Notes on Vector Calculus (following Apostol, Schey, and Feynman)
Frank A. Benford May, 2007
1. Dot Product, Cross Product, Scalar Triple Product.
The standard inner product in ‘8 is the “dot product,” defined as follows.If +,œ+ß+ßáß+and œ,ß,ßáß, a"#8bab"#8 then 8 +,†´+,Þ (1.1) " 33 3œ" The standard norm in ‘8 is defined in terms of the dot product as
+´++†œ+##€+€â€+Þ# (1.2) ll È É"# 8 In ‘‘#$ and the norm of a vector is its length. If ??œ", then is said to be a unit ll vector. Some special notation is used in ‘# and ‘‘$#. A point in is sometimes written BßC and a point in ‘$ is sometimes written BßCßD. Alternatively, the symbols BC, , and abab D replace the indices "ß #ßand 3. For example, we might write + œ+ß+ß+ for a vector abBCD in ‘$. The symbols 3, 45, and denote the three standard unit coordinate vectors. Hence BßCßDœB3€C4€D5and +œ+ß+ß+œ+3€+45€+ . ababBCDBCD Suppose that ?B is a unit vector. For any vector , B†?œœB?Bcos))cos llllll where ) is the angle between ? andB. This means that B†?B is the component of in the direction of ?, and B†??B is the orthogonal projection of onto the subspace of scalar ab multiples of ?? (called the subspace of vectors “spanned” by )Þ This is illustrated below.
B
) ! ?B†?? ab
Vector Calculus. Page 1 The cross product of two vectors +,œ+ß+ß+ and œ,ß,ß, in ‘$ is defined as aBCDbabBCD +‚,´+,•+,3€+,•+,45€+,•+, aCDDCbaDBBDbabBCCB ââ345 (1.3) ââ œââ+B++CD. ââ ââ,B,,CD ââ It may be shown that +‚, equals the area of the parallelogram determined by +, and ll (i.e., +,sin ) where ) is the angle between + and ,, !Ÿ)1Ÿ‚), and that +, is llll orthogonal to the plane determined by + and ,. More precisely, the direction of +,‚ is determined by the “right-hand” rule as follows: if the right hand is held with the thumb stuck out and with the fingers curled in the direction of rotation of +, into , then the thumb points in the direction of +,‚. In other words, if the index finger of the right hand is pointed forward and shows the direction of +, and if the middle finger is bent to show the direction of ,, and if the thumb is perpendicular to the plane determined by the index and middle finger, then the thumb points in the direction of +‚‚,. Because +, is orthogonal to both +, and , it follows that +†‚+,œ,†‚+,œ!Þ abab Also, it's clear that +‚œ+!+for any vector . Cross products have the following algebraic properties. +‚,œ•‚,+ ab +‚,€-œ+‚,€‚+- ababab 0+‚,œ00+‚,œ‚+, ababab (where 0 is any scalar). It may be shown that +‚,‚-œÐ+†-Ñ,•Ð+†Ñ,- (1.4) ab for any three vectors +, ,-, and .
The scalar triple product of any three vectors +, ,-, and is defined as the scalar +‚,-†Þ ab It may be shown that +‚†,- is the volume of the parallelepiped determined by +,, , kkab and -. This suggests (and itmay be shown to be true) thata cyclic permutation of the three vectors does not affect the scalar triple product; that is, +‚,†-œ,‚-†+œ-‚†+,.(1.5) ababab The commutativity of the dot product then implies that the dot and cross products in a scalar triple product may be interchanged: +‚,†-œ+†,-‚Þ (1.6) abab
Vector Calculus. Page 2 Finally, it may be shown that the scalar triple product +†‚,- may be written in terms of a determinant as follows: ab
ââ+B++CD ââ +†,-‚œÞââ,B,,CD (1.7) abââ ââ-B--CD ââ
2.The Gradient.
Let H be a subset of ‘‘8. Definition: a scalar field on HH is a mapping from into ; a vector field on HH is a mapping from into ‘8.
Suppose that HH be a subset of ‘:8 and that is a differentiable scalar field defined on . For any point < œBßBßáßB in H8, the -tuple ab"#8 `:``:: f::<<´grad ´ßßâß (2.1) abab Œ•`B"`B#8`B (where each partial derivative is evaluated at <) is called the gradient of ::. We'll write f or grad : if the point where the partial derivatives are to be evaluated is clear. The collection of vectors f:Ð<ÑH constitutes a vector field over .
Example 1. Let :Ð<<Ñ´´<œB##€B€â€B#. Then ll È"# 8 " `:`<"B###•#3 œœ#B3B"€B#8€â€Bœ.(2.2) `B33`B# fœ:<<<•" , (2.3) abll a unit vector in the direction of <. As :: is differentiable, the derivative of at Vector Calculus. Page 3 Suppose, now, that < is a differentiable vector-valued function that maps an interval of real numbers +ß, into H©‘8. For any >−+ß, we write cdcd <Ð>ÑœBÐ>ÑßBÐ>ÑßáßBÐ>ÑÞ ab"#8 The parameter > is commonly interpreted as time. The vector <Ð>Ñ traces out a curveor “path” in ‘8 as > varies over +ß,. The vector of derivatives cd > =Ð>Ñœ Vector Calculus. Page 4 f: Potential functions. The meaning of “potential function” varies from author to author. Broadly speaking, there are two definitions, one used by mathematicians and the other used by physicists. Mathematicians. Let J be a vector field defined on a set H©‘8. If there exists a scalar field : defined on H such that J œf::, then is said to be a potential function for J. Physicists. Let J be a vector field defined on a set H©‘8. If there exists a scalar field Y defined on H such that JJœ•fYY, then is said to be a potential function for . In mechanics, the notion of a potential function is applied almost exclusively to force fields. A vector field J is said to be a force field if J< may be interpreted as the force acting ab on a particle at the point <. If there exists a potential function for a force field JJ, then is said to be conservative (for reasons that will be explained later). In other parts of physics, the use of “potential function” is broadened. For example, in electrostatics the force on a charge ; is given by ;ÞII where is a vector field called the “electric field” If there exists a scalar field Y such that Iœ•fYY, then is said to be an electrostatic potential. The two notions of “potential function” differ principally in a sign convention; clearly YÐ<<Ñœ•::ÐÑY. I will attempt to use the symbols and consistently to denote, respectively, the mathematician's and physicist's meaning of “potential function.” Example 2. In ‘$P, let Y<<œ<, where P is an integer and <´ . Generalizing Example 1, it may be showanb that ll •fYœ•P Example 3: The Newtonian potential. Newton's law of gravitation says that the force which a particle of mass Q exerts on a particle of mass 7 is a vector of norm K7QÎ<# and directed from the particle of mass 7 towards the particle of mass QK, where is a proportionality constant and < is the distance between the two particles. Hence, if the particle of mass Q7 is placed at the origin and the particle of mass is located at <œB3€C45€D7, then the force acting on the particle of mass is given by Vector Calculus. Page 5 K7Q Jœ•< Example 4: Central forces. A central or “radial” force field J in ‘$ is one that can be written in the form JÐcentral force ll < is directed radially, either towards the origin (if 2Ð<Ñ•!), or away from the origin (if 2Ð<Ñž!), and the magnitude of the force at any point depends only on the distance from the center to that point. Proposition. Every central force field is conservative. Proof. Define LÐ<Ñœ2Ð<Ñ. 3. Divergence and Curl The symbol ff is called “del” or “nabla.” It is useful to think of as a vector operator: ``` fœßßâßÞ Œ•`B"`B#8`B In ‘$, we write ``` fœ3€45€Þ `B`C`D “Multiplication” by `/`BB33 means “take the partial derivative with respect to .” That is, if Vector Calculus. Page 6 ::Ð<ÑœÐB"ßB#8ßáßBÑ is a scalar field, ``: :´Þ Œ•`B33`B Playing with this operator as if it were a real vector often (but not always) yields results that turn out to be true. For the true results, then, this device has heuristic utility. For example, suppose that J is a vector field defined on H©‘$. For anypoint <´B3€C45€DH in we'll write JÐ<ÑœJBÐ<Ñ3€JCDÐ<Ñ4€JÐÑ<5. Operating in a purely formal manner, we may form both a dot product and a cross product of fÞ and J These operations yielda scalar `J`J `J f†J ´BD€€C (3.1) `B`C`D and a vector ââ345 ââ ââ``` f‚´J ââ`B`C`D ââ ââJJJ (3.2) ââBCD `J`J`J`J`J `J œD•CC3€B•DB45€• , Œ`C`D•Œ`D`B•Œ•`B`C where all partial derivatives are to be evaluated at the point <. Amazingly, both these objects are meaningful and useful. The scalar f†JJ is called the divergence of and is also written “div J.” The vector f‚JJ is called the curl of and is also written “curl J .” We will give geometric interpretations of f†JJ and f‚ after our discussion of line and surface integrals. However, two simple examples at this stage will start to give the reader some idea of the meaning of the divergence and the curl. Example 1. Suppose JÐ<Ñ´<œB3€C45€D . That is, this vector field is radially directed, and JÐ<Ñœ<<, the distance from the origin to . Hence, llll `B`C`D div J<ÐÑœ€€œ$Þ `B`C`D Example 2. Consider a rigid circular disk rotating around an axis through its center and perpendicular to the plane of the disk. Without loss of generality, we may set up the coordinate system so that the disk rotates in the BC-plane, and the axis of rotation coincides with the D coordinate. Let = denote the angular speed of the disk (in radians per second). Physicists find it convenient to let = œ =5 denote the angular velocity of the disk: that is, Vector Calculus. Page 7 angular velocity is a vector with magnitude = directed along the D axis. Let <œB34€C denote a point on the disk. The speed of that point depends on = and on <´ < according ll to the equation @œ<=. In more detail, the velocity @ of that point (a vector) is given by ââ345 ââ @œ=‚<œââ!!