Useful Mathematics

Useful Mathematics - 'B.l‘ Some Useful Mathematics...

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Unformatted text preview: 'B.l‘ Some Useful Mathematics Differential Calculus with Several Variables The fundamental equation of differential calculus for a function with independent variables x. '1‘. and : is a ' e" ' 'i ci'f' : (11+ alrf dz {8-1) I“ 1:: ‘ ill. 1.: (l: .r._i where (WK/axle: (ii/'/i.it')\_:. and (Elf/dz)” are partial derivatives. A partial derivative with respect to one independent variable is obtained by the ordinary procedures of differentiation. treating all other independent variables as though they were constants, An example of Fq. (8—1) is 3p ‘tP ‘iP JP : dT + (“I + d” (Bel (if tin r” I.” a” Tl- This equation represents the value of an infinitesimal change in pressure that is produced when we impose arbitrary infinitesimal changes dT. u'l'. and (in on the system. An approximate version of Eq. (B-Z) can be written for finite increments in P. T. l'. and n. equal respectively to AP. AT. Ai'. and An: up JP '9) AP «2 (97) AT + 17:) A!’ + An (3-3) ,)T F” iii , [fl WI T} where 7% means “is approximately equal to." Equation (B-3l will usually be more nearly correct ifthe finite increments AT. All and An are small. and less nearly correct if the increments are large. An Identity for a Change of Variables The expression for the differential of a function U is _ ‘iL" ‘iti 'iL" at; : dT+ (— ar + (IN (8-4) ' l ,u L” I)! 8” Till if T. l'. and n are used as the independent variables. 11' T. 1”. and n are used as the independent variables. then db“ is given by 8U HIE” r'lt-‘Y EU : - i I. I —' J 7 '5 r (HT)Pan—i—(ap)mcfl +(an)T.Pdn (13.) 1001 1002 B Some Useful Mathematics In a nonrigorous fashion. we “divide‘ Eq, (8-4) by (ET and specify that P and n are fixed. Of course. you cannot correctly do this. since (IT is infinitesimal. but it gives the correct relationship between the derivatives. Each "quotient" such as dU/dT is interpreted as a partial derivative with the same variables fixed in each “quotient.” The result is. holding T and a fixed: (“U i (HU iiT +(ilU) ill") + fig) (B 6 Phi P." T.” i".n T.l’ P.” ) The derivative of T with respect to T is equal to unity. and the derivative of n with respect to anything is equal to zero if n is fixed. so that 8U) 77 BU +(3U) (31’) (B 7) PJ: l'.ri TJr I‘Jt - Equation (8-7) is an example of the variable—change identity. The version for any particular case can be obtained by systematically replacing each letter by the letter for any desired variable. The Reciprocal Identity 1f the role of the independent and dependent variables are reversed keeping the same variables held constant. the resulting derivative is the reciprocal of the original derivative. An example ofthis identity is 3V) _ 1 (B 8) UP M— (art/an” This identity has the same form as though the derivatives were simple quotients. instead of limits of quotients. The Chain Rule 1f the independent variable of a function is itself a function of a second variable. this rule can be used to obtain the derivative of the first dependent variable with respect to the second independent variable. For example. if U is considered to be a function of P. V. and r1. while P is considered to be a function of T. I". and n. then '31.! ii i ‘i i— = i—P (8-9) ‘lf :1» RP 1m ‘lT v." The same quantities must be held fixed in all of the derivatives in the identity. We can also obtain the differential of a quantity which is expressed as a function of one variable. which is in turn given as a function of other variables. For example. if :fi“) and u : 1.r(__\'._1‘. 5); 3f _ E 8i: (all (in) (8-10) The differential off can be written _ (if {in Ha an i : ‘7 — 2'. r i’ ' g .: B-ll if d“ [(811)-12er + (8.“) _ r) + (82)“ d] i l Second Derivatives and Euler’s Reciprocity Relation A second derivative is the derivative of a first derivative. lff is a differentiable function of two independent B Some Useful Mathematics 1003 variables. x and i. there are four second derivatives: 821' _ 6 (6f 3}“ 8.