3 0 x I -l 0 1 2 0 1 ~t 2 — — 21 , 1 < t < 3. - -01 2 TTJ . Show that C is an arc of the parabola y ~ x2 - Ax + 3 and use this fact to give another pammetrization of C. if — 7 3s 4 3, r “ 0 A' Ex. 2. What curve is represented by the parametric equations? ( b) x ( /.) » sin( 2/.), ?/( /.) = cos(2t ), 0 < t < 2x (a) x( t ) 2 cos(f ) , j/ (l.) = 2 sin(l) , 0 < /. < 27r - \ C\ ecU ra-AI <5 'l- &r ' 4rft>t trstA 0 A (JL oicbs UjiM» ^ in ^W 0 cflonW- cW ^ wi ^ lrftbjrv. CvrcX P+ rJtU \ -VvAtft tA * ¥ 0,6) cWW\AK AptcWO t\r . ( tv Vector Equation: ^ It iXt ) = fp + tjl Parametric Equations: x ( l ) = x$ + jjt , y( l ) To Lines in R 2 and R 3 V z( L ) = zo +,ct = j/o + 61, = (x ( f ) , t/ ( t ), z( t)) are tlie position vectors of the points ( x( t ) , y( t ) , z { t ) ) on the line rjx = (xn , j/o *o) is the position vector of a fixed point Pn = (xo, ?/n , rn ) on the line j7 {a , 6, c) is a direction vector of the line, i .e. i7 is any vector parallel to the line Rmk. For lines in R2 , just delete all of the third coordinates above. fjt ) r W- 0+ 1 1 . . . - — - Ex 3 l’ind parametric equations For the line that passes through the points ( 2, 0, 1) and ( 1, 1, 1). ^ < r fvam • -Ui , H\ /\ - *i ~ 5 "t I '' D = i, i 3 ^ - I -I t — . . Ex 4 Find parametric equations for the line segment from (3, 1, 2) to ( 5, 4, 1). tf -- < -s -a , + i , i -O V 3- 8 ^ l + 3-t I . 2 t ^, Ex. 5 Find parametric equations for the line that passes through the point ( 2, 0, 1) and is parallel to lire 5. and z ( t ) line given by parametric equations x ( i) 3i, y( t ) + 4. = ^ 2. » =< 3 * —— > 3, 0 | t °> * = l -H - t + 1, -t + l . . Ex 6 Find parametric equations for the tangent line to the parametrized curve x ( t ) at the point ( 0, 3). 0 i~~\ x ' r t W i t -i'? t 1 t-h < b 'D - - T c vs - ' k" ^ A;*r . v» c. J 3 Ht “ A T. L . — t - 2i CWC- vj = 3 / y( L ) 2 , I Planes in R 3 Vector Equation: n (r * - ro ) 0 IA Scalar Equations: / ( « (a- - T0) + l>( y ~ Vo ) + c{z - *o) =0 - or ax + by + cz A P(.r, >, r ) =d jj = (x, y, 2) are the position vectors of points (x, y, 2) in the plane jrjj sr (XQ , yo , 20 ) is the position vector of a fixed point P0 = (io , 1/0 , Zo ) in the plane Jl — {«, b,r ) is the normal vector. i.e. / * r - rj / .Vrj. Z«) & y any vector orthogonal to the plane Ex. 7. Find an equation of the plane that passes through the point (1, 0, 2) and contains the line x = 2 A + 2, y = 2 t , r = 4 f . \> " < l 0 ( 2 iv V V-o ,v AU t j - , -j B tf * - i 2 2. -I -1 0 2. < , , .« I rXaNL 4 w« 1 (< t 4 \n 4 z 2( * 1 ~ X , 0 - 6 / 2-O = < H d ^ ( ^, ^ f > (1 0 , fl V pUll -- — pH A * 0 Xl Or AOt - i'i + - ( 4 = 2. + ' ^ ^0 . 