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化工應用數學 Vector Analysis 授課教師： 林佳璋
Introduction 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 the original idea of vector. Any number of variables can be grouped into a single symbol in two ways: (1) Matrices (2) Tensors~ introduced to indicate a general type of number of which vectors are a special case The principal difference between tensors and matrices is the labeling and ordering of the many distinct parts.
Tensors 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) . 3D9 Zmn 4D16 Zmn The symbolism for the general tensor consists of a main symbol such as z with any 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.
Tensors of Zero Rank • 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: • all physical properties are examples of scalars: • density, mass, specific heat, thermal conductivity, viscosity, diffusion coefficient, thermal conductivity, etc. • scalar point functions: • temperature, concentration and pressure which are all signed by a number which may vary with position but not depend • upon direction. • energy and time are also scalars.
Tensors of First Rank • The tensor of first rank is alternatively names a “vector”. • 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 which • are normally called components. • Vectors have both magnitude and direction can be • represented in three dimensions by a straight line in space. • Examples: • force, velocity, momentum, angular velocity, weight and area etc.
Scalars and Vectors Mass is a scalar and depends only upon the substance involved and not upon its environment, whereas weight is a force resulting from the action of gravity upon the mass. Weight is thus a vector since it acts in a direction chosen by the gravitational filed in which the scale mass is situated. Pressure is a scalar quantity because it acts equally in all directions and thus has no special direction. Scalar pressure becomes vector force when a surface is defined to sustain the pressure; the direction of the force vector is associated with the orientation of the surface and thus surface area is a vector quantity. The results of various products between vectors and scalars will be better understood later when multiplication of vectors has been defined especially the reasons why momentum is a vector, whereas energy is a scalar.
Tensors of Second Rank The one tensor of second rank which occurs frequently in engineering is the stress tensor. This of course has a double subscript and has both a magnitude and two directions associated with it. In three dimensions, the stress tensor consists of nine quantities which can be arranged in a matrix form:
y z xy xz pxx x Tensors of Second Rank 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 plane acting in the y direction”.
Addition of Vectors Any vector can be represented geometrically by a straight line with an arrow, the length of the line representing the magnitude of the vector and the direction of the line as indicated by the arrow representing the direction of the vector. The addition of two vector is defined geometrically by the well-know ”triangle of forces”. The start of the line representing the second vector is superimposed on the end of the line representing the first vector. The sum of the two vectors is represented by the line joining the start of the first vector to the end of the second vector as illustrated in the following figure. A+B B A
Subtraction of Vectors The product of two scalars is obtained according to the rules of ordinary arithmetic and to multiply a vector by a scalar, the magnitude of the vector is multiplied by the scalar and the direction remains unchanged. Multiplication of a vector by a negative number also involves reversing the direction of the vector. Thus to subtract one vector from another it is only necessary to reverse the direction of the vector to be subtracted and add the other to it as illustrated in the following figure. B A A-B
Components The process of addition described above can be reversed, allowing a vector to be resolved into equivalent constituent parts called “components”. If all components of a vector are added together using the above rule, the original vector is returned. A vector can be resolved into components in an infinite variety of ways but the resolution into components in three specified non-coplanar direction is unique. z y Az The standard set of unit vectors which define the cartesian coordinate system is i, j, k in the x, y, z directions respectively. Ay Ax x
Position Vectors The position of a point in space can be specified relative to an origin by defining the vector joining the origin to the point. Such a vector is called a “position vector”. The above definition of position is independent of any coordinate system and only requires the definition of origin. A vector defining the position of a point is usually given the symbol r which can be resolved into components in the same way as any other vector once a coordinate system has been defined. Thus The point can therefore be specified by its coordinates either by equation (*) in a vector form or by equation (**) in a scalar form. The latter is more usual and it is implied that x is measured in a direction defined by i and similarly for y and z. Thus when a point is specified by (x, y, z) it is vitally important that the order of the coordinates should not be disturbed.
Addition and Subtraction of Vectors Returning to the addition of two vectors A and B, and referring to the following figure, it can be seen that the sum of the vectors can be obtained by summing the separate components. A+B B A For two vector to be equal, their difference must be zero, and from equation (*), A can only equal B if Thus a single vector equation represents three simultaneous scalar equations.