“ _ by at F ' \ an) {it iii _ 8X {11' l ‘ . - r) a a (6211 ‘— (ef— (B-izc) 3.1“ 1. {it [it v V 1 — - . - - ‘71 i: 2) (8-1a a2 t div av i We refer to the second partial derivatives in Eqs. (8-! 2a) and (B-12b) as mixed second partial derivatives. The Euler reciprocity relation is a theorem of mathematics: If," is differentiable, then the two mixed second partial derivatives in Eqs. (B-lZa) and (B-12b) are the same function: (B—l2a) (B—l2b) 8*)" 691‘ By 8x = 8x {21' (13-13) For a function of three variables, there are nine second partial derivatives. six of which are mixed derivatives. The mixed second partial derivatives obey relations exactly analogous to Eq. (13-13). For example! 32V i3 V t = .( . (3-14) (if 3P n rip 0T " The same third independent variable is held fixed in both derivatives, as shown by the subscript. The Cycle Rule Ifx. y. and z are related so that any two ofthcm can be considered as independent variables, we can write the cycle rule: a 3x_‘,d_i‘zdz_l_ (') We obtain this identity in a nonrigorous way. The differential d2 can be written 62 d: t dz = dx -l— dy (8-16) dx 1, (it x We consider the special case in which 2 is held fixed so that :12 : 0, and “divide” Eq. (B—16) nonrigorously by aft-x The “quotient” a'x/dy at constant 2 is interpreted as a partial derivative at constant 2. and the “quotient” dy/dy equals unity. We obtain 0_ 8:: + E B17 _(i§lx J (3,“); (3y); ( _ ) We multiply by (av/32)“. and apply the reciprocal identity to obtain 82 at a}, T] (B-IS) which is equivalent to Eq. (B-IS). it!" 1 004 B2 Curve giving y = ytx) or x : x( y) | yé—l-v X———> Figure 13.1. A Curve Giving y as a Function of x or Giving x as a Func— tion of y. B Some Useful Mathematics Bil) gives the differential of a function. Exact and lnexact Differentials Equation ( e a general differential in terms of which is called an exact differential We can also writ (ix. (2'); and d:: du : Ltr. _r. :) “’1‘ + il-l'(.\'._r. :l d_v+ N(11_r.:) d:- (3-19) . M. and are some functions ofr. _r. and 3. A general difi'crential form like a Pfaffian form. ll‘the functions L. M. and N are e function. then the differential tilt is an where I. that of liq. (Big) is sometimes called not the appropriate partial derivatives ofthe sam inexact differential. and has some different properties from an exact differential. To test the differential (in for exactness, we can see it‘the appropriate derivatives all. M. and N are mixed second derivatives of the same function and obey the Euler reciprocity relation: ‘JL 'lM I (exact differential) (8-203) dy I: d.\’ H at. BN ‘ _ I (f) : (exact differential) (3-20m d; m _d.t‘ 17: (B-Ztlc) ‘iM ‘lN : (exact differential) ( z \‘._l (7“ L: 8—20) is not obeyed. then (in is an inexact If any one of the conditions of Eq. ( t is an exact differential. difi'crential. and if all of them are obeyed then o’t Integral Calculus with Several Variables There are two principal types of integrals of functions of several variables. the line integral and the multiple integral. Line Integrals For a differential with two independent variables. du : M(x. '1') (1x + N(.r. 7r) (1y a line integral is denoted by J du : 1 [Min 71') alt + N{x.)) Liv] tB-Zl) —Jr plane. This curve givesr as a function ol‘x say that the integral is carried out along M by the function ofx rye. llthese where the letter L‘ denotes a curve in the .r and .r as a function ol‘y. as in Figure B. 1. We this curve (or path). To carry out the integral. we replace _1' in given by the curve and replace .r in N by the function ofy given by the cu functions are represented by _r(.r) and All]: leb'). _r. ) Jr (BHZZl \ . I (In : I A. M(_.t".}‘(.\')l (it -l— are the coordinates of the initial point ofthc line integral and .r3 and y; ntegral is now an ordinary integral and can l form has three or more independent in a space of all where x, and 71y are the coordinates ol'the final point. Each i be carried out in the usual way. If the differentia variables. the procedure is analogous The curve must be a curve Fit??? B Some Useful Mathematics Region at integration 2 of xfrom 31 to 32 ylrom b. to l)2 zfrorn c. to a; Figure 5.2. An Integration Region in Cartesian Coordinates with Constant Limits. 