Boosting Vector Differential Calculus with the Graphical Notation

Joon-Hwi Kim∗ Department of Physics and Astronomy, Seoul National University, Seoul, South Korea

Maverick S. H. Oh† and Keun-Young Kim‡ Department of Physics and Photon Science, Gwangju Institute of Science and Technology, Gwangju, South Korea (Dated: January 9, 2020) Learning vector calculus techniques is one of the major hurdles faced by physics undergraduates. However, beginners report various difficulties dealing with the index notation due to its bulkiness. Meanwhile, there have been graphical notations for tensor algebra that are intuitive and effective in calculations and can serve as a quick mnemonic for algebraic identities. Although they have been introduced and applied in vector algebra in the educational context, to the best of our knowledge, there have been no publications that employ the graphical notation to three-dimensional Euclidean vector calculus, involving differentiation and integration of vector fields. Aiming for physics students and educators, we introduce such “graphical vector calculus,” demonstrate its pedagogical advan- tages, and provide enough exercises containing both purely mathematical identities and practical calculations in physics. The graphical notation can readily be utilized in the educational environ- ment to not only lower the barriers in learning and practicing vector calculus but also make students interested and self-motivated to manipulate the vector calculus syntax and heuristically comprehend the language of tensors by themselves.

I. INTRODUCTION will increase their virtuosity in index gymnastics and pro- mote them to develop concrete ideas of coordinate-free As an essential tool in all fields of physics, vector calcu- tensor algebra. Lastly, the graphical notation of vector lus is one of the mathematical skills that physics under- calculus serves as an excellent primer for graphical tools graduates have to be acquainted with. However, vector in modern physics such as perturbative diagrams in field calculus with the index notation can be challenging to theories as a conceptual precursor to Feynman diagrams. beginners due to its abstractness and bulkiness. They We anticipate that this “user’s manual” of graphical vec- report various difficulties: manipulating indices, getting tor calculus we provide will lower the barriers in learning lost and being ignorant about where to proceed toward and practicing vector calculus, as Feynman diagrams did during long calculations, memorizing the vector calcu- in quantum field theory. lus identities, etc. Meanwhile, there have been graphical languages for tensor algebra such as Penrose graphical notation,1 birdtracks,2,3 or trace diagrams4 that are in- II. GRAPHICAL VECTOR ALGEBRA tuitive and effective in calculations. Although they can be readily applied to three-dimensional Euclidean vec- A. Motivation and Basic Rules tor calculus, publications covering vector calculus in a graphical notation remain absent in our best knowledge. We have two vectors, A~ and B~ . We can make a scalar Previous works3–9 only dealt with linear “algebraic” cal- from these two by the dot product. In the ordinary index culations and did not consider vector differential and in- notation, we write B~ A~ = BiAi. Now, let us give some tegral “calculus.” artistic touch to it. · In response to this, for physics learners and educators, i i we introduce the “graphical vector calculus,” advertise BiAi = BiA = BiA (1) how easy and quick the graphical notation can derive vec- tor calculus identities, and provide practical examples in The “B-atom” and the “A-atom” are pairing their “elec- arXiv:1911.00892v2 [physics.ed-ph] 8 Jan 2020 the physics context. Here, we consider differential calcu- trons” (repeated index i) to form a “covalent bond!” lus only; vector integral calculus might be covered in a Analogous to chemistry, depict a “shared electron pair” following paper, as it also frequently appears in physics. by a line connecting two “atoms.” See the supplementary material10 for a brief discussion. B~ A~ = B A (2) Pedagogical advantages of the graphical notation are · numerous. First of all, it evidently resolves the aforemen- tioned difficulties of a beginner. It serves as an intuitive Vectors A~ and B~ are graphically represented as a box language that is easy to acquire but does not lack any with a line attached to it. The inner product is depicted essential elements of vector calculus compared to the or- by connecting the two lines of the two boxes. Further- dinary index notation. In addition, students who are ac- more, an additional insight from this is that scalars will quainted with the index notation would also benefit from be graphically represented as objects with no “external” learning the graphical notation. The graphical notation lines. B A only has an “internal” line; no lines are 2 connected to the outside. It is isolated so that if the en- So even if a diagram is drawn to look a little bit stiff, tire diagram is put inside a black box, no lines will poke please remember that it is “dancing” freely behind the out from it. In other words, scalars do not have free scene! Also, a line can freely pass under boxes, as you indices. can see in the second equality in Eq.6. In addition, intersections of lines have no significance; think of them Scalars: f = f just overpassing each other. When such intersections oc- (3) cur, we will always draw it in a manner that no ambiguity Vectors: A~ = A arises if one follows the “law of good continuation.” That is, “ ” is an overlap of “ ” and “ ,” not “ ” and The basic observations here are summarized in Table I. “ .” Index Language Graphical Language An n-index quantity A box with n attached lines The name of a quantity The character written inside B. Meet the Kronecker Delta the box Pairing (contracting) two Connecting two ends of lines The diagram for B~ A~ can be interpreted from a dif- indices · Free indices External lines ferent perspective. The last diagram in Eq.5 seems like Contracted (dummy) indices Internal lines two vectors and are “plugged into” a -shaped B A object. TABLE I. Translation between the index language and the graphical language.