=œ=•C34€B ââab ââBC! ââ where @´Þ@Note that @@ is a vector field. We now ask: what is the curl of ? From eq. (3.2), ll ââ345 ââ curl ââ``` @œf‚œ@5ââ`B`C`D œ#=œ#Þ= ââ ââ•==CB! ââ In words, the curl of linear velocity is just twice the angular velocity of the disk. So far in our play with f we've only considered first derivatives. When we consider second derivatives, four of the possible combinations turn out to be meaningful and useful. (1) f†fœ::divgrad . Working formally, we find abab ````:``:: f†f:œ3€4€5†3€€45 abŒ`B`C`D•Œ•`B`C`D ```###::: œ€€Þ `B#`C##`D It turns out that this scalar field is very useful in physics. The operation f†f: is ab called the Laplacian of : and is written f##::. If f<<œ! for all in some volume ab H, the scalar function : is said to be harmonic. The Laplacian of a vector field J is defined “component-wise”: if JœJB3€JCD45€J then #### fJœfJB3€fJCD45€fJÞ Ð#Ñf‚f:œcurlgrad :. For any vector @ and scalar00, we know that @@‚œ ababab @‚@0:œ!!. This suggests that f‚f , the curl of a gradient, should equal . abab This turns out to be true under some weak conditions: if : is a scalar field with continuous second-order mixed partial derivatives, then curl(grad :)œÞ!Conversely, it may be shown that if curl JœH!B for all points in an open convex set , then there exists a scalar field :: defined on H such that JœfÞ (3) f†(f‚JÑœdivcurl JÞ For any vectors + and ,, we know that +†‚œ+, 0. abab This suggests that f†(f‚JJÑœdivcurl œ!. This is in fact the case: if all the ab mixed partial derivatives of a vector field JJ are continuous, then f†(f‚Ñœ divcurl JKœ!. Conversely, if H is an open interval in ‘$, and f†œ! throughout ab Vector Calculus. Page 8 Hœ, then Kcurl JJ for some vector field .[An “open interval” in ‘$ is the Cartesian product of open intervals. That is, an open interval in ‘$ has the form +ß,‚Ð+ß,Ñ‚Ð+ß,Ñ where +•,ß+•,ß and +•,.] abBBCCDDBBCCDD (4) f‚f‚JœcurlcurlJ. Equation (1.4) may be written +‚,-‚œ ababab ,Ð+†-Ñ•Ð+†,Ñf-. If we substitute for + and ,, and J- for , we obtain f‚f‚Jœff†J•f†fJœff†JJ•f# (3.3) abababab which holds if all mixed partial derivatives are continuous. In other words, curlcurl Jœgraddiv JJ•f# . abab (There are other ways the right-hand side of eq. (1.4) may be written, but these lead to meaningless formulae when f is substituted for +, and .) 4. Line Integrals. Let < be a vector-valued function that maps an interval of real numbers +ß, into H©‘8. cd If << is continuous on +ß,8, then is said to be a continuous path in -space. The path is cd said to be smooth if Let Other notations for line integrals. If denotes the graph of , the line integral G<'J<†. is also written as J†.< and is called the integral of J along GÞ If +<œÐ+Ñ and 'G ,œ<Ð,Ñ, then the line integral is sometimes written as ,,J†. Vector Calculus. Page 9 ,,88 J†.<œJ<<Ð>ÑBwwÐ>Ñ.>œJÐ>ÑBÐ>Ñ.>Þ ""3333 ( ((++3œ"cd3œ" cd In this case, the line integral is also written as . In # the ' J".B"€J#.B#€â€J88.B ‘ path < is usually written as a pair of parametric equations BÐ>ÑßCÐ>Ñ , and the line integral ab J†.<< is written J.B€J.C. Similarly, in ‘$ the path is usually written as a 'G 'G BC triple of parametric equations BÐ>ÑßCÐ>ÑßDÐ>Ñ and the line integral J<†. is written ab 'G J.B€J.C€J.D. 'G BCD Basic properties of line integrals. Line integrals share many of the fundamental properties of ordinary integrals. For example, they have a linearity property with respect to the integrand: αJ€"K†.<œα"J†.<€K<†. (ab(( and an additive property with respect to the path of integration: J†.<œJ†.<€J<†. (G((GG"# where the two curves G"# and GG make up the curve . Change of parameter. As evaluation of the integral J<†. makes use of the parametric 'G representation <Ð>ÑG, it might seem that an alternative parameterization of the curve would yield a different value of J†.<. In fact, the value of J<†. is invariant with respect ''GG to the parameterization of G8 up to a change of sign. Let < be a continuous path in -space defined on an interval +ß,1, and let be a differentiable real-valued function defined on an cd interval -ß. such that (1) 1w is never zero on -ß., and (2) 1 maps-ß. onto +ß,. Then cdcdcdcd the function ë<À-ß.Ä‘8defined by cd ë<<Ð?Ñ´1? cdab is a continuous path having the same graph G as <.Two paths << and ë so related are said to be equivalent. If 1wž! everywhere on -ß.G, we say that << and ë trace out in the same cd direction, and if 1w•! everywhere on -ß.G, we say that << and ë trace out in opposite cd directions. In the first case, the change of parameter function 1 is said to be orientation- preserving, and in the second case it is said to be orientation-reversing. Theorem 4.1. Let << and ë be equivalent piecewise smooth paths. Then we have J†.<œJ<†.ë ((GG if << and ë trace out G in the same direction, and Vector Calculus. Page 10 J†.<œ•J<†.ë ((GG if << and ë trace out G in opposite directions. Line integrals with respect to arc length. In some circumstances the arc-length function = provides a natural and convenient parameterization of G2, the graph of <. Suppose that is a scalar field defined and bounded on G. The line integral of 2 with respect to arc length along G is denoted by 2.= and defined by 'G , 2.=´2<Ð>Ñ=wÐ>Ñ.>ß (4.2) ((G+cd whenever this integral exists. In particular, consider the scalar field given by 2<Ð>Ñ´J R 2= obtained when the curve G is partitioned into R4 segments, where the th segment is of length ?=44 and contains the point <. If G is a closed path, the line integral J†X.=œJ<†. **GG is called the circulation of J around G. The concept of work in mechanics. Consider a particle which moves along a curve in ‘$ under the action of a force field J<. If the curve is the graph of a piecewise smooth path , then the work done by is defined to be the line integral J'J<†.Þ The principle of work and energy. Suppose a particle of mass 7 moves freely through space under the action of a force field J . If the speed of the particle at time > is @Ð>Ñ, then " # its kinetic energy is defined to be # 7@Ð>Ñ . We may show that the change in the particle's kinetic energy in any time interval is equal to the work done by J during that time interval. Proof. Let <Ð>Ñ denote the position of the particle at time >, for all >−+ß,. We want to cd Vector Calculus. Page 11 show that <Ð,Ñ "" J<†.œ7@Ð,Ñ# •7@+#.(4.3) (<Ð+Ñ ##ab The motion of the particle at any time is governed by Newton's second law of motion, which says J<Ð>Ñœ7<@wwwÐ>Ñœ7Ð>Ñ cd where @@Ð>Ñ denotes the velocity vector at time >, and @Ð>Ñ´Ð>Ñ . Hence ll ".". J<Ð>ц <Ð,Ñ, w"##,""# J†.<œJ<<Ð>цÐ>Ñ.>œ7@Ð>Ñ+œ7@Ð,Ñ•7@+ , ((<Ð+Ñ+cd#•‘##ab as was to be shown. Independence of the path. Suppose that J is a vector field that is continuous on an open connected set H©‘8. [For the definition of “open connected set” see Apostol, pp. 332- 333.] In general, the line integral , J<†. (+ depends not only on the end points + and ,<, but also on the path ÐÞÑ that connects them. For some vector fields J, however, ,J†.<< doesn't depend on ÐÞÑ, and in this case we '+ say the integral is independent of the path from + to ,. If the integral ,J<†. is '+ independent of the path from + to , for all + and , in H, then we'll say that ,J<†. is '+ independent of the path in H. Let G be a piecewise smooth closed path in HH, where is an open connected set in ‘8. Let + and , be two distinct points on the path G. If the integral ,J<†. is independent of '+ the path from + to ,J, then the circulation of around G is zero: J<†.œ!Þ *G If the integral , J†.< is independent of the path from + to ,+ for every pair of points and '+ ,J, then the circulation of around G is zero, J<†.œ!ß (4.4) *G Vector Calculus. Page 12 for every piecewise smooth closed path GH in . Conversely, if eq. (4.4) holds for every piecewise smooth closed path G in H, then ,J<†.H is independent of the path in . '+ The second fundamental theorem of calculus for real functions states that: , :wÐBÑ.Bœ::Ð,Ñ•Ð+Ñ (+ provided that :w is continuous on some open interval containing both +, and . An analogous result holds for line integrals. Theorem 4.2 (The second fundamental theorem of calculus for line integrals). Let : be a differentiable scalar field with a continuous gradient f:‘ on an open connected set H© 8. For any two points +, and joined by a piecewise smooth path <ÐÞÑHin we have , f:†.<œ::Ð,+Ñ•ÐÑÞ (4.5) (+ Corollary. Equation (4.5) implies that ,f:†.< is independent of the path from +, to . '+ As eq. (4.5) holds for every pair of points + and ,< in H, it follows that ,f:†. is '+ independent of the path in H. Hence, f:†.<œ! (4.6) *G for every piecewise smooth closed path G in HÞ In words, the circulation of a gradient around any piecewise smooth closed path in H is zero. The conservation of mechanical energy in a conservative force field. Suppose a particle of mass 7 moves freely through space under the action of a force field J . We have previously shown that the work done by J over an interval of time equals the change in the kinetic energy of the particle during that time interval. To be precise, if Ò+ß,Ó is the time interval, then <Ð,Ñ "" J<†.œ7@Ð,Ñ# •7@+# (4.3) (<Ð+Ñ ##ab where <Ð>Ñ denotes the location and @Ð>Ñ denotes the speed of the particle at any time >−Ò+ß,Ó. Assume now that JJ is a conservative force field, so œ•fY for some potential function Y . Then <<Ð,ÑÐ,Ñ J†.<œ•fY†.<œY<<Ð+Ñ•YÐ,Ñ .(4.7) ((<<Ð+ÑÐ+Ñ cdcd Combining equations (4.3) and (4.7) and rearranging, we find "" 7@Ð,Ñ# €Y<<Ð,Ñœ7@+# €YÐ+Ñ .