1005 independent variables. giving each one of the other independent variables as a function of one variable. There is an important theorem of mathematics concerning the line integral of an exact differential. Ifa'z is an exact differential. it is the differential ofa Function. If T, l". and n are the independent variables. then this function is z = :(T. V. n) and a line integral ofdz is equal to the value of: at the endpoint ol'the integration minus the value of 2 at the starting point: 8: ii: V 3: id- = l. (a) d” (alt. ‘” t ‘1’” 22m. nan—gin. VH1.) (rs—23) where T 2. V2, and 212 are the values of the independent variables at the final point of the curve, and T]. V]. and n1 are the values at the initial point of the curve. Since many different curves can have the same initial and final points. liq. ('B-23} means that the line integral depends only on the initial point and the final point. and is independent of the curve between these points. It is said to be path-independent. However. the line integral of an inexact differential is generally path-dependent. That is. one can always find two or more paths between a given initial point and a given final point for which the line integrals are not equal. Multiple Integrals Iff :fix.y. z) is an integrand function. a multiple integral with constant limits is denoted by ‘1: h: ": [(al.az. [31.1521]. c2) = i [ J fix}: 2) d2 dy dx (B—24) hi .a3. .1. The integrations are carried out sequentially. The left-most differential and the right- most integral sign belong together, and this integration is done first. and so on. Variables not yet integrated are treated as constants during the integrations. in Eq. (B-24). z is first integrated from c. to (‘2. treating x and y as constants during this integration. The result is a fiJnction of x and ‘1'. which is the integrand when y is then integrated from bE to £72. treating x as a constant. The result is a function ofx, which is the integrand when x is then integrated from a! to :22. in this multiple integral the limits of the 2 integration can be replaced by fiJnctions of .1‘ and y, and the limits of the y integration can be replaced by functions ofx. The limit functions are substituted into the indefinite integral in exactly the same way as are constants when the indefinite integral is evaluated at the limits. If the variables are cartesian coordinates and the limits are constants. the region of integration is a rectangular parallepiped (box) as shown in Figure 3.2. lfthe limits for the first two integrations are not constants. the region of integration can have a more complicated shape. The integration process can be depicted geometrically as follows: The product dx dy dz is considered to be a volume element. which is depicted in Figure B3 as a little box of finite size (the box of dimensions dx by dv by dz is infinitesimal). This volume element is also denoted as (fir. If (.t‘._\'.:) represents a point in the volume element. then the contribution of the element of volume to the integral is equal to the value of the function at (say. :) times the volume of the volume element: (Contribution of the volume element air dy dz) :f(x. y. :) dx d}: dz will 4" 1006 B Some Useful Mathematics Volume element of dimensions dx by dyby dz | X Figure 3.3. An lnfinitesimal Volume Element in Cartesian Coordinates. The integral is the sum ofthe contributions ol‘all the volume elements in the region of integration. If an integral over a volume in a three—dimensional space is needed and spherical polar coordinates are used. the volume element is as depicted in Figure 8.4. The length of the volume element. in the :— direction is equal to air. The length of the box in the H direction (the direction in which an infinitesimal change in U carries a point in space) is equal to r all) iftl is measured in radians. since the measure of an angle in radians is the ratio ofthe arc length to the radius. The length ofthc volume element in the cf) direction is r sinttl') deb. which comes from the fact that the projection of r in the x- _1' plane has length rsinttll. as shown in the figure. The tolume of the element of volume is thus d3r : r3 sin(Ul dip cm dr {13—25) Volume element of volume r2 sin(9) do d9 dr """-rsin(6)dm X Figure 3.4. An Infinitesimal Volume Element in Spherical Polar Coodinates. B Some Useful Mathematics B3 1007 where dir is a general abbreviation for a volume element in any coordinate system. An integral over all of space using spherical polar coordinates is We rt 2n N I :J l [ fir. (l. (1‘))!