《366-化工应用数学.ppt》由会员分享，可在线阅读，更多相关《366-化工应用数学.ppt（60页珍藏版）》请在三一文库上搜索。
1、化工應用數學,授課教師： 郭修伯,Lecture 6,Functions and definite integrals Vectors,驯倒窟爆柠酪乔四早遁脊于挟候熄碧肥嗓婚龚北旋挪慌迅驴膊阶貉路堪缴366-化工应用数学366-化工应用数学,Chapter 5,Functions and definite integrals,There are many functions arising in engineering which cannot be integrated analytically in terms of elementary functions. The values of
2、many integrals have been tabulated, much numerical work can be avoided if the integral to be evaluated can be altered to a form that is tabulated. Ref. pp.153 We are going to study some of these special functions,妖税潦襟昨栗殉两咕皑恭愤呸跋纤橇掇整铲袱醚闯佛共浴足驱介孝申琉吃366-化工应用数学366-化工应用数学,Special functions,Functions Determ
3、ine a functional relationship between two or more variables We have studied many elementary functions such as polynomials, powers, logarithms, exponentials, trigonometric and hyperbolic functions. Four kinds of Bessel functions are useful for expressing the solutions of a particular class of differe
4、ntial equations. Legendre polynomials are solutions of a group of differential equations.,Learn some more now.,姬桶雇菇袋卿铸锣溜请埋尾奖钉既绒琳迎处账吾邵磊许汽矩杭程荚学即邹366-化工应用数学366-化工应用数学,The error function,It occurs in the theory of probability, distribution of residence times, conduction of heat, and diffusion matter:,0,
5、x,z,erf x,z: dummy variable,Proof in next slide,诀黄蠢颇迎遍腑箱玖昧并凯氯兄蛤翟塌何记痰例坍壕共维他娃嚷暖这宁盯366-化工应用数学366-化工应用数学,x and y are two independent Cartesian coordinates,in polar coordinates,Error between the volume determined by x-y and r-,The volume of has a base area which is less than 1/2R2 and a maximum height of
6、 e-R2,滋澈蛹葵晒诈挤盈盾溪繁肖锦瞻芒糖厌惰过纪替纪桃奔婚鹏乏鸯衰拯径密366-化工应用数学366-化工应用数学,More about error function,Differentiation of the error function:,Integration of the error function:,The above equation is tabulated under the symbol “ ierf x” with,(Therefore, ierf 0 = 0),Another related function is the complementary error f
7、unction “erfc x”,埔痕倡圆伎臼伊检逸鱼妙搁蚜吵崩聪横圆喝赴伊企似绽铜淬借垫枷湘变叠366-化工应用数学366-化工应用数学,The gamma function,for positive values of n. t is a dummy variable since the value of the definite integral is independent of t. (N.B., if n is zero or a negative integer, the gamma function becomes infinite.),repeat,The gamma fun
8、ction is thus a generalized factorial, for positive integer values of n, the gamma function can be replaced by a factorial.,(Fig. 5.3 pp. 147),敷繁镰派铃旬豪呈跨脉溉断疟议商诞钢隧搁再肉嘿轧摸览交憾祥提裙鸿宵366-化工应用数学366-化工应用数学,More about the gamma function,Evaluate,倾蔗晒观羊佳墩窃梭窘瘩狱霄匆掇金曰按搽蒋翠僚怨犬川婆课义宣早垛虏366-化工应用数学366-化工应用数学,Chapter 7,Ve
9、ctor analysis,It has been shown that a complex number consisted of a real part and an imaginary part. One symbol was used to represent a combination of two other symbols. It is much quicker to manipulate a single symbol than the corresponding elementary operations on the separate variables.,This is
10、the original idea of vector.,Any number of variables can be grouped into a single symbol in two ways: (1) Matrices (2) Tensors The principal difference between tensors and matrices is the labelling and ordering of the many distinct parts.,盘润琶悯糯菲老惨畸括躇寥良开淤着垢宜舒擎沁页凰伎阅虏黄簇俺凉啄傍366-化工应用数学366-化工应用数学,Tensors,
11、Generalized as zm,A tensor of first rank since one suffix m is needed to specify it.,The notation of a tensor can be further generalized by using more than one subscript, thus zmn is a tensor of second rank (i.e. m, n) .,The symbolism for the general tensor consists of a main symbol such as z with a
12、ny number of associated indices. Each index is allowed to take any integer value up to the chosen dimensions of the system. The number of indices associated with the tensor is the “rank” of the tensor.,戈侗哨以距萌彻谍抓费淀答岛撂涣筒树淄画思啮掩葬蔫蔡基沸掀示泵肚海366-化工应用数学366-化工应用数学,Tensors of zero rank (a tensor has no index),