(4.8) ##cdabcd Vector Calculus. Page 13 The function Y << gives the potential energy of the particle at . Equation (4.8), then, says that the sumab of the kinetic and potential energy of a particle is a conserved quantity if the particle moves under the action of a conservative force field. (This explains why such force fields are said to be “conservative.”) A converse of Theorem 4.2 is also true. Theorem 4.3. Let J be a continuous vector field defined on an open connected set H©‘8. If the line integral ,J<†. is independent of '+ the path in H[or if eq. (4.4) holds for every piecewise smooth closed path GH in ] then there exists a differentiable scalar field :: on H such that JÐ<<ÑœfÐÑ. Suppose that J is a vector field that is continuous on an open connected set H©‘8. Theorems 4.2 and 4.3 give us necessary and sufficient conditions for JJ to be a gradient: is a gradient if and only if ,J†.H Theorem 4.4. Let JÐ<ÑœJÐ<ÑßJÐ<<ÑßáßJÐÑ be a continuously differentiable ab"#8 vector field defined on an open connected set H©‘8. If J is a gradient, then `J `J 3 Ð<<Ñœ4ÐÑ (4.7) `B43`B for all 3 and4 in "ß#ßáß8 and all As a corollary of Theorem 4.4, we have formula (2) of Section 3: for any continuously twice differentiable scalar field :‘ defined on an open connected set H© $, curl f::œf‚fœÞ! ab The proof is left to the reader. If the set H of Theorem 4.4 is assumed to be convex, then eq. (4.7) gives sufficient conditions for J to be a gradient. Theorem 4.5. Let JÐ<ÑœJÐ<ÑßJÐ<<ÑßáßJÐÑ be ab"#8 a continuously differentiable vector field defined on a convex open connected set H©‘8. If Vector Calculus. Page 14 `J `J 3 Ð<<Ñœ4ÐÑ `B43`B for all 3 and4 in "ß#ßáß8 and all <−HH, then there exists a scalar field : defined on ef such that JJœf:. For the proof, see Apostol, pp. 351-352. Corollary: Suppose that is a continuously differentiable vector field defined on a convex open connected set H in ‘$. If curl Jœf‚œJ! everywhere in H, then there exists a scalar field :: defined on H such that Jœf . 5. Surface Integrals There are several ways to specify a “surface” in ‘$. (1) Implicit representation. The set of all points BßCßD that satisfy an equation of the form JBßCßDœ!. (2) Explicit abab representation. Sometimes one can solve JBßCßDœ! for one of the variables in terms ab of the other two. For example, suppose it's possible to solve for D in terms of BC and . The solution Dœ0ÐBßCÑ is said to be an explicit representation of the surface. (3) Parametric representation. We have 3 equations expressing B, CD, and as functions of two parameters ?@ and : Bœ\Ð?ß@ÑßCœ]Ð?ß@ÑßandDœ^Ð?ß@Ñ (5.1) where ?ß@ is allowed to vary over some connected set X in the ?@-plane. Sometimes we'll write thabe three parametric equations of eq. (5.1) in a single vector form: <Ð?ß@Ñœ\Ð?ß@Ñ3€]Ð?ß@Ñ45€^Ð?ß@ÑÞ (5.2) The image of X under the mapping << is called a parametric surface and is denoted ÐXÑ. We assume that \, ]^, and are continuous. If the mapping < is one-to-one, the image <ÐXÑ is called a simple parametric surface. Note that an explicit representation of a surface is obtained from a parametric representation with the functions \Ð?ß@Ñœ?, ]Ð?ß@Ñœ@, and ^Ð?ß@Ñœ0Ð?ß@Ñ. The fundamental vector product. If \, ], and ^X are differentiable on , we consider the two vectors `<`\`]`^ œ3€€45 `?`?`?`? and `<`\`]`^ œ3€45€Þ `@`@`@`@ The cross product of these two vectors is referred to as the fundamental vector product of Vector Calculus. Page 15 the representation <. ââ345 ââ ââ`\`]`^ ``<<ââ R´‚œââ`?`?`? `?`@ ââ ââ`\`]`^ ââ ââ`@`@`@ â`]`^ââ`^`\ ââ`\`] â (5.3) âââââ â â`?`?ââ`?`? ââ`?`? â œââ34€€âââ â5 â`]`^ââ`^`\ ââ`\`] â â`@`@ââ`@`@ ââ`@`@ â ââââââ `]ß^`^ß\`\ß] œab3€€ab45ab. `?ß@`Ð?ß@Ñ`?ß@ abab If Ð?ß@Ñ is a point in X at which both `<Î`? and `<Î`@Á are continuous and R!, then the image point <Ð?ß@Ñ is said to be a regular point of <<. If Ð?ß@Ñ is not a regular point, then it is said to be a singular point of <<. A surface ÐXÑ is said to be smooth if all of its points are regular points. In the case of an explicitly represented surface <ÐBßCÑœB3€C45€0ÐBßCÑ we have `<<`0``0 œ3€5and œ€45 `B`B`C`C so ââ345 ââ`0`0 Rœââ"!`0Î`B œ•3•45€Þ (5.4) ââ`B`C ââ!"`0Î`C ââ Note that `0##`0 Rœ"€€ (5.5) ll Ë Œ`B•Œ•`C in this case. As each vector `<Î`? and `<Î`@ is tangent to the surface Vector Calculus. Page 16 Let V denote a rectangle with base ?? and height ?@ in X, where ???@ and are “small.” The image <ÐVÑ is approximately a parallelogram with sides ``<< ???@and. `?`@ The area of this parallelogram is `<`<``<< ??‚?@œ‚???@Þ ¾`?`@¾¾¾`?`@ Hence ``<< Rœ‚ ll ¾¾`?`@ may be thought of as a local magnification factor for areas. The area of a parametric surface. Let W´<ÐXÑ. The computation given above suggests the following definition. The area of W, denoted EÐWÑ, is defined by the double integral ``<< EÐWÑ´‚.?.@Þ (5.7) (( ¾¾`?`@ X If W is defined explicitly, this integral becomes `0##`0 EÐWÑ´"€€.B.C (( Ë Œ`B•Œ•`C X where X is the projection of W onto the BC-plane. Definition. Let W´<<ÐXÑ be a parametric surface described by a differentiable function defined on a region X in the ?@-plane, and let 1W be a scalar field defined and bounded on . The surface integral of over is denoted by the symbol [or by ], 1W''1.W'' 1BßCßD.W W <ÐXÑ ab and is defined by ``<< 1.W´1Ò<Ð?ß@ÑÓ‚.?.@ (5.8) (((( ¾¾`?`@ <ÐXÑ X whenever the double integral on the right exists. Note: the symbol .W used in a surface integral always denotes a differential element of surface area, whereas the symbol .= used in a line integral always denotes a differential element of arc length. (Later we'll use the symbol .Z to denote a differential volume element. That is, .Z is just shorthand for .B.C.D.) Vector Calculus. Page 17 Any surface W may be represented parametrically in different ways. It may be shown that the value of does not depend on the parameterization. ''1.W W Although it's necessary to go back to eq. (5.8) to actually calculate a surface integral, intuitively we may think of this surface integral as the limiting value of a Riemann sum. Suppose we approximate the surface Wby a polyhedron of Pj faces, where the th face has area ?Wj and is tangent to W at ÐBjßCjjßDÑ. Now consider the sum P 1ÐBßCßDÑW?. " jjjj jœ" If we let in such a way that max, this Riemann sum approaches . PÄ∞?Wj Ä!''1.W ef W If the surface W is represented explicitly by Dœ0ÐBßCÑ, the surface integral may be written `0##`0 1.Wœ1ÒBßCß0ÐBßCÑÓ"€€.B.C (((( Ë Œ`B•Œ•`C W X The flux of a vector field through a surface. Let WœX< be a simple parametric ab surface, let 8J be the unit normal vector to W defined by eq. (5.6), and let be a vector field defined on W. At any point on W† the dot product J8J is the component of in the direction of 8. The surface integral ``<< J†8.WœJ†8‚.?.@œJR†.?.@ (5.9) ((((¾¾`?`@ (( W XX is called the flux of J through the surface. This kind of surface integral occurs frequently in applications. The flux of a vector field through a surface is meaningful regardless of the nature of J , but perhaps the situation where flux is easiest to interpret is when J@ÐBßCßDÑœ3ÐBßCßDÑÐBßCßDÑ where 3ÐBßCßDÑand @ÐBßCßDÑ denote the density and the velocity, respectively, of a fluid at the point ÐBßCßDÑ. Then the flux measures the mass of fluid passing through the surface per unit time. See the discussions in Feynman and Schey for more on the intuitive meaning of “flux.” Suppose that W is represented explicitly by Dœ0ÐBßCÑ. From eq. (5.4) we have `0`0 Rœ•3•45€Þ `B`C Now write the vector field J in terms of its components: Vector Calculus. Page 18 JÐBßCßDÑœJBÐBßCßDÑ3€JCDÐBßCßDÑ45€JÐBßCßDÑÞ It follows that the flux integral in this case may be written `0`0 J8†.Wœ•JB•JCD€J.B.C ((((”•`B`C W X where JB,JCD, andJ are evaluated at ÐBßCß0ÐBßCÑÑ. 6. The Divergence Theorem Mathematical solids. To a mathematician, a “solid” is a particular kind of subset of ‘$. As my mathematical dictionary quaintly defines it, a “geometric solid” is “[a]ny portion of space which is occupied conceptually by a physical solid; e.g., a cube or a sphere.” The key word here is “conceptually.” A mathematical solid, unlike a physical solid, has no solidity. For example, a spherical bubble trapped in a block of ice is a mathematical solid. In this document we'll implicitly assume various things about the solids of interest. In particular, we'll assume of any solid of interest Z that (1) ZZ is a connected set, (2) is bounded, and (3) the boundary of Z is a regular surface in the sense of section 5, or the union of several such surfaces. In addition, this surface must be “orientable”; for a definition, see Apostol, page 456. The boundary of a solid partitions ‘$ into two parts, an interior (the solid) and an exterior, and it's not possible to pass from the interior to the exterior along a continuous path without going through the surface. Open and closed surfaces. Vector calculus deals with two different kinds of surface: open, and closed. An open surface is bounded by an edge that we'll assume is a piecewise smooth curve. For example, a piece of paper is an open surface. A closed surface is not bounded by an edge, but itself forms the boundary of a solid. The surface of a beach ball is an example of a closed surface. Unit normal vectors. Suppose <ÀXÄW‘$ is a parametric representation of a surface . At any regular point <Ð?ß@Ñ there are two unit vectors that are normal to the surface: R``<< 8R´where ´‚ "R `?`@ ll and 88#"´•Þ In calculating a “flux integral” J8†.W (( W Vector Calculus. Page 19 it is necessary to specify which of these two unit normal vectors is to be used. If the surface W is closed, a universal convention is that the outward facing normal unit vector is used. Let J be a differentiable vector field defined on H©‘$, say JÐ<ÑœJBÐ<Ñ3€JCDÐ<Ñ4€JÐ<Ñ5where.