2 Functions of Several Variables ( from Stewart , Calculus, Chapter 14 ) — Functions of Two Variables: / : D f R , D C R 2 . The graph of the function / (x, y ) is the set of all points (x, y , r ) 6 R3 such that (x, t/ ) 6 D and ; = / (a y) . The grapli of a function of two variables is a surface in R 3. ^ Functions of Three Variables: / ; D R, OCR3. The graph of a function f { :r , y , r ) of three variables would lie in R'1. For a visual representation , we often look instead at the level surfaces of f , i.e. the surfaces in R 3 with equations f ( x , y , z ) = k for constants k . . Continuity: A function / (x , y ) is continuous at (a , b ) if . lint - (x ,y) »(a,( ) f ( x, y ) Graph of J ( x , y ) = f ( a , b ). Partial Derivatives «* *) - / (x + h , y ) - / (x , y ) 21 h and sj " £S f ( x , y + h ) - / (x , y ) h Shortcut for Finding Partial Derivatives respect to x. •To find /x , regard y as constant and differentiate /f (x,, yy )) with with respect to y. To find / , regard x as constant and differentiate { x „ • if (x, y ) 0 if (x, y ) . Ex 9. Consider the function / (x, y ) = (1) Show that / is continuous at (0, 0). ^= (( ,, )). 00 0 0 X * IT 0OS & \ i* 4 r3 CoS 39 f (X V) = |i /v> Sues. / A Mr 0 TW* . ' _ r 0 rx * ” r-* o Mr3 ( OJ2 <9 - lt*a tWuj " / ~\' Cod 4 6 - I W * ' & * «4 \ i v> 4 K COS 3 © - 0 «- ( Mo So> 0 *b „ (2) Find /x (0, 0 ) and / (0, 0). MO - 1'ina V*0 - U Mo So h 4Vi3 - jinj - U1 ^ (V ) ^ So i v, Mo - 6 Vi ' I C.3 M£L \ » oo Mo ^^ C°|C "" liM . * o «4 , & *) . MV, ivd- ' Hok Zo 51© 4's"' ) . &+„* { i( Milv » * i& N&r, if ( v ) T4 (o, o) * Ex 10. Let / (x, j/ ) 0 if (x,!/ ) (0, 0). (1) Is / continuous at (0, 0)? Justify your answer. . \ W> Sin ,} ) ' * & *** IXs U = x~* 0 } ( (x, *XH P,O = So , 3 h> C 2.+ iirvi tiai. 4 ^ oVU On y „^ * (or ^ 2' * * TVM - mM 4+ ZinJL 4+mrvV ^ (2) Find /x (0, O) and / (0, 0). ^ ^ \°~ =m -Pfo WM') = 50 0 . Wo ^ ^ |« m ^^ 1^ - t ' ** = VM 6 \\ v\ ( r: Uw \r* t> __ 1_ , ^ MoOV I- = 0 fT ' , ai \ = ^ r"> ^ ^ 0 id ~ W £ oo rn 1 . aJ f °i *0 4x (°|0} ^ 0 " " ^ A4 C6 O4 ^OJJ aUrv « j dt -HiK/vt ey» s - • At> ci *4 ^ 4 ( « , V,WM Vf 4 o ^ X\*\ ^ O 7.M + Q , -f 0 4- Mo ^ ^ \v>? dUftaJj ,0 )0 -4 , ft ^ j maiVvAr y - MX . -ft * + ( > )* * XS =X *U3 +v« 4»**V'on a 4Vt onyn ^WCiny ) V = ^ ^ / Sv*ee - -U * *-A 4 Co* A Air b ^ mx ^* 0\ tT> . Level Curves and Level Surfaces Def. The level curves of a function f of two variables are the curves in the xj/ plane with equation f ( x, y ) k for any constant k . Level curves are the projection onto the xy plane of traces of the surface in the planes z = k ( A contour map is a collection or level curves. ) - = . - Def. The level surfaces of a function f ( x , y , r ) are the surfaces with equations f ( x , y , z ) k . (at £j*Hwi < h ) Horizontal irnco wc raised level curvet Directional Derivatives and the Gradient Vector Directional Derivatives Given a unit vector u ( a , b ) , the vertical plane through ( xo, J/o ) in the direction of tZ intersects S in a curve <7, The slope of the tangent line to C at the point {xo, i/o ) gives the rate of change of f at (xo, yo) in the direction of the unit vector u . This is called the directional derivative of / in the direction of u , and is denoted D ,l J { x{i , ya ). = f ctyt Ot ' - \ DlnVtx+ \vcj r 2> 0 Note that the partial derivatives fx and fv are the directional derivatives of / in the directions of i and j , respectively. . Thm If / is a differentiable function then / has u directional derivative in the direction of any unit vector tZ and Dcf V/ • u - The Gradient Vector or Properties of the Gradient : Let 0 be the angle between a unit vector u and V / at a point P Def — V / • u » \Vf cos(0) . 