Meanwhile, for scalar multiplication, addition, and (7) subtraction, we do not introduce new notational rules B A to represent them but just borrow the ordinary notation; that is, they are denoted by juxtaposition and by “+” Then, what does the -shaped object represent? It is a and “ ” symbols. “machine”11 that takes two vectors as input and gives a − scalar; it is the inner product “ ,” or in the index nota- · Scalar multiplication: fg = f g tion, “δij.” Plugging lines into the machine corresponds to contraction of indices. fA~ = A f (4) Addition/subtraction: f g = f g ± ± BiδijAj = = B A = ; (8) A~ B~ = A B ··· ± ± B A When two objects are juxtaposed, their relative position δ = = i j = . (9) g g ij ··· is irrelevant, such as f g = = = etc. i j f f ··· However, it should be noted that in Eq.2, B~ is de- In the second line, we turned on the “index markers” picted as a box with a line attached at its right side. It to avoid confusion that which terminal of the line corre- turns out that it is okay to not care about which side a sponds to the index i and j, respectively. line stems from a box for denoting vectors. A line can A comment should be made about the symmetry of start from the left side, right side, upper side, lower side, the Kronnecker delta. The fact that δij = δji is already or anywhere from the box, as if it freely “dangles” to be reflected in the design of our graphical notation, that is freely repositioned. For example, the appearance of δij with the dancing rule of equivalent diagrams. In the graphical notation, δij is an undirected A line, so that there is no way to distinguish its “left” and B A = A B = = = , (5) “right” terminals. For instance, see the first equality of B ··· B A Eq.5. If you want to write this symmetry condition without “test vectors” plugged in, observe the second and so on. It can be seen that an arbitrary rotation does form of B~ A~ in Eq.5 and the last form in Eq.6. It can not affect the value of a graphical equation. Moreover, be seen that· an arbitrary rearrangement of boxes also does not. For example, Eq.5 can be further deformed as the following. = . (10)

Turning on the index markers,12 A B B = = = A B (6) B A A i j = i j , (11) 3 or giving one more touch, clank . (12) i j = i j B A − B A

The left hand side assigns i to the left terminal of the - shaped and j to the right terminal; the right hand side FIG. 1. A minus sign pops out with a “clank!” sound when assigns i to the right terminal and j to the left terminal. you swap-then-yank the two arms of a cross product machine. Just pretend for a moment that the index assigned to The plaintext equation corresponding to this action is “A~ × the left terminal should be placed first when reading the B~ = −B~ × A~.” -shaped in Eq. 12 in the index notation; then, we have δij = δji. D. Triple Products

C. Meet the Cross Product Machine Having introduced the graphical notation for the cross product, let us now graphically express triple product Now, move on to the next important structure, the identities. First, a scalar triple product C~ A~ B~
can cross product. The cross product is a machine that takes · × be drawn by connecting the free terminals of and Eq. two vectors as a input and gives a vector. Hence, two C lines are needed for input and one line for output. 13:

C A~ B~ = = = = (13) = . (14) A B B B A × ··· A B C A B A

al- Please do not forget the diagrams are dancing and Eq. 13 The cyclic symmetry of the scalar triple product is ready reflected in its graphical design is showing just three snapshots. There are infinitude of : it looks the same under threefold rotation. possible configurations that A~ B~ can be drawn. Also, × note that the third diagram is read as A~ B~ as well as the first one. The lines attached to the cross× product C A B machine ( ) should be read counterclockwise from the = = core (the small dot) of the machine: . The left and A B B C C A (15) ~ right arms of the cross product machine is connected to A l and B~ respectively in both the first and third diagrams in C~ A~ B~
= A~ B~ C~
= B~ C~ A~
Eq. 13, so they are equivalent. Continuous deformations · × · × · × do not affect the value of a diagram. This is the economy of graphical notations: redundant However, how about discontinuous deformations? In plaintext expressions are brought to the same or at least case of the inner product, yanking a twist, a discontin- manifestly equivalent diagram. uous deformation that yields a cusp during the process, did not affect the value because the inner product is sym