<´B3€C45€D We've defined the divergence of J by eq. (3.1): `J`J `J div JJ´f†´BD€€C. (3.1) `B`C`D This expression may seem totally dependent on the chosen system of coordinates. Miraculously enough, it turns out that div J has a physical interpretation that is completely independent of the coordinate system. To explain this interpretation, we need to introduce the concept of “flux density” at a point. Let Zdenote a mathematical solid with surface W, let ?ZZ denote the volume of , and suppose that < is in the interior of ZZ. For example, could be a sphere, or a rectangular parallelepiped. By the “flux density of JJover Z” I mean the ratioof the flux of through WZ to the volume of : " J8†.W. ?Z(( W By “the flux density of J Vector Calculus. Page 20 Some authors simply define div J to be what I've called the flux density. While this approach has some conceptual advantages, it also complicates the exposition. The Divergence Theorem. The Divergence Theorem, also called Gauss' Theorem, relates a triple integral over the interior of a solid to an integral over the surface of that solid. Theorem 6.1 (The Divergence Theorem). Let Z be a solid in ‘$ bound by an orientable closed surface WW, and let 8J be the outwardly directed unit vector on . If is a continuously differentiable vector field defined on Z , then div J.ZœJ8†.WÞ (6.2) (((ab (( ZW For a proof of the Divergence Theorem, see Apostol, pp. 457-459. Given that we may interpret div JJ as a “flux density,” we see that eq. (6.2) says just that the flux of through the surface of a solid ZZ is the integral of the flux density of J over the interior of . Exercise. Prove the following proposition. Let Z be a solid in ‘$ bound by an orientable closed surface WW, and let 8J be the outwardly directed unit vector on . If is a continuously differentiable vector field defined on Z , then curl J8†.Wœ!Þ (6.3) ((ab W Exercise. Use the Divergence Theorem to prove eq. (6.1). We can gain some insight into the Divergence Theorem if we combine eq. (6.1) with what I call the “shared surface” theorem1. Let ZW be a solid bounded by a surface . Suppose we divide Z into two solids Z" andZW# by inserting a surface "#, which becomes part of the surface of both Z" andZ#. We'll say that WW"#" is a “shared surface.” Let denote the part of W that still bounds Z", and let W## denote the part of WZ that still bounds . Hence, ZœZ"∪Z#,WœW"∪W#, the boundary surface of Z" is W"∪W"#, and the boundary surface of Z# is W#∪WZ"#. For example, let be the rectangular parallelepiped Z´ÐBßCßDÑÀ!ŸBŸ#ß!ŸCŸ"ß!ŸDŸ" ef and insert the square surface W´ÐBßCßDÑÀBœ"ß!ŸCŸ"ß!ŸDŸ"Þ "# ef Then ZZ"# and are cubes: Z´ÐBßCßDÑÀ!ŸBŸ"ß!ŸCŸ"ß!ŸDŸ" " ef 1 As the Divergence Theorem is used to prove eq. (6.1), this analysis may seem more than a little ass- backwards. Point taken! But this analysis has heuristic utility as it increases our insight into why the Divergence Theorem is true. Vector Calculus. Page 21 and Z´ÐBßCßDÑÀ"ŸBŸ#ß!ŸCŸ"ß!ŸDŸ" # ef that share the face WZ"#. Returning to the general case, if J is a vector field defined over , then the flux of J out of Z" may be written J†8".WœJ†8"".W€J8†.W (((((( W"∪W"#WW""# where 8J"" denotes an outwardly directed unit normal vector for Z. Similarly, the flux of out of Z2 may be written J†82.WœJ†822.W€J8†.W (((((( W22∪W"#WW"# where 82 denotes an outwardly directed unit normal vector for Z22. As 88" and are outward normal vectors to Z" and Z#, respectively, it follows that 882œ•W"on "#. Hence, the flux out of ZW# through the shared face "# is just the negative of the flux out of ZW" through "#. In symbols, J†82.Wœ•J8†" .WÞ (((( WW"#"# Hence the sum of the fluxes out of the two solids ZZ"# and is given by J†8" .W€J8†2.WÞ (6.4) (((( WW" 2 The flux of J out of the whole solid Z is given by J8†.W (( W where 8 is a unit normal vector on W. But this flux can be rewritten as J†8.WœJ†8".W€J8†2.W (6.5) (((((( WWW" 2 because 8œ8" on W" and88œW## on . Comparing eqs. (6.4) and (6.5), we conclude: the flux of J out of the whole solid Z is equal to the sum of the fluxes out of the two component solids Z" andZ#, and this is true because the fluxes from ZZ"# and across the shared surface cancel. This conclusion holds if the original solid Z is partitioned into any number of component solids ZZ",#R,á,ZZ. The shared surface theorem. Suppose a mathematical solid with Vector Calculus. Page 22 surface W is partitioned into any number of component solids ZZ",#R,áZ, with surfaces WW", #R, á, WZ. If J is a vector field defined on , then R J†8.WœJ8†.W (6.6) " j ((jœ" (( WWj where 88 is a unit normal vector on W and jj is a unit normal vector on W for jœ"ßáßR. In words, the flux of J out of the original solid Z is equal to the sum of the fluxes out of the RZ component solids. This conclusion follows from the fact that partitioning into component subsolids creates internal shared surfaces, and all the fluxes across shared surfaces cancel. We may combine eqs. (6.1) and (6.6) to gain some insight into the Divergence Theorem. Let ZW be a solid in ‘$ bound by an orientable closed surface , let 8 be the outwardly directed unit vector on W, and let J be a continuously differentiable vector field defined on Z . The expression on the right-hand side of eq. (6.2) J8†.W (( W is the flux of J out of Z. We now partition ZR into a large number of component “subsolids” ZZ",#,á,ZR with surfaces WW", #R, áW, . From the shared surface theorem, R J†8.WœJ8†.W (6.6) " j ((jœ" (( WWj where the terms in this equation are explained above. Let ?ZZjj denote the volume of for jœ"ß#ßáßRZ. If ? j is small enough, it follows from eq. (6.1) that J†8j.W¸ZdivJ R J†8.W¸divJ This approximation becomes an equality if we let RÄ∞ and max?ZjÄ!. But the sum on the right-hand side of eq. (6.8) is just a Riemann sum for ef div . This ''' J .Z Zab completes our heuristic “proof” of the Divergence Theorem. Vector Calculus. Page 23 7. Stokes' Theorem. Let J be a vector field defined on H©‘$. In the previous section we used the concept of a “flux density” to give a geometric meaning to div J . In this section we'll introduce the concept of “circulation density” to give a geometric meaning to curl J . To be precise, let <´ÐBßCßDÑ be a point in Hß, let 8 be a unit vector in ‘C$, let <8< be the plane through ab that is normal to 8, let Gß be a piecewise smooth closed path in C<8< that encloses , and ab let ?WG denote the area of the region enclosed by . Now consider the “circulation integral” M´JX†.=.(7.1) *G By convention, in calculating MG the path is traversed in a counterclockwise direction as viewed from the tip of 8 when 8< is based at . The circulation M given by eq. (7.1) is a scalar defined as an integral over the whole path G, whereas the curl of J< at is a vector defined at the point < alone. What can M possibly tell us about f‚J To appreciate the utility of #Ð<ß8Ñœ, it's best to see some examples. First, let 85, so CÐ<8ßÑ is parallel to the BC-plane.Let G be the rectangle with base ??BC, height , and centered around < in the plane CÐ<5ßÑ. An instructive and easy calculation shows that `J `J #Ð<5ßÑœ C •ÞB `B`C From eq. (3.2), this is the 5 component of curl J< at . Hence, f‚JÐ<ц5œ#Ð<5ßÑÞ cd Similar calculations with planes parallel to the BD-plane and the CD-plane yield f‚JÐ<ц3œ##Ð<ß3Ñand.f‚JÐ<ц4œÐ<4ßÑ cdcd These results suggest (but don't exactly prove) the following: for any unit vector 8, f‚JÐ<ц8œ#Ð<8ßÑÞ (7.2) cd In words, f‚JÐ<Ñ is a vector whose component in the direction of 8 is equal to #Ð<8ßÑ, the circulation density of J at < in the plane CÐ<8ßÑ. Vector Calculus. Page 24 Jordan curves. A path >‘ in 8 is specified parametrically by a continuous vector valued function #ÀÒ+ß,ÓÄ‘>8. If ##Ð+ÑœÐ,Ñ, the path is closed. If is closed and ##Ð>"ÑÁÐ>#Ñ for every >"#Á> in Ð+ß,Ó, then > is said to be a simple closed curve. Geometrically, a simple closed curve doesn't intersect itself. A simple closed curve in a plane is called a Jordan curve. Every Jordan curve > partitions the plane into two disjoint open connected sets having > as their common boundary. One of these sets is bounded and is called the interior of >>. The other is unbounded and is called the exterior of . “Counterclockwise” traversal. Let > be a Jordan curve in the BCV-plane, and let denote the interior of >>. We need to define (somewhat informally) what it means to traverse in a “counterclockwise” direction. First, we define “upright” to mean: in the direction of positive values of D. Definition: an upright pedestrian walking on > is moving in a counterclockwise direction if V is on his or her left. Green's Theorem (for a plane region bounded by a piecewise smooth Jordan curve). Let T and UW be scalar fields be scalar fields that are continuously differentiable on an open set in the BC-plane. Let GV be a piecewise smooth Jordan curve, and let denote the union of G and its interior. Assume that V©W. Then the following equation is true: `U`T •.B.CœT.B€U.C (7.3) ((*Œ•`B`C G V where the line integral is taken around G in the counterclockwise direction. Stokes' Theorem is a direct generalization of Green's Theorem. Let W be a surface in ‘$ bounded by a curve GW, and let J be a vector field defined on . Stokes' Theorem states that the circulation of J around G is equal to the surface integral of curl J8†W over , ab where 8 is a suitably chosen unit normal vector at each point of W. Stokes' Theorem. Let W be a smooth simple parametric surface, say Wœ<ÐXÑX, where is a region in the ?@-plane bounded by a piecewise smooth Jordan curve >. Assume also that < is a one-to-one mapping whose components have continuous second-order partial derivatives on some open set containing X∪G>>. Let denote the image of under <, and let J be a continuously differentiable vector field defined on W. Then curl J†8.WœJX†.