Properties of Addition Both the associative law and the commutative law of algebra for the addition of vectors are valid. That is, The distributive law is also valid for the multiplication of the sum of two vectors by a scalar. Thus There is no divergence therefore from the normal rule of algebra when addition and subtraction of vectors is considered.
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 Referring to figure, where the origin is O and C is the position vector of any other point on the line through A and B, the vector joining A to B is B-A. Similarly, the vector joining B to C is C-B. If ABC is to be straight line, then these two vectors have the same direction and can only differ in magnitude. Therefore where m is variable scalar and C is the variable vector. O
Geometrical Applications Equation (**) is the standard equation of a straight line with m the independent variable and C the dependent variable. Equation (***) is in the form of a linear relationship between three vectors, the general form of which is Comparison of equations (***) and (****) shows that equation (***) is the special case in which p+q+r=0. Thus, the general result that if three position vectors are related by a linear equation then they are coplanar. If the sum of the three coefficients is also zero, then the end points of the vectors are colinear.
Geometrical Applications Prove that medians of a triangle are concurrent. A N M P B C L Figure illustrates the problem with reference to the triangle ABC, where L, M, N are the mid-points of the sides. It is assumed that BM and AL intersect at P and CN does not necessary pass through P. Taking C as origin, and denoting the position vectors of A and B by A and B, then the equation of the lines BM and AL can be written as follows using the results of above example. Thus the equation of the line through B and M is and as the parameter s varies, the vector determines the various points along the line BM. Similarly, the equation of AL is
Geometrical Applications Since P is the point of intersection of AL and BM, the position vector of P is given by =’, or Equation (****) relates two vector which have different directions and the equality can only be satisfied by both vectors having zero magnitude. The solution of equation (*****) is and the vector CP becomes 1/3(A+B) from equation (*). The vector CN is the sum of vector along CB and BN; and thus the position vector of N is given by Therefore the vectors CN and CP have the same direction (A+B) and thus CN must pass through P.
Geometrical Applications Generalized Vector Method for Stagewise Processes. It has been shown by Lemlich and Leonard that many of the existing difference point constructions for determining the number of theoretical steps in a stagewise process are variations of a more general vector method. They have also shown that the comprehensive vector method suggests many hitherto untried constructions and this example from their work derives the standard rectangular diagram from the general vector diagram. 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 considering three separate scalar balances, one vector balance can be taken by using a set of cartesian coordinates in the following manner.
Geometrical Applications Using x to measure M, y to measure H and z to measure C, then any process stream can be represented by a vector thus, Similarly, a second stream can be represented by 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. Hence, to perform a calculation, when either of the streams OM or ON is determined, the other is obtained by subtraction from the constant OR.
Geometrical Applications It will now be shown that the intersection of these vector with plane x=1, gives the usual enthalpy-concentration diagram for the Ponchon-Savarit method. The constant line OR will cross the plane x=1 at point P which will be a fixed point. The variable vectors OM and ON will also cross the plane at A and B respectively, and since OMRN is parallelogram, APB will be a straight line. Let Then by the triangle rule of addition,
Geometrical Applications The constant line OR cross the plane x = 1 at point P point A is : point B is : point P is : Compare with these with the original interpretation of the coordinate axes, the appropriate two-dimensional coordinates are the enthalpy and the mass fraction of the important component.
A Asin B Acos Multiplication of Vectors Scalar or dot product: The scalar product of two vector is defined as the product of the magnitude of the vector with the magnitude of the component of the other vector resolved along it. This product is signified by placing a dot between the vector to be multiplied together. Symbolically, where is the angle between the vector as shown in figure. The result of the operation is a scalar quantity. One physical interpretation of this product is the calculation giving the workdone by a force during a displacement. The fundamental definition of the work done is the magnitude of the force multiplied by the distance moved by its point of application in the direction of the force. Alternatively, the work done is the displacement multiplied by the component of force causing the displacement. That these two alternatives are equivalent is obvious from the definition. Work and hence energy are thus scalar quantities which arise from the multiplication of two vectors.
Multiplication of Vectors When a vector is multiplied by itself in this fashion, the scalar result can be written in the equivalent forms The inverse process to multiplication, division, is not uniquely defined for the following reason. If an equation such as is satisfied, then any of the following three conclusions can be drawn. (a) The vector A is zero. (b) The vector B is zero. (c) =90, and A and B are mutually perpendicular. The third possibility arise because cos may be zero. It is therefore possible for the scalar product of two vectors to be zero when both vectors are finite. Division thus have no meaning, for if equation (*) could be divided by a non-zero A, the inevitable conclusion would be that B is zero which is not necessarily true.