“ sinttl) dd) all) rt’r (B-Zbl l) - l) - 0 Since the limits are constants. this integral is carried out in the same way as that of Eq. (B-24). with the (f) integration done first and the (l integration done next. For other coordinate systems, a factor analogous to the factor r3 sintti) must be used. This factor is called a Jacobian. For example. for cylindrical polar coordinates. where the coordinates are 3. d) (the same angle as in spherical polar coordinates. and p (the projection of r into the tar plane), the Jacobian is the factor p. so that the element of volume is p dp d: dqi. Vectors A vector is a quantity with both magnitude and direction. The vector A can be represented by its cartesian components. A}. .41, and A_.: A : 14, +141. + k4: (8—27") where i. j. and k are unit vectors in the x. v, and 2 directions. respectively. The dot product. or scalar product. of two vectors is a scalar quantity equal to the product of the magnitudes of the two vectors times the cosine of the angle between them: A - B : lAllBl cos(x) = ABcostcxl (B-28) where at is the angle between the vectors. The scalar product is commutative: A-Bzfl-A (8-29) The scalar product of a vector with itself is the square of the magnitude of the vector: A-A = W3 : .43 (3-30} The scalar products ofthe unit vectors are i'j:i-k=j-k:0 (B-3la) i-i=j-j:k'k:l (B-Blb) Ifthe vectors A and B are represented by cartesian components as in Eq. (PS—27). Eq. (8-31) implies that six of the nine terms in the product A - B vanish. leaving A - B : are. + AVB‘ + A13: (8-32) The cross product. or vector product. of two vectors is a vector quantity that is perpendicular to the plane containing the two vectors with magnitude equal to the product ofthe magnitudes of the two vectors times the sine ofthe angle between them: |AXB| : lAllBlsin(atl (8-33) The direction of' the product vector is the direction in which an ordinary (right-handed) screw moves it if is rotated in the direction which the vector on the left must be rotated - i 1008 B Some Useiul Mathematics to coincide with the vector on the right. rotating through an angle less than or equal to 180 . The cross product is not commutative: Asz—BXA (8-34) The cross products of the unit vectors are ixizt]. jxj:0. kxk:0 (835a) ixj=k. ixkzej. jxk:i (IS-35b) In terms of cartesian components. we can deduce from Eq. (B735) that A x B : HAVE: - AZBL] + j[/t:BX — AIBz] + MAKE, i rifle] (8—36) The product of a vector and a scalar is a vector whose magnitude is equal to the magnitude of the vector times the magnitude of the scalar Its direction is the same as the direction ofthe first vector ifthe Scalar is positive. and its direction is the opposite of the direction oithe first vector if the scalar is negative. Vector Derivatives The gradient is a vector derivative of a scalar function. lff is a function ofx. “1‘. and 2. its gradient is given by Vf' : + + (3-37) dr dt‘ d: The symbol for the gradient operator, is called "del." At a given point in space. the gradient points in the direction of most rapid increase ofthe function. and its magnitude is equal to the rate of change of the function in that direction. The gradient ofa vector Function is also defined. and the gradient of each component is as defined in Eq. (8-37). The gradient of a vector quantity has nine components. and is called a dyadic or a eartesian tensor. Each of its components is multiplied by a product of two unit vectors. The divergence of a vector function F is denoted by V - F and is defined by _ 6Fr HF. 8F. V -P 2 4- + {838) (it _ di‘ _ dz The divergence is a scalar quantity. lfthe vector function represents the flow velocity of a fluid. the