= (7.4) ((*ab G W where 8 is the unit normal vector defined by eq. (5.6), and the path > is traversed in the counterclockwise direction when the line integral is evaluated. This statement of Stokes' theorem is taken from Apostol, where a proof may be found. Remark 1. This statement of Stokes' theorem makes explicit use of the parameterization Wœ<ÐXÑ and the parameterization of >. As noted previously, the value of a surface Vector Calculus. Page 25 integral doesn't depend on how the surface is parameterized, and the value of a line integral doesn't depend on parameterization up to a sign. Hence eq. (7.4) is true almost regardless of how W and > are parameterized. It might seem preferable, therefore, to state Stokes' theorem in a way that makes no explicit reference to a particular parameterization. Significant complication arise, however, if one attempts to rephrase Stokes' theorem without making explicit use of these parameterizations. In particular, eq. (5.6) gives us a convenient way to ensure that the normal unit vectors 8 are all on the same “side” of W, and it's difficult to see how this condition could be guaranteed without using the parameterization. Also, it's much easier to define “counterclockwise” for the Jordan curve > in the ?@-plane than for the closed path G in ‘$. Remark 2. Stokes' theorem reduces to Green's theorem if W is a region in the BC-plane. To see this, write JÐBßCßDÑœTÐBßCßDÑ3€UÐBßCßDÑ45€VÐBßCßDÑÞ If W is a region in the BC-plane, then 85œW everywhere on , and hence `U`T curl J8†œ•Þ ab `B`C Also, the closed curve G lies entirely in the BC-plane, so the line integral in eq. (7.4) becomes JX†.=œT.B€U.CÞ *(GG Remark 3. The surface WG is said to be a “capping surface” of the closed curve . For any given closed curve G, there are an infinite number of capping surfaces. Some are as tight and “minimal energy” as a soap film on a wire frame. Others billow out to Betelgeuse or beyond. To me, the most amazing thing about Stokes' theorem is that it says that the value of the surface integral curl J8†.W ((ab W is invariant over all surfaces WG that cap , so long as J is defined and continuously differentiable on WG. Now imagine a large capping surface on a small closed path . If we let G shrink down to a point, the circulation JX†.= necessarily decreases to zero. )G This gives us another way to prove eq. (6.3). (I learned of this method from Feynman.) Remark 4. Equation (7.2) tells us that curl J8† may be interpreted as a “circulation density.” On the other hand, we recognabize that the line integral in Stokes' theorem is the circulation of JJ around G. Hence, Stokes' theorem tells us that the circulation of around GG is equal to the integral of circulation density over any surface that caps . Vector Calculus. Page 26 There are several ways to extend the conclusion of Stokes' theorem (eq. (7.4)) to more general surfaces than are stated in the hypotheses of the theorem. In particular, one may knit several surfaces together along their edges, so long as the directions of integration along any edge shared by two surfaces is opposite. This is easiest to explain by an illustration. Consider the two rectanglesWW"#and with a common edge shown below. WW"# Let G" and G#denote the borders of W" and W#, respectively, let G"#œG"#∩G denote the shared edge,letW´W∪W, and let GœG∪G•G denote the border of WÞ We "#ab"#"# want to compute the sum of the circulations of J around GG"# and , i.e., J†X.=€JX†.=,(7.5) **GG"# where the direction of integration is counterclockwise (as indicated by the arrows shown in the figure), and compare this sum to the circulation of J around G, i.e., JX†.=, *G also integrated in a counterclockwise direction. Now consider the contributions to J†X .= and JX†.=G attributable to integration along the shared edge "#. ))GG"# Because J†X .= and JX†.= are integrated in opposite directions along the ))GG"# shared edge, we see that their contributions to the sum (7.5) just cancel (see Theorem 4.1), so J†X.=œJ†X.=€JX†.=Þ (7.6) *G**GG"# Now, Stokes' theorem applies to both WW"# and : J†X.=œcurl J8†".W (7.7) *G" ((ab W" and Vector Calculus. Page 27 J†X.=œcurl J8†#.W (7.8) *G# ((ab W# where 8" and 88# are normal unit vectors to WW"# and , respectively. If we now define to be 88" on WW" and ## on, we see that curl J†8.Wœcurl J†8"#.W€curl J8†.W.(7.9) ((ab((ab((ab WWW"# Combining eqs. (7.6) through (7.9), we see that eq. (7.4) holds for the composite surface W. This kind of argument may be extended to any kind of surface that may be construed as the union of simpler surfaces knit together along part of their edges. The only requirement for this argument to go through is that it be possible for “counterclockwise” to be defined for each subsurface in such a way that the direction of integration along any arc that is a shared edge will be opposite. For example, surfaces with “holes” can be treated by introducing “cross-cuts.” A picture is worth a thousand words here, and I advise the reader to consult almost any text on advanced calculus. We can knit together surfaces in more complicated ways. Consider the two rectangles X" andX# in ?@-space shown below. XX"# Let >>"and# denote the boundaries of X" andX#, respectively, and let X´X"#∪X . Suppose that the image <ÐXÑ in ‘$ is the cylinder shown below where (in effect) the long strip XX has been bent around until the image of the left edge of " has been brought into coincidence with the image of the right edge of X#. The images <<Ð>>"#Ñ and ÐÑ coincide on two arcs: the image <Ð>>"∩#Ñ of the short vertical line where XX"# and join, and the common image under < of the left edge of XX"# and the right edge of . An argument similar to that given above shows that Stokes' equation applies to this cylinder, where the total circulation is the sum of the line integrals taken over the upper and lower rims of the cylinder, and in the directions indicated in the diagram. Vector Calculus. Page 28 The argument works because the line integrals along arcs that are common to two regions are always in opposite directions, so they cancel. In summary, the sum of the line integrals over the two component surfaces is just equal to the line integral over the exterior edge (or edges) of the amalgamated surface because the contributions to line integrals over interior (and therefore shared) arcs just sum to zero. Now suppose that the mapping < gives, in effect, the strip X a half twist before the image of the left edge of XX"# and the image of the right edge of are brought into coincidence. The image <ÐXÑ is called a Möbius band. Stokes' equation fails to hold in this case because the direction of integration of the two line integrals is necessarily in the same direction along some arc that is common to <<ÐX"#Ñ and ÐXÑ. The Möbius band is an example of a nonorientable surface. 8. Some concluding remarks. Remark 1. Green's theorem, Stokes' theorem, and the divergence theorem are all extensions of the second fundamental theorem of calculus. Each of these theorems states that the integral of some function over a “region” of ‘$ is equal to the integral of a related function over the boundary of that region. For Green's theorem and Stokes' theorem, the region is a surface and the boundary is a closed curve. For the divergence theorem, the region is a mathematical solid and the boundary is a closed surface. Remark 2. The divergence (eq. (3.1)) and curl (eq. (3.2)) were defined for a vector field J that's defined on a subset of ‘$. That's adequate for electromagnetism, the subject for which these tools were essentially invented. However, the dot product is naturally extended to ‘‘88 (see eq. (1.1)), and it's natural to extend the definition of divergence to . If J is a vector field defined on a subset of ‘8, say JÐ<Ñ´JÐ<ÑßJÐ<ÑßáßJÐ<<Ñwhere ´BßBßáßB a"#8bab"#8 then 8 `J divJÐ<Ñ´f†J<ÐÑ´ 3 (8.1) " 3œ" `B3 Vector Calculus. Page 29 where all the partial derivatives are evaluated at <. This plays an important role in (for example) the kinetic theory of gases. Similarly, the gradient is naturally defined on ‘8, and plays an important role in many fields, including economics. It follows that the “Laplacian” operator f# , defined as the divergence of a gradient, is naturally defined on ‘:8: for any scalar field BßBßáßB , ab"#8 8 `#: f#::´divgrad œÞ (8.2) "# ab3œ" `B3 On the other hand, both the cross product (eq. (1.3)) and the curl are meaningful constructs only in ‘$ , so far as I can tell. Remark 3: Some commuting and some non-commuting operators. The Laplacian operator f# is defined as the divergence of a gradient. If J ´JßJßJ is a vector field abBCD in ‘$, the Laplacian of J is defined “component-wise”: f#J´f#Jßf##JßfJÞ (8.3) ˆ‰BCD For any vector field JJ in ‘$, the gradient of the divergence of graddiv JJœff† abab is a meaningful vector fieldÞ It occurs, for example, in the formula for the curl of a curl: f‚f‚Jœff†JJ•fÞ# (8.4) abab The unwary student might naively assume that graddiv is equal to divgrad. This would be a gross error! Among other differences, garadadbbiv is a vectorab, wheabreas divgrad is a scalar. In words, the operators “grad” aabnd ab“div” do not commute. abab On the other hand, consider the two operators “curl” and “Laplacian.” For any vector field J defined on a subset of ‘$ , the following formula f‚f##JJœff‚ (8.5) ˆ‰ ab is true. In words, “curl” and “Laplacian” do commute. Feynman passes eq. (8.5) off with the casual remark “[s]ince the Laplacian is a scalar operator, the order of the Laplacian and curl operations can be interchanged.” I don't buy this; so far as I can see, eq. (8.5) needs a proof. The work is grungy but straightforward, and it all works out in the end. Remark 4: spherical coordinates. The gradient, curl, and divergence were defined in terms of derivatives with respect to BßCßDand , the coordinates of a point relative to the $$ standard coordinate system of ‘‘ÞFor example, if J´ÐJBßJCDßJÑ is a vector field in , then Vector Calculus. Page 30 `J`J `J div J ´BD€C€Þ `B`C`D We gave geometric interpretations of the gradient, curl, and divergence that showed that these operations have physical meanings that are independent of the coordinate system used to locate points in ‘$ . For some problems, it's useful to express grad, div, and curl in