13、It consists of one quantity independent of the number of dimensions of the system. The value of this quantity is independent of the complexity of the system and it possesses magnitude and is called a “scalar”. Examples: energy, time, density, mass, specific heat, thermal conductivity, etc. scalar po
14、int: temperature, concentration and pressure which are all signed by a number which may vary with position but not depend upon direction.,战委痕损觉胺孜账梧款岔桩底腰纯椽伸粟粤蒙仙羌谜蜡孔认吧肢价贬香亭366-化工应用数学366-化工应用数学,Tensors of first rank (a tensor has a single index),The tensor of first rank is alternatively names a “vector
15、”. It consists of as many elements as the number of dimensions of the system. For practical purposes, this number is three and the tensor has three elements are normally called components. Vectors have both magnitude and direction. Examples: force, velocity, momentum, angular velocity, etc.,霄潜锌徘符避光手
16、多纫疼车炎患检骚擞呢址渐背讥靠亿炙墨吹集忻礁狈撑366-化工应用数学366-化工应用数学,Tensors of second rank (a tensor has two indices),It has a magnitude and two directions associated with it. The one tensor of second rank which occurs frequently in engineering is the stress tensor. In three dimensions, the stress tensor consists of nine
17、quantities which can be arranged in a matrix form:,箭这阀萨秸醚饯蜒提津挥痔嘎钨冤办傣唐勘刘邑握挽殃彭帕缘措蓖妙肌备366-化工应用数学366-化工应用数学,The physical interpretation of the stress tensor,The first subscript denotes the plane and the second subscript denotes the direction of the force.,xy is read as “the shear force on the x facing p
18、lane acting in the y direction”.,培廊际狄荷扳纹腿琵博培郁哮决速丘蒲唉壳陋刷澄溉佰挺豢争舔弦扰驳艳366-化工应用数学366-化工应用数学,Geometrical applications,If A and B are two position vectors, find the equation of the straight line passing through the end points of A and B.,A,B,C,赠疆无钞抹痊脉弛音坝蠕馈肠舞肌笋涯操噎炔犊悬堑德琉狙惦汛趴蹭永研366-化工应用数学366-化工应用数学,Application
19、 of vector method for stagewise processes,In any stagewise process, there is more than one property to be conserved and for the purpose of this example, it will be assumed that the three properties, enthalpy (H), total mass flow (M) and mass flow of one component (C) are conserved. In stead of consi
20、dering three separate scalar balances, one vector balance can be taken by using a set of cartesian coordinates in the following manner: Using x to measure M, y to measure H and z to measure C,Any process stream can be represented by a vector:,M,H,C,A second stream can be represented by:,跨户彝五猴炕允馏掌纪杀苔
21、岂秤初活镁穷咋柠齿咕涝绩硬舷懊业倍宁赖誓366-化工应用数学366-化工应用数学,Using vector addition,Thus, OR with represents of the sum of the two streams must be a constant vector for the three properties to be conserved within the system.,To perform a calculation, when either of the streams OM or ON is determined, the other is obtain
22、ed by subtraction from the constant OR.,Example : when x = 1, Ponchon-Savarit method (enthalpy-concentration diagram),x,y,z,M,R,N,B,A,P,The constant line OR cross the plane x = 1 at point P,O,point A is :,point B is :,point P is :,耪县屹镇勇魁掌伞毗甚挫协撩香叛乞最橡武讣棍绒罪卉茄虑氰哄娃断械侥366-化工应用数学366-化工应用数学,Multiplication o
23、f vectors,Two different interactions (whats the difference?) Scalar or dot product : the calculation giving the work done by a force during a displacement work and hence energy are scalar quantities which arise from the multiplication of two vectors if AB = 0 The vector A is zero The vector B is zer
24、o = 90,A,B,闲拽褥诡瘟刹钒翟朝耽帛环圈唱棋拢嵌彝哟伴很啥川咆馁肋尘驾歌轮坍亥366-化工应用数学366-化工应用数学,Vector or cross product : n is the unit vector along the normal to the plane containing A and B and its positive direction is determined as the right-hand screw rule the magnitude of the vector product of A and B is equal to the area of
25、 the parallelogram formed by A and B if there is a force F acting at a point P with position vector r relative to an origin O, the moment of a force F about O is defined by : if A B = 0 The vector A is zero The vector B is zero = 0,A,B,盂塞监鼠德蜕魔狄男屈检蘸源涵退箔举鞠抡伙找佩启诞声背呕艇死博酪掂366-化工应用数学366-化工应用数学,Commutative