divergence is a measure of the spreading out of the streaming curves along which small elements of the fluid flow. A positive value of the divergence corresponds to a decrease in density along a curve following the flow. See the discussion of the equation of continuity in Section l 1.2. The curl is a vector derivative ofa vector function. The curl ofF is somewhat similar to the vector product (cross product) of two vectors, and is denoted by V X F and defined by HF. 8F. as a}: HP. 3F V F2. . i __ - .r ! __L . _ = _39) X iii-l (azflfliiaz) (axlfllinl (ax-l] {B The curl of a vector Function is a measure of the turning of the vector as a function of position. and has also been called the “rotation.” The curl ofF has also been denoted as curl F and rot F. The divergence of the gradient is called the laplacian. The laplacian of a scalar function f is given in cartcsian coordinates by 2r — ‘7” + 15+ i”. (13-40) _ fix2 di'- 3* V B Some Useful Mathematics 1009 13.4 The vector derivative operators can be expressed in other coordinate systems In spherical polar coordinates. the gradient of the scalar function!" is s 3i)" 1 3f- 1 Eif- Vf _ 8’ T'Jre" T‘ 7' + e'” rsintti) fir/J (8—41) where e, is the unit vector in the r direction (the direction of motion if r is increased by a small amount. keeping {1‘ and q‘) fixed). efl is the unit vector in the (i direction. and 94, is the unit vector in the (f: direction. In spherical polar coordinates. the laplacian is i. 1 a .d/‘ 1 a . sir 1 air ~:7fib; —.H-—+ - 3-42 V" r2 Lair ctr] +sin((l) swim ) an] smith) aazi ( } Solution of a Differential Equation in Chapter 12 Equation (12.5-5) can be put into the form 2 = M ({[B] + N d! z 0 (Fl-43) where M and N are functions oft and [B]. the concentration of substance B. The equation shown in Eq. (8-43) is called a Pfaffian differential equation. An equation of this type is called an exact differential equation ifdz is an exact differential. We can determine whether 0’: is exact by finding out whether M and N conform to the Euler reciprocity relation. if (12 is exact. M and N must be derivatives ofthe function :. and must obey the Euler reciprocity relation shown in Eq. (8—13}: at; ‘ #2 a“: zw a; ‘ 3: ans] _ are] a: ‘ 8[B] We multiply Eq, (12.5-5) by dt and recognize that (d[B]/dr)dr : d[B] to obtain an equation in Pfaffian form: d[B] + {Mn} — k1[A]0€'k=')dt: 0 (13—45) (3—44) This equation is not an exact differential equation. since it corresponds to M equal to l and N equal to the expression in parentheses. The derivative of M with respect to t equals zero. and the derivative of N with respect to [B] equals k2. However. if the equation is multiplied by the factor 9’9”. we get the exact differential equation ctr d[B] + (Haws a t; [Air-iatrrt: ") a: = 0 (13-46) as can be checked by differentiation. A factor that converts an inexact Pfafhan differential equation into an exact differential equation is called an integrating factor. Since multiplication of any equation on both sides by the same factor yields a valid equation containing the same variables. this equation has the same solution as did the original equation. Finding an integrating factor for a particular equation is not always easy. However, mathematicians have shown that if one integrating factor exists. there is an infinite number ofother integrating factors, so that trial and error might lead to a usable integrating factor. Consider the special case that no B or F is present at time I: 0. We denote the differential in Eq. (8-46) by dz and perform a line integral of dz on the path shown in Figure 8.5. The result ofthe integration must equal zero. since the differential equals zero if it satisfies the differential equation and the function : must therefore have the ...
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This note was uploaded on 01/15/2012 for the course CHM 4411 taught by Professor Ohrn during the Spring '08 term at University of Florida.

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Useful Mathematics - 'B.l‘ Some Useful Mathematics...

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