terms of derivatives relative to alternative coordinate systems. In particular, one can find such expressions relative to cylindrical and spherical coordinates. A full discussion may be found in Schey. Here I'll just state the formulas for f0# in spherical coordinates. I need this formula for the discussion of “spherical waves” in the appendix. A point< in ‘$ is located in spherical coordinates by a triple of numbers <ßß9) where ab < ! is a distance, and 9) and are angles. Specifically, <œ < is the distance from the ll origin to <<, )9 is the angle between and the DB-axis, and is the angle between the -axis and the projection of < onto the BC-plane. The angle ) corresponds to “latitude” in geography, except that ) is measured from the north pole rather than the equator. The angle 9 corresponds to “longitude,” with the B-axis essentially in the role of “prime meridian,” except that 9 is only measured in an “eastward” direction (i.e., counterclockwise as seen from the North Pole.) With spherical coordinates, a scalar field is expressed as a function of <, 9), and . With these conventions, the following may be shown. Let 0<ßß9) denote a scalar field. Then ab "``0"``0"`0# f##0œ<€sin ) €Þ(8.6) <#`<Œ`<•<#sin)`)Œ•`)<`#sin##)9 Although this formula is impressively complicated, in a problem with spherical symmetry it quickly reduces to a much simpler expression. Vector Calculus. Page 31 References My primary sources for this document are as follows. Apostol, Tom. Calculus, Vol. II (2nd ed.). Xerox College Publishing, 1969. Feynman, Richard, Robert Leighton, and Matthew Sands. The Feynman Lectures on Physics: The Electromagnetic Field (Vol. II). Addison-Wesley, 1964. Schey, Harry Moritz. div, grad, curl, and all that: an informal text on vector calculus (2nd ed.). W. W. Norton & Company, 1992. With regard to mathematics, Apostol is the most rigorous of these authors in his presentation, Feynman is the most relaxed, and Schey is intermediate in rigor. Apostol is always careful to state precisely the conditions under which a given result is true, and always gives rigorous proofs. While this is all good in a mathematical reference book, it makes his text rather difficult reading for undergraduates. In contrast, Feynman is quite cavalier in his statement of theorems. Schey is careful in his statements of theorems, but his proofs are generally heuristic and informal instead of being rigorous. This document is closest in rigor to Schey. With regard to physics, my primary source was Feynman (of course), but I found useful material in the other two books. I also extracted some physics from the following text. Stauffer, Dietrich, and H. Eugene Stanley. From Newton to Mandelbrot: A Primer in Theoretical Physics. Springer-Verlag, 1991 Vector Calculus. Page 32 Appendix: Supplementary Material. 1. Linearity of grad, div, and curl. The gradient, divergence, and curl are all linear operators. That is, if +, and are scalars, : and <‘ are scalar fields on $, and JK and are vector fields on ‘$, then f+:€,<œ+f:<€,fß (A1.1) ab f†+J€,Kœ+f†JK€,f†ß (A1.2) ababab f‚+J€,Kœ+f‚JK€,f‚Þ (A1.3) ababab 2. Product Differentiation Formulas. The following identities are all generalizations of the rule in elementary calculus for differentiating the product of two functions. Let :< and be scalar fields on ‘‘$$, and let J be a vector field on . Then f:<œ:f<€<:fß (A2.1) ab f†:Jœ::f†JJ€†fß (A2.2) abab f‚:Jœ::f‚JJ€f‚Þ (A2.3) abab 3. “Irrotational” and “Solenoidal” Vector Fields. If J is a vector field in ‘$ and curl Jœ!J on some set H©‘$, then is said to be irrotational. We know that if J œf:: for some scalar field with continuous second-order mixed partial derivatives, then JJ is irrotational. Conversely, it's known that if is irrotational at all points in an open convex set H, then there exists a scalar field :: defined on H such that JœfÞ If J is a vector field in ‘‘$$ and div JJœ! on some set H© , then is said to be solenoidal. We know that if all the mixed partial derivatives of a vector field K are continuous, then J´curl KJ is solenoidal. Conversely, if is solenoidal everywhere in some open interval HH, then there exists a vector field K defined on such that JKœcurl . Suppose that J is a continuously differentiable vector field defined on an open interval H in ‘$ . It's known that every such vector field may be written in the form Jœ€GK where G is solenoidal and KG is irrotational [Apostol, p. 452]. As is solenoidal, it follows that GœcurlL for some vector field LK. Similarly, as is irrotational, it follows that K œf:: for some scalar field . Hence, we can write JLœcurl€f:.(A3.1) To find LJ and : given , we take the curl and divergence of each side of eq. (A3.1) and Vector Calculus. Page 33 make use of the linearity of curl and divergence. This yields the following partial differential equations for : and L: fœ#:div J and curlcurl Lœff†L•f#LJœÞcurl abab 4. “Central” Vector Fields. We previously defined “central” force fields. We now wish to extend this vocabulary to general vector fields. A vector field J defined on ‘8 is said to be central if it can be written in the form JÐ<Ñœ1Ð<Ñ<< where <´ . The purpose of this section is to record some of the properties of central vector fields.ll We previously showed that every central force field is conservative; that is, if J is a central force field defined on some set H©‘$, then Jœ•fYYfor some potential function . This result clearly isn't restricted just to force fields: if J is a central vector field defined on H©‘8, then J œfH:: for some scalar field defined on . As a corollary, we see that every central vector field is irrotational. We next want to find f1<< for an arbitrary function 1Þ. Let œÐBßBßáßBÑ and allbab "#8 let <œ <. Recall eq. (2.2): ll ` We may use this result to find the divergence of a central vector field. Suppose that J<œ1Ð<Ñ is a central vector field in ‘$.From eq. (A2.2), Example 1 of Section 3, and eq. (A4.1), we find divJœ1Ð<Ñdiv <<€†f1Ð<Ñœ$1Ð<Ñ€<1wÐ<ÑÞ (A4.2) ab In particular, consider the central vector field given by J<œ Vector Calculus. Page 34 interesting case that Pœ•$, we have divJœ!unless 5. Inverse Square Laws. If the magnitude of a central force fieldJ< at a point −‘$ is inversely proportional to <# (i.e., inversely proportional to the square of the distance from ll the origin), then the force field is said to obey an “inverse square law.” Let Jœ J and ll let <œÞ< The best known examples of inverse square laws are Newton's law of gravitatllion KQ7 Jœ <# (which gives the gravitational force between a point mass of Q7 and a point mass of ) and Coulomb's law ";;! Jœ kk# %<1%! which gives the magnitude of the electrical force acting between stationary charges ;; and !. (K and %! are constants.) To give the direction as well as the magnitude of these forces, let /´<•"<< be a unit vector in the direction of . Then Newton's law may be written ";;!!";; Jœœ#$/<< %1%!!<%<1% (where the charge ;;! is at the origin, the charge is at A force that obeys an inverse square law can be written JJ Jœ„""/<œ„ (A5.1) <<#$< where J" denotes the magnitude of the force at unit distance. The plus sign is used if the force is repulsive, and the negative sign is used if the force is attractive. Note that J YÐ<Ñœ„ " (A5.2) < is a potential function for the force given by eq. (A5.1); i.e., J œ•fYÞ Also, note that Vector Calculus. Page 35 eq. (A4.4) applies if J obeys an inverse square law. I'll only consider repulsive forces in the remainder of this section, but the same results apply to attractive forces with suitable modifications of language. Let JJ satisfy eq. (A5.1) with a plus sign. Let's compute the flux of out of the surface of •" a sphere of radius < centered at the origin. Let 8´œ< J"JJ""# .Wœ.Wœ†%11<œ%JÞ" (A5.3) ((<#<<##(( WW Note that this flux is independent of the radius <. That is, the flux of J through the surface of any sphere centered at the origin equals %J1". Combining this result with the observation that div J = 0 except at the origin yields the following theorem. Theorem. Suppose that J obeys the inverse square law of eq. (A5.1) (with a plus sign). Let Z denote a solid that includes the origin !J as an interior point. Then the flux of through the surface of Z equals %J1". Proof. As ! is an interior point of Z, we can find %%ž! such that a sphere with radius centered at ! will be entirely contained within Z. Let Z%% denote this sphere, and let WÐÑ ab denote the surface of Z%%. Let W denote the surface of Z and define Zw´Z•ZÐÑ. ab You may think of Zww as being ZZ with a bubble removed. The surface of equals W∪WÐ%%Ñ. We'll say that W is the exterior surface of Zw and WÐÑ is the interior surface of ZZww. As div J = 0 throughout , it follows from the divergence theorem that the total flux of JJ out of ZZww must equal zero. Therefore, the total flux of into through the interior surface WÐ%Ñ must equal the total flux of J out of ZWw through the exterior surface . w But the flux of J into Z through the interior surface WÐ%1Ñ is just equal to %J", the flux of J out of ZÐÑ%. 6. Maxwell's Equations. In the following 4 equations, > denotes time, I, FN, and are $$ vector fields in ‘, 3 is a scalar field in ‘%, and - and ! are constants. In somewhat more detail, I œ the electric field, F œ the magnetic field, N œ current density, and 3 œ charge density. Maxwell's equations in differential form are as follows: Vector Calculus. Page 36 3 f†Iœß (A6.1) %! `F f‚I œ•ß (A6.2) `> `IN -# f‚Fœ€ß (A6.3) ab`> %! f†F œ!Þ (A6.4) If we apply the divergence theorem to the first and last of these equations, and apply Stokes' theorem to the second and third, we obtain Maxwell's equations in integrated form: (1) The flux of I through a closed surface WW equals the total charge contained within divided by %!