A Asin ’ B Acos Multiplication of Vectors Vector or cross product: Referring to figure in which is the angle between the two vectors A and B, then the vector product of A and B is defined by where the symbol () is used to denote a vector product, n is a 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 right-hand rule is applied to the sense of rotation from A to B in measuring the angle , thus defining the positive direction of n. Referring to figure, this rule indicates that n acts into the plane of the paper, thus AB is also a vector into the plane of the paper.
A Asin B Multiplication of Vectors This definition is unique, because if were described in the opposite sense as indicated by ’, the positive direction of n would be reversed, the sign of sin would also be reversed, the vector AB would still act into the plane of the paper. However, the vector BA has the same magnitude as AB but the direction of n is reversed due to describing by moving from B to A. Hence and the order of terms in the product is important. The magnitude of the vector product of A and B is equal to the area of the parallelogram formed by A and B. Figure shows the base of the parallelogram of length B and height Asin, giving an area of ABsin, or |A||B| sin.
F r O P rsin Multiplication of Vectors The moment of a force is defined by a vector product in the following manner. If there is a force F acting at a point P with position vector r relative to an origin O as illustrated in Figure, then the moment of F about O is defined by In order to define the positive normal n, the vector r and F must have a common origin. If P is taken as this origin and the right-hand screw rule applied to equation (*) then n is seen to act out of the plane of the paper. Using the fundamental definition of L as the product of the force and the perpendicular distance of the origin from the line of action of the force gives The direction of the vector L which is outwards from the plane of the paper also corresponds with the fundamental definition of a moment.
Multiplication of Vectors Again, vector division is impossible, since if then either (a) The vector A is zero. (b) The vector B is zero. (c) is zero, and A and B are parallel.
Properties of Multiplication Commutative law: The commutative law is valid for the scalar product. The commutative law is not valid for the vector product. Distribution law: The distributive law is unreservedly valid for all vector products.
Properties of Multiplication Associative law: In equations (*) and (**) each dot product yields a scalar which can be treated as any other scalar, so that equation (*) is a product of two scalars and in equation (**) the magnitude of the vector C is increased by the scalar multiplier (A.B). In equations (***) the vector product must be taken first because a vector product cannot be formed between the vector A and the scalar (B.C). Bracket are usually essential when more than one vector product is involved, and then the associative law is not valid because
Unit Vector Relationships It is frequently useful to resolve vectors into components along the axial direction in terms of the unit vector i, j, k. All operations are then performed on the unit vectors and the results of products between them are all standard. By the definition of the vector product, ij is a vector normal to the plane of i and j and hence must lie in the k direction. A convention is now needed which determines whether ij is in the positive or negative k direction. If ij=k, the set of a axes is said to be “right-hand”, since k is defined by the right-hand screw rule from the vector product of i and j; whereas if ij= -k, the set of axes is left hand. It is normal to use a right-hand set of axes.
Unit Vector Relationships The use of unit vector relationships is illustrated by the straightforward application to the simple products of the two vectors
Example 空間二平面方程式為x+y+z=1, 2x+cy+7z=0，若此二平面互相垂直，則c為何值? 此平面與ar=0之平面平行 兩平面垂直
Example 試求通過A(1,1,1), B(5,4,3), C(10,8,6)三點之平面 A(1,1,1) 所求之平面為
Example 試求通過三頂點A(1,0,0), B(0,1,0), C(0,0,1)三角形之面積 三角形ABC面積=
AB C B A Scalar Triple Product Because AB is a vector, further products can be taken with a third vector. The scalar product of AB with C is best considered geometrically as shown in the following figure, where is the usual angle between the two vectors (AB) and C. By definition, is a scalar whose magnitude is the magnitude of (AB) multiplied by the component of the vector C resolved along (AB). But the magnitude of (AB) is the area of the parallelogram formed on A and B, and since the vector (AB) must be perpendicular to the plane of A and B, then the resolved part of C along (AB) must equal the height of the parallelepiped as show in the above figure.
