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 +,+ß+ßáß+and ,ß,ßáß, a"#8bab"#8 then 8 +,´+,Þ (1.1) " 33 3" The standard norm in ‘8 is defined in terms of the dot product as +´+++##€+€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 ‘$. The symbols 3, 45, and denote the three standard unit coordinate vectors. Hence BßCßDœB3€C4€D5and +œ+ß+ß+œ+3€+45€+ . ababBCDBCD Suppose that ?B is a unit vector. For any vector , B†?œœB?Bcos))cos llllll where ) is the angle between ? andB. This means that B†?B is the component of in the direction of ?, and B†??B is the orthogonal projection of onto the subspace of scalar ab multiples of ?? (called the subspace of vectors “spanned” by )Þ This is illustrated below. B ) ! ?B†?? ab Vector Calculus. Page 1 The cross product of two vectors +,+ß+ß+ and ,ß,ß, in ‘$ 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, +‚,†-,‚-†+-‚†+,.(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: +‚,†-+†,-‚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 +,-Þââ,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 < BßBßáßB 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´´<B##€B€â€B#. Then ll È"# 8 " `:`<"B###•#3 œœ#B3B"€B#8€â€Bœ.(2.2) `B33`B#<abˆ‰ It follows that fœ:<<<•" , (2.3) abll a unit vector in the direction of <. As :: is differentiable, the derivative of at <? with respect to any vector exists and is denoted :wÐ<?àÑ. It may be shown that ::wÐ<à?ÑœfÐ<?ц Þ (2.4) If ? is a unit vector, :wÐ<?àÑ is said to be a directional derivative; it's the rate of change of : with respect to distance in the direction of ?. In this case, :wÐ<à?<Ñœf:)ÐÑcos (2.5) ll where ) is the angle between f:Ð<Ñ and ?. That is, f::Ð<цf?< is the component of ab in the direction of ?Þ As cos )is maximal when ):œ!f, it follows that points in the direction at which : increases fastest, and f::ÐÑ< gives the rate of change of in that direction. ll 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 <wÐ>ÑœBwÐ>ÑßBwwÐ>ÑßáßBÐ>Ñ ab"#8 is called the velocity vector and is tangent to the curve at each point. The norm of the velocity vector <wÐ>Ñ measures the speed at which the curve is traversed. The unit tangent ll vector X Ð>Ñ is defined as <wÐ>Ñ X´XÐ>Ñ´Þ (2.6) <wÐ>Ñ ll The arc-length function = is given by > =Ð>Ñœ<wÐ77Ñ. (2.7) (+ll with derivative given by =wÐ>Ñœ<wÐ>Ñ .(2.8) ll Combining eqs. (2.6) and (2.8), we find thatthe unit tangent vector may be interpreted as the rate of change of < with respect to =: <wÐ>Ñ.<<Î.>..B.B X´œœœ"8ßáßÞ (2.9) =wÐ>Ñ.=Î.>.=Œ•.=.= In ‘$, we write <Ð>ÑœBÐ>Ñ3€CÐ>Ñ45€DÐ>Ñ , so .B.C.D Xœ3€45€Þ .=.=.= Now suppose that :: is a differentiable scalar field defined on H. Let 1´‰<ÞThen 1À+ß,Ä‘ and for each >−+ß, cdcd 1Ð>Ñ´:<Ð>ÑÞ cd Under these assumptions, the function 1 is differentiable, and the derivative 1wÐ>Ñ is given by the following chain rule: 8 `: 1wÐ>Ñœf:<<Ð>ц wÐ>ÑœBw Ð>Ñ (2.10) " 3 cd 3œ" `B3 where each partial derivative is evaluated at <Ð>Ñ. The dot product Vector Calculus. Page 4 f: <XÐ>ÑÐ>Ñ cd is called the directional derivative of :: along the curve. Some authors write .Î.= for this directional derivative, as 8 `::.B . f: X 3 Þ " 3" `B3 .= .= 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 f::, 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 JJ•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<P•#< Hence Y is a potential function of the force field J<œ•P<P•# . The equipotential surfaces of Y are concentric spheres centered at the origin. 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 J•<<where <´Þ <$ ll Using Example 2, we see that J<œ•fY where Yœ•K7Q<•".