Þ (Both this statement and eq. (A6.1) are known as “Gauss' law.”) (2) The clockwise circulation of I around a closed loop G is equal to the rate of change of the flux of F through any surface that caps GÞ (Both this statement and eq. (A6.2) are known as “Faraday's law.”) (3) -G# times the counterclockwise circulation of F around any closed loop equals the rate of change of the flux of I through any surface WG that caps , plus the total flux of electric current through W divided by %!. (4) The flux of F through any closed surface is zero. 7. Electrostatics and Magnetostatics. If the charge density 3 and the current density N in Maxwell's equations do not depend on time, then the two time derivatives equal zero, and Maxwell's equations reduce to two pairs of equations: Electrostatics: 3 f†Iœß (A7.1) %! f‚œI!.(A7.2) Magnetostatics: N -#f‚Fœß (A7.3) ab%! f†F œ!Þ (A7.4) Vector Calculus. Page 37 Equation (A7.3) is known as “Ampere's law.” In this static situation, the electric field I appears in only the first two equations and the magnetic field F appears in only the second two equations. Hence, if charges and currents are static, then electricity and magnetism are distinct and separate phenomena. Notice that in electrostatics, the electric field I is irrotational. Hence, there exists a scalar field F such that I œ•fÞF (A7.5) The scalar field F is called the electrostatic potential. By substituting eq. (A7.5) into eq. (A7.1), we see that F satisfies Poisson's equation: 3 f#Fœ•Þ (A7.6) %! The specialization of Poisson's equation obtained when 3 œ!, i.e., f#F œ!ß (A7.7) is called Laplace's equation. Before turning to the subject of magnetostatics, let's examine the electrostatic potential in a little more detail. We may write Coulomb's law as ";;""";; Jœœ#$/<< %1%!!<%<1% where the charge ;;" is at the origin, the charge is at ";"""; Iœ#$/<<œÞ %1%!!<%<1% Note that I œ•fF where "; F < œ ". ab %<1%! Generalizing, the electric field at << produced by a point charge ;"" at is given by ";" I""œ•$<< %1%!<<•ab ll" and I""œ•fF where "; F <œÞ" "ab %•1% << !"ll Now suppose we have 7 point charges ;"ß;#ßáß;7 at points <"ß<<#7ßáß . By the Vector Calculus. Page 38 principle of superposition, the electric field I< at any point is the vector sum of the electric fields produced by the individual point charges. That is, 7 ";3 IÐ<Ñœ$ <<•Þ3 (A7.8) %1%! " <<•ab 3œ" ll3 Now let F3 denote the potential function associated with the point charge ;33 at <, "; F<œ3 for 3œ"ßáß7 3ab %•1% << !3ll and define 77"; FF<<´œÞ3 (A7.9) ab""3ab %•1% << 3œ"!33œ" ll By the linearity of the gradient, 77 ";3 •fFFœ•f33œ$<•<œI<ÐÑÞ ""ab%1%!<<•ab 3œ"3œ" ll3 In summary, the principle of superposition applies to potential functions as well as to force and electric fields. We may extend these results from point charges to a continuous distribution of charge over ‘$. This yields an electric field w "3ÐÑ< w IÐ<Ñœ$ <<•.Z (A7.10) %1%! ((( <<•w ab ll and an associated potential function "3ÐÑ We now turn to the subject of magnetostatics. From eq. (A6.4), we see that the magnetic field is solenoidal. The physical meaning of this is often stated as “there are no magnetic monopoles.” As FE is solenoidal, it follows that there exists a vector field such that Vector Calculus. Page 39 Fœf‚EEœÞcurl(A7.12) The vector field E is called the vector potential. In magnetostatics, we may combine eqs. (A7.12) and (A7.3) to see that E satisfies curlcurl Eœff†E•fœ#EN. (A7.13) abab ! where " (A7.14) .! ´Þ# -%! The vector potential EE is not uniquely determined by eq. (A7.12). Let be a vector field that satisfies eq. (A7.12), let < be any scalar field defined on ‘<$w, and let EE´€fÞ Because the curl of a gradient is always !, it follows that f‚Ewœf‚E€f<œf‚EFœÞ ab In short, we have a considerable amount of freedom in how the vector potential E is chosen. In particular, it's possible (and convenient) to impose the restriction f†EEœdiv œ!Þ With this restriction, eq. (A7.13) simplifies to # fENœ•Þ.! (A7.15) Hence, the vector potential E in magnetostatics may be found by solving a vector version of Poisson's equation. That is, eq. (A7.15) is really three equations: one for each of the components of E. By comparing eqs. (A7.6), (A7.11), and (A7.15), we see that an explicit solution of (A7.15) is given by . N<ÐÑw E< œ! .ZÞ (A7.16) ab %•1((( < 8. Conservation of Charge (and Other “Stuff”). Let Z be a mathematical solid in ‘$ with a boundary surface W, and let 3BßCßDß> denote “charge density” at any point BßCßDZ in abab at time >Þ The total amount of “charge” inside Z> at time is therefore given by UÐ>Ñ´3ÐBßCßDß>Ñ.ZÞ ((( Z Hence, the rate of change of U is given by `3 UwÐ>Ñœ.ZÞ (A8.1) ((( `> Z (The operation of differentiating under the integral sign is justified if `3Î`> is continuous.) Vector Calculus. Page 40 On the other hand, the only way the amount of charge in Z can change is if there is a current across the border of Z . If we let N denote current density as before, it follows that UwÐ>Ñœ•N8†.W (A8.2) (( W where 8 is a outward unit normal. (The integral on the right hand side of eq. (A8.2) is the flux of current across W, and the negative sign is motivated by the observation that a positive flux of current across WZ implies a decrease in charge inside .) Applying the divergence theorem to the right hand side of eq. (A8.2), we find UwÐ>Ñœ• f†N .ZÞ (A8.3) ((( Z Combining eqs. (A8.1) and (A8.3), we find that `3 .Zœ• f†N .ZÞ (A8.4) (((`> ((( ZZ But as the solid Z is quite arbitrary, it follows that `3 œ•f†N `> at all points where 3 is defined. This equation is usually written `3 €f†N œ!Þ (A8.5) `> Equation (A8.5) is called a continuity equation as it expresses a conservation law: in this case, the conservation of charge. However, it applies in any situation where there is some kind of “stuff” that is conserved where fields 3 andN may be defined that quantify the density of stuff at a point and the movement of stuff through space. For example, this analysis applies to the study of heat. The derivation of eq. (A8.5) given above was intended to motivate its' interpretation as an expression of the conservation of some “stuff.” It is also possible to derive eq. (A8.5) directly from Maxwell's equations. By taking the divergence of both sides of eq. (A6.3) we obtain `"I f†€f†N œ! (A8.6) `> %! as the divergence of a curl is always zero. Now, ``I f†œf†I `>`>ab Vector Calculus. Page 41 as we may exchange the order of time and space derivatives. But f†IœÎ3%! from eq. (A6.1), so `I "`3 f†œÞ `>%! `> Substituting this relation into eq. (A8.6) yields eq. (A8.5). The point of this demonstration is to show that the conservation of charge is a consequence of Maxwell's equations. 9. Waves. We next want to take up the topic of electromagnetic radiation. This requires a brief review of the physics of waves. The Wave Equation. Based on physical consideration, a wave propagating at speed - along the B-axis may be modeled by the hyperbolic partial differential equation `##<<"` œÞ (A9.1) `B#-##`> This is the (one-dimensional) wave equation. It's easy to show that any function of the form < œ0ÐB•->Ñ is a solution. This represents a wave propagating to the right. Another solution is < œ0ÐB€->Ñ, which represents a wave propagating to the left. Equation (A9.1) is linear. This implies that if <<"# and are two solutions to the wave equation, then any linear combination of <<"# and is also a solution. Sinusoidal waves and fundamental wave vocabulary. The sinusoidal waves are solutions of (A9.1) of fundamental importance. These solutions can be written in the form <=ÐBß>ÑœEcos OB•> (A9.2) ab where -œ==ÎO. The three coefficients EßOß and are named and interpreted as follows. E is the amplitude of the wave and measures its vertical size relative to the BO axis. is called the “wave number” and specifies how the wave varies with space. If the unit of space is the “meter,” then O specifies the number of radians per meter. = is the “angular frequency” and specifies how the wave varies with time. If the unit of time is the “second,” then == specifies the number of radians per second. The combined expression OB•> is called the “phase” of the wave. We can relate O and = to properties of waves that may be more familiar to the reader. Suppose we look at a snapshot of the wave taken at a particular moment (so > is fixed). The wavelength - of the wave is the distance (in meters) between peaks. This is the change in B required to change the phase by #1, so #1 - œÞ O Now fix B> and consider how < varies with time. The period ! is the amount of time required for the phase to change by #1, so Vector Calculus. Page 42 #1 >œÞ ! = Putting these two equations together, the speed of the wave (in meters per second) is given by -#1=ÎO -œœœ >! #1=ÎO as stated above. Let / denote the frequency of the wave measured in cycles per second. As there are #1 radians per cycle, it follows that = " / œœß #>1 ! which makes sense, as >! is the number of seconds per cycle. The wave equation in space. As noted above, eq. (A9.1) is the wave equation for a wave propagating along the B--axis. The equation for a wave propagating at speed in ‘$ is "`#< fœ#< .