Scalar Triple Product The magnitude of AB.C is thus the volume of the parallelepiped with edges parallel to A, B and C. The result is positive or negative according as is acute or obtuse. Since the volume of a parallelepiped can be expressed as the product of any base with the corresponding height, and the order of vectors in a dot product is irrelevant, the scalar tripe product can be written in six equivalent ways, thus where the three vector remain in the cyclic order A, B, C and the positions of the dot and cross are arbitrary. The other six ways of writing this triple product have the same numerical value as the above expressions but have the opposite sign due to reversing the order of terms in the vector product. Thus There are seven ways in which the parallelepiped can have zero volume. Any of the three vectors may be zero, any pair of vectors may be equal or parallel, or the three vectors may lie in the same plane.
Vector Triple Product The vector product of AB with a third vector C can be taken, but in this case the order of multiplication position of brackets is of vital importance. It can be seen that the vector AB is perpendicular to the plane of A and B. When the further vector product with C is taken, the resulting vector must be perpendicular to AB and hence in the plane of A and B. The resulting vector triple product can therefore be resolved into components along A and B, thus where m and n are scalar constants to be determined. Multiplying throughout equation (*) by C using a scalar product gives Considering AB as a combined vector, the left-hand side of equation (**) is a scalar triple product with the vector C appearing twice. Hence the left-hand side is zero.
Vector Triple Product Since each term in equation (*) contains the three vectors, must be just a number. Also, because equation (*) is valid for any vectors A, B, and C, it must be valid for a particular set of values for A, B, and C. Thus letting A=i, B=C=j, and using unit vector relationships gives Using these results in equation (*) gives Similarly it can be shown that The vector appearing outside the brackets on the right-hand side are the vectors appearing inside the brackets on the left-hand side. The positive term is the dot product of the extreme vectors multiplied by the central vector.
Differentiation of Vectors If a vector r is a function of a scalar variable t (say time), then when t varies by an increment t, r will vary by an increment r. Just as in ordinary calculus, r is a variable associated with r but it needs not have either the same magnitude or direction as r . If r tends to zero as t tends to zero, then defines the first derivative of r with respect to t. If r is interpreted as a position vector resolved into its components, then and taking the limit as t0,
Differentiation of Vectors Differentiation of a vector with respect to a scalar is thus similar to ordinary differentiation and the rules as applied to products of vectors are unchanged, As t varies, the end point of 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,
Differentiation of Vectors Therefore, dr/ds is a unit vector in the direction of the tangent to the curve. If A is a vector of constant magnitude but variable direction, then A2 will be a constant. and the vector dA/dt is perpendicular to A. It has been shown above that dr/ds is a unit vector and hence differentiating again with respect to s and using equation (*), then d2r/ds2 must be perpendicular to the tangent dr/ds. The direction of d2r/ds2 is the normal to the curve, and the two vectors defined as the tangent and normal define what is called the “osculating plane” of the curve.
Partial Differentiation of Vectors • Temperature is a scalar quantity which can depend in general upon three coordinates defining position and a fourth • 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.
Scalar Field and Vector Field • A dependent variable such as temperature, having these 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 constitute “vector field”. • Examples are velocity, heat flow rate, and mass transfer rate.
Hamilton’s Operator It has been shown that the three partial derivatives of the temperature were vector gradients. If these three 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 upon any scalar quantity and yield a vector gradient.
Hamilton’s Operator The nature of T can be further illustrated by taking a scalar product of T with an infinitesimal increment dr in the position vector r. Thus and using the properties of products of unit vectors, this simplifies to If vector division were possible ( which it is not) then equation (*) could be rearranged to show that T is the ratio of dT to dr, which is the generalized first derivative of a scalar variable with respect to an independent vector variable. The expression of dT/dr does not exist to represent this quantity, but T is the vector equivalent of the generalized gradient.
T dr Physical Meaning of T A variable position vector r to describe an isothermal surface Since the vector dr can lie in any direction in the tangent plane to the isothermal at r, and T can only have one value at a point r, then T must be perpendicular to the tangent plane at r. This direction is the line of most rapid change of T. Thus T is a vector in the direction of the most rapid change of T, and its magnitude is equal to this rate of change.
Example 某曲面之方程式為f(x,y,z)=x2+y2+z2=32，求在P點(4,4,0)之單位法線向量? (4,4,0) 若一曲面方程式為x2+y2/4+z2/9=3，求過曲面上P(1,2,3)點之切面? (1,2,3) 切面之方程式為
Divergence of a Vector The operator is of vector form, a scalar product can be obtained as : application The equation of continuity : 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.
Divergence of a Vector 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