9- 0 (1) The maximum value of the directional derivative Def ( P ) at a point P is |V / ( P)| and it occurs when u has the same direction as the gradient vector V / ( P) . So V / ( P) points in the direction of its ilfc, Vbdb'Am Penic maximum rate of increase of f at P. ^ lCcrt',0 r* \ bfyTwupvCj" (2) V / (x, y ) is perpendicular to the level curves of a function f ( x , y ) . Similarly, V / (X , J/ , : ) is perpendicular to the level surfaces of a function f { x , y , z ) . ... r V/ U Piw,} - levelcurve . / u rl r K I . Ex. 11 Find the directional derivative of the function f ( x , y , z ) ” direction of the vector (4, 2, 1) . * = , ,-ri \|+ ( • ^ 2 eTU* — at the point (0, 1, 1) in the RT S - -t- < Vi >7 hi * j/ 0 1 ^z - , ( > ? t*3 * o' VZi ( -1 M =0 Ex. 12. Let f ( x , y ) — 2 .r 2 + .n/ 2. Find the unit vector that points in the direction of the maximum rate of f a r . o+ 74 increase at the point (1, 2 ). What is the rate of change of / in this direction? = < 'i t v s’ , iuj D ' " ' t = If -Jv, cWjf o * f ' iw * . . +Wi ckw,*,.' \» " ^ ~ x VvoraH. .£X+- -VW. 'o o- - + I 1® * u l\ V -( • - M 13 . Ex 13. The temperature at the point ( x , y , z ) is T ( x , y , z ) = i sin (7ra:j/ ) + ln ( z2 + 1) + GO. (1) Find a vector pointing in the direction in which the temperature increases most rapidly at the point (2, i , \ y . - VT -<^ ^ , (t - CPi ( l-q ) f ' ^ ' (2 ) Let v = - i + 2J + 2fc. direction of v? tk f ,1 ^ 0 „ xcoiCrx 2- \ ) z U 5 C-IT ) lA » -VU eX\ ^ <+ » » 0 f ^ 0 What is the rate of change of the temperature at the point (2, -1, 1) in the u t 11 = {iTuT : 3 i . ( i -» uV vrU -MVi . X j (1 + = -* h 1* i \ Tangent Planes Tangent Planes: The tangent plane to a surface 5 at a point P tangent line to any curve C on S passing through P . = (xu , ya , ro) is the plane containing the Equations of Tangent Planes If S is the graph of a function, i.e. z = / (:/:, y ) is given explicitly as a function of x and y: f ( x0 , !/o ) = f x ( xo , l/u ) (x Xu ) + /v (xo , yu ) ( y - V o ) Z If S is the graph of a level surface F ( x , y , z ) = k and VF(xo, j/o o ) # 0: F*(x0 , yo , zo ) ( x x 0 ) + Fy (x0, i/o , za ) { y y0 ) + Fz ( x0 , y0 , z0 )( z *o ) 0. Rmk . This second equation is just the usual scalar equation of a plane, with n VF. This equation ran be used in the first case as well. - - - .- - — - = . . Ex 14 Let f ( x , y ) = y + sin ( x / y ). Find an equation of the tangent plane to the graph of z the point (0, 3, 3). f* hi • ^ 3 j\ ® *-i * S \ rV . -\*\-o1®+ = 1 Vb / * 2 =1 *i » X3l = I + 1 rXW za 0 = l ( =7 Two ' ° D‘'/ <, = ± ^ ( ± l j 1/J. J f ) 2