(A9.3) -##`> Some authors write eq. (A9.3) as ñ< œ! where the “wave operator” ñ (also called the d'Alembert operator or “quabla”) is defined as "`# ñ ´f# •Þ (A9.4) -##`> Equation (A9.3) is linear, so any linear combination of solutions is also a solution. Sinusoidal “plane” waves. The reader may confirm that one solution of eq. (A9.3) is given by <=<ß>œEcos O<†•> (A9.5) abab where < œBßCßDß ab O œÐOBßOCDßOÑß -œ=ÎÞO ll The wave number O in eq. (A9.2) has been replaced by a “wave vector” O whose components give the number of radians per meter in the directions of the three coordinate $ axes. If <" and <# are two points in ‘ such that <"•<#" is perpendicular to O<, then and <# are on the same “wave front”; i.e., for any time >, Vector Calculus. Page 43 O†<"#•==>œO<†•>Þ It follows that the wave specified by eq. (A9.4) propagates in the direction of O, the wave fronts of eq. (A9.4) are planes perpendicular to O (which is why we call this solution “plane waves”), and O gives the number of radians per meter in the direction of propagation. If ll - does not depend on = (which is the case with light, for example), then it's convenient to write Oœ=Î-. ll Spherical waves. Although plane waves are mathematically and conceptually agreeable, they are physically problematic: it's difficult to imagine a mechanism that can generate a plane wave that is not physically infinite in some way. Therefore, we now consider solutions of eq. (A9.5) whose wave fronts consist of spheres expanding at speed - away from the origin. Specifically, consider " < œ0Ð<•->ÑÞ (A9.6) < We wish to show that eq. (A9.6) satisfies eq. (A9.3). It's convenient to use spherical coordinates for this problem. Because < has no dependence on )9 or , the equation for f#< becomes "``< f##<œ<Þ <#`<Œ•`< The reader may use this formula to confirm that (A9.6) satisfies the wave equation. Notice that the amplitude of these waves are inversely proportional to <. 10. Electromagnetic Radiation. We now consider solutions of Maxwell's equations in “free space.” In a region of ‘3$ where there is no charge and no current (so œ!œ and N!), Maxwell's equations become f†I œ!ß (A10.1) `F f‚I œ•ß (A10.2) `> `I -# f‚Fœß (A10.3) ab`> f†F œ!Þ (A10.4) The situation here is “dual” in some sense to the situation considered in electrostatics and magnetostatics, where we allowed (constant) charge density 3 and (steady) currents N , but required that IF and not vary with time. The “trivial” solution of these equations is IœœF!, but we're interested in the possibility of non-trivial solutions. To start, rewrite eq. (A10.2) as Vector Calculus. Page 44 `F œ•f‚ÞI `> ab Now differentiate with respect to time. Under normal conditions, which we assume here, we can exchange the order of differentiation, so ``#FI œ•f‚Þ `>#Œ•`> Substituting eq. (A10.3), we obtain ` #F œ•-#f‚f‚ÞF `># cdab From eqs. (8.4) and (A10.4), f‚f‚Fœff†F•f##FFœ•fß abab so `#F œ-f##F. `># We'll rewrite this as "`#F fœ#F (A10.5) -##`> which we recognize as having the form of a “vector” wave equation. The reader may show that eq. (A10.5) is actually three equations, one for each component of F: "`##F"`F#"`F f#FœBDßf##FœCßand fFœÞ B-#`>#CD-#`>#-##`> An exactly parallel derivation starting with eq. (A10.3) and using eq. (A10.1) yields "`#I fœ#I (A10.6) -##`> which actually means "`##I"`I#"`I f#IœBDßf##IœCßand fIœÞ B-#`>#CD-#`>#-##`> In summary, Maxwell's equations in free space permit solutions for each component of I and F that have the form of waves traveling with speed -, the speed of light. These waves are electromagnetic radiation, the most familiar example being light itself. We can say more about the nature of electromagnetic radiation. To begin, let's consider “plane wave” solutions for I and F. Without loss of generality, suppose that IF and Vector Calculus. Page 45 propagate in plane waves in the direction of the B-axis, so the wave fronts of IF and are perpendicular to the B-axis. This implies that IF and can have no dependence on CDor , so we can write IœI<ß>œIÐBß>ÑßIÐBß>ÑßIÐBß>Ñ ababBCD and FœF<ß>œFÐBß>ÑßFÐBß>ÑßFÐBß>ÑÞ ababBCD Without going into the details (see Feynman, Chapter 20), eqs. (A10.1) - (A10.4) imply that IBBœ! and Fœ!, so we may write IœI<ß>œ!ßIÐBß>ÑßIÐBß>Ñ ababCD and FœF<ß>œ!ßFÐBß>ÑßFÐBß>ÑÞ ababCD That is, all the variation in IF and is in a plane perpendicular to the direction of propagation of the waves. To make further headway, let's consider a “trial solution” of the following form: I4œ!ß0ÐB•->Ñß!œ0ÐB•->ÑÞ ab That is, I is a wave traveling to the right and the 5I component of is zero.It follows that 345 `F Ô×``` œ•f‚I5œ•œ•0wÐB•->ÑÞ `> abÖÙ`B`C`D ÕØ!0ÐB•->Ñ! Hence, the BC and components of F are constant over time. As above, the only physically interesting solution of these equations is FBCœFœ!. Hence, F is zero except in the direction of 5, and `F D œ•0wÐB•->ÑÞ `> •" Integrating, we obtain FD œ-0ÐB•->Ñ plus a constant of integration. On physical grounds again, it may be shown that the constant of integration is zero, so we conclude in this case that FBß>œ-•"0ÐB•->ÑÞ Dab We may repeat this analysis under the assumption that the C component of I is zero, I5œ!ß!ß0ÐB•->Ñœ0ÐB•->Ñ . ab Vector Calculus. Page 46 We may also repeat both analyses under the trial solution of a wave traveling to the left: IßIœ0ÐB€->Ñß!and IßIœ!ß0ÐB€->ÑÞ aCDbabaCDbab Our results are summarized in the following table. IICDFFCD 0ÐB•->Ñ!!-•"0ÐB•->Ñ 0ÐB€->Ñ!!•-•"0ÐB€->Ñ !0ÐB•->Ñ•-•"0ÐB•->Ñ! !0ÐB€->Ñ-•"0ÐB€->Ñ! By the linearity of the wave equation, the general formula for I as a plane wave moving along the B-axis is an arbitrary combination of the components given in the columns headed IICD and , and the implied solution for F is the same combination of the components given in the columns headed FFCD and . For example, if a wave propagating to the right is written Iœαα""0ÐB•->Ñ45€220ÐB•->Ñß (where αα" and # are arbitrary constants, and 00"# and are arbitrary functions), then " Fœαα0ÐB•->Ñ5•40ÐB•->ÑÞ -’“""## Similarly, if a wave propagating to the left is written Iœαα""0ÐB€->Ñ45€220ÐB€->Ñß then " Fœ•αα0ÐB€->Ñ54€0ÐB€->ÑÞ -’“""## Note that IF and are perpendicular to one another in both cases. An important class of solutions to these equations are the sinusoidal waves. To fix ideas, suppose I is a wave propagating to the right along the B-axis and oscillating with angular frequency =. Then we may write I as IœEcosOB•=>45€Ecos OB•=α>• CDabab where Oœ=Î-•. The parameter α is a “phase shifter” that may vary from 11 to . If αœ!, then the two components of II are “in phase” and the path of in the CD-plane is a straight line segment from EßE to •Eß•EÞIf α1œ„ , then the two components aCDbabCD of II are ")!° out of phase, and the path of in the CD-plane is a straight line segment " from ECß•ED to •ECDßEÞ If α1œ„#, then the path of I is an ellipse with semi-axes abab" ECD andE. For fixed B, if α1œ># the path is traversed in a clockwise direction as " increases, and if α1œ•>#, the path is traversed in a counterclockwise direction as increases. (These directions of traversal are reversed if >B is fixed and is allowed to Vector Calculus. Page 47 increase.) The phase shifter α and the two amplitudes EECD and control the “polarization” of IF. In any case, the value of implied by this equation is " Fœ•EcosOB•=>•α=45€Ecos OB•>Þ -’“DCabab 11. Solving Maxwell's equations. In section 7 of this appendix we solved Maxwell's equations for electrostatics and magnetostatics. We found that I œ•fF where the “electrostatic potential” F satisfies Poisson's equation 3 f#Fœ•ß %! and FEœf‚ where the “vector potential” E satisfies a vector version of Poisson's equation # fENœ•Þ.! (A11.1) To get eq. (A11.1) we needed to impose a restriction on E, namely f†E œ!Þ (A11.2) We now show how this analysis may be extended to solve Maxwell's equations in general. For reference, here are Maxwell's equations. 3 f†Iœß (A11.3) %! `F f‚I œ•ß (A11.4) `> `IN -# f‚Fœ€ß (A11.5) ab`> %! f†F œ!Þ (A11.6) As before, we begin with eq. (A11.6). As F is solenoidal, it follows that we may write FEœf‚ (A11.7) for some vector field E called the vector potential (as before). Vector Calculus. Page 48 Next, substitute eq. (A11.7) into eq. (A11.4). This yields ``E f‚IEœ•f‚œ•f‚Þ `>abŒ•`> Hence, `E f‚I!€œÞ Œ•`> In the language introduced above, IE€`Î`> is irrotational. Therefore, there exists a scalar field F called the scalar potential such that `E I €œ•fÞF `> We rewrite this as `E I œ•fF •Þ (A11.8) `> As before, there's some flexibility in our choice of E. For given F and IE, suppose and F satisfy eqs. (A11.7) and (A11.8). If we make the substitution EEwœ€f< for some scalar field <, then eq. (A11.7) will still be satisfied, but eq. (A11.8) will not. However, if we make the simultaneous substitution `< EEwwœ€f Equations (A11.7) and (A11.8) express I and FE in terms of vector potential and a scalar potential F. We now substitute eqs. (A11.7) and (A11.8) into eqs. (A11.3) and (A11.5) to obtain equations for EN and F3 in terms of the “sources” and . This yields ` 3 f#F €f†Eœ• `>ab%! and # # "`E "`F fE•œ•.!NE€ff†€ -#`>##Œ•-`> # where .%!!´"Î-Þ To simplify the mathematics, we impose a gauge transformation (the “Lorentz gauge”) such that Vector Calculus. Page 49 "`F f†E œ•Þ (A11.10) -# `> Substituting eq. (A11.10) into the preceding two equations, we obtain "`#F3 # (A11.11) fF•##œñF œ• -`> %! and "`#E f#E•œñ.ENœ•Þ (A11.12) -##`> ! Equation (A11.12) is actually three equations, one for each component of E: ñEBœ•.!NBßñECœ•.!NCßand.ñ.EDœ•N!D Recall that an equation of the form ñ< œ! is said to be a “wave equation.” Given this, you shouldn't be too surprised to learn that an equation of the form ñ<5œ is called a “wave equation with a source term.” A wave equation with a source term effectively combines a wave equation "`#< f#<•œ! -##`> and Poisson's equation f#<5œÞ To summarize, we've replaced Maxwell's four equations with the four equations FEœf‚ `E Iœ•f•F `> 3 ñF œ•ßand %! ñ.ENœ•Þ! These four equations contain the same physical content as Maxwell's equations, and in many circumstances are easier to handle. I refer the reader to Feynman for the physical interpretation of E and F. Vector Calculus. Page 50