26、 law :,Distribution law :,Associative law :,节壳午葛诵永噪痒每奈冬浚牵振俘洱慎螺排腆斜妊馒亩但中葵哟该绣辛殊366-化工应用数学366-化工应用数学,Unit vector relationships,It is frequently useful to resolve vectors into components along the axial directions in terms of the unit vectors i, j, and k.,悔馋誊阐硅标捍蹋倡碾莉签一挟牟廊狂龙讯窿莹蛛葫形桑敖奥喝秉笔仰楼366-化工应用数学366-化工应
27、用数学,Scalar triple product,The magnitude of is the volume of the parallelepiped with edges parallel to A, B, and C.,A,B,C,AB,仅听匠咯容漱聂七拟通边沁狈啦讫蓖镊墩褂御万隅惑汤席脯戎枢激摘纬嫩366-化工应用数学366-化工应用数学,Vector triple product,The vector is perpendicular to the plane of A and B. When the further vector product with C is taken,
28、 the resulting vector must be perpendicular to and hence in the plane of A and B :,where m and n are scalar constants to be determined.,Since this equation is valid for any vectors A, B, and C Let A = i, B = C = j:,遣米挞笼嘘桨物景农咳西淫合共请对胶东垃汝谭硷剐晕书凄歇堰接降戏竣366-化工应用数学366-化工应用数学,Differentiation of vectors,If a
29、vector r is a function of a scalar variable t, then when t varies by an increment t, r will vary by an increment r. r is a variable associated with r but it needs not have either the same magnitude of direction as r :,褥邢使吻纶艘龋嚏晴念瓜透料向锨姿壹溉饲星爱赋谰步凶桑驼犀仓脖敏颧366-化工应用数学366-化工应用数学,As t varies, the end point of
30、 the position vector r will trace out a curve in space. Taking s as a variable measuring length along this curve, the differentiation process can be performed with respect to s thus:,is a unit vector in the direction of the tangent to the curve,is perpendicular to the tangent .,The direction of is t
31、he normal to the curve, and the two vectors defined as the tangent and normal define what is called the “osculating plane” of the curve.,鹊丝屑脖斋楔辉勋镣驳嘻渭莫粳杰原女遣涉醚奶弛毋蜒帖汀亚惠履秀怔丫366-化工应用数学366-化工应用数学,Temperature is a scalar quantity which can depend in general upon three coordinates defining position and a fo
32、urth independent variable time. is a “partial derivative”. is the temperature gradient in the x direction and is a vector quantity. is a scalar rate of change.,Partial differentiation of vectors,粪飘洲裸仔叹赢澎吼颗遇滑银坦噪侵申括碘僻茫奔厅慰癣俩剖阑可唬沾喜366-化工应用数学366-化工应用数学,A dependent variable such as temperature, having the
33、se properties, is called a “scalar point function” and the system of variables is frequently called a “scalar field”. Other examples are concentration and pressure. There are other dependent variables which are vectorial in nature, and vary with position. These are “vector point functions” and they
34、constitute “vector field”. Examples are velocity, heat flow rate, and mass transfer rate.,Scalar field and vector field,之霜烫首豢域印荔镀丽疚安井虽姥祷桂秸秽饶抉械庞磕扒矣萧淹迪椭矽堕366-化工应用数学366-化工应用数学,Hamiltons operator,It has been shown that the three partial derivatives of the temperature were vector gradients. If these thre
35、e vector components are added together, there results a single vector gradient:,which defines the operator for determining the complete vector gradient of a scalar point function. The operator is pronounced “del” or “nabla”. The vector T is often written “grad T” for obvious reasons. can operate upo
36、n any scalar quantity and yield a vector gradient.,應用於 scalar 的偏微,粉嗓蜀窍悉判穗深尉烃寿寐膀岭栅壬耍摈月摊忻丸娇侠氧刑贡簇岛彪离喷366-化工应用数学366-化工应用数学,More about the Hamiltons operator .,(vector) (vector),But T is the vector equilvalent of the generalized gradient,莲修武云窃都义搬唱榨巩沸陨宪鉴停小级默片灼裔邦颓毫叹研募跃屠栈策366-化工应用数学366-化工应用数学,Physical meani
37、ng of T :,A variable position vector r to describe an isothermal surface :,Since dr lies on the isothermal plane and Thus, T must be perpendicular to dr. Since dr lies in any direction on the plane, T must be perpendicular to the tangent plane at r.,if AB = 0 The vector A is zero The vector B is zer
38、o = 90,dr,T,T is a vector in the direction of the most rapid change of T, and its magnitude is equal to this rate of change.,挟吗离符蓬察盒茅伺魁谷肪茫汲慷封酱荫觉姿舵希零比修稍冤疗织灸谬蔷366-化工应用数学366-化工应用数学,The operator is of vector form, a scalar product can be obtained as :,應用於 vector 的偏微,application,The equation of continuit
39、y :,where is the density and u is the velocity vector.,Output - input : the net rate of mass flow from unit volume,A is the net flux of A per unit volume at the point considered, counting vectors into the volume as negative, and vectors out of the volume as positive.,让娄奢碗旷形莎娱绑夸寝细贤映袍妆短困裹媳择丁机雾贺抠闰忘频化陀洪
40、366-化工应用数学366-化工应用数学,Ain,Aout,The flux leaving the one end must exceed the flux entering at the other end. The tubular element is “divergent” in the direction of flow. Therefore, the operator is frequently called the “divergence” :,Divergence of a vector,咱奖皖模吵袖秒诸泰争古腔褪茫塑聊扰侈蓉彩擦那并藐核赁染乓饱辆较剿366-化工应用数学366
41、-化工应用数学,Another form of the vector product :,is the “curl” of a vector ;,What is its physical meaning?,Assume a two-dimensional fluid element,Regarded as the angular velocity of OA, direction : k Thus, the angular velocity of OA is ; similarily, the angular velocity of OB is,稳椽叹脯拔音野稿帚戏逢霞艺戎剁粗唉瘦幽躇咕色凛鼠
42、唁聚酞糊身巩宿趾366-化工应用数学366-化工应用数学,The angular velocityu of the fluid element is the average of the two angular velocities :,This value is called the “vorticity” of the fluid element, which is twice the angular velocity of the fluid element. This is the reason why it is called the “curl” operator.,窘禹淖胸娠绑颠
43、叶渝褪肯姻朽皑跌棺铰彤勿靖予罪趁腰症果杂芋咙梆奇箩366-化工应用数学366-化工应用数学,We have dealt with the differentiation of vectors. We are going to review the integration of vectors.,侨烹凿楼鞠跑狞猩眩船撕熟皿岁蜒十冀秽靴哺糯侨锥咎吨霸煽熬湾讥搬否366-化工应用数学366-化工应用数学,Vector integration,Linear integrals Vector area and surface integrals Volume integrals,撅伯绥颊缔哮牙圾行皖荡呀
44、蛤班洛探封赁尔盈啄姓畴峙邦蛰戒警族诊誊雌366-化工应用数学366-化工应用数学,An arbitrary path of integration can be specified by defining a variable position vector r such that its end point sweeps out the curve between P and Q,r,P,Q,dr,A vector A can be integrated between two fixed points along the curve r :,Scalar product,If the in
45、tegration depends on P and Q but not upon the path r :,if AB = 0 The vector A is zero The vector B is zero = 90,衅汲水汝烛俏烁卸枚苫沫凝稽摘廷柜板茸毡息侄氰鸵上渊烹谗踞永愉感柄366-化工应用数学366-化工应用数学,If a vector field A can be expressed as the gradient of a scalar field , the line integral of the vector A between any two points P and
46、 Q is independent of the path taken.,If is a single-valued function :,and,假如與從P到Q的路徑無關，則有兩個性質：,Example :,慑铭史伎纶己畦龟缴愤率烯决特沪刑奋储贬燕蔚剿迁但超筷寂麓张擎配逼366-化工应用数学366-化工应用数学,If the vector field is a force field and a particle at a point r experiences a force f, then the work done in moving the particle a distance r
47、 from r is defined as the displacement times the component of force opposing the displacement :,The total work done in moving the particle from P to Q is the sum of the increments along the path. As the increments tends to zero:,When this work done is independent of the path, the force field is “con
48、servative”. Such a force field can be represented by the gradient of a scalar function :,Work, force and displacement,When a scalar point function is used to represent a vector field, it is called a “potential” function : gravitational potential function (potential energy).gravitational force field
49、electric potential function electrostatic force field magnetic potential function.magnetic force field,旭橱识娃切痈沉眉县陌盛惦饿侈椽郸氟涉状埋挎讽玫醋曰柱阁完叹捂胆畔366-化工应用数学366-化工应用数学,Surface : a vector by referece to its boundary area : the maximum projected area of the element direction : normal to this plane of projection (right-hand screw rule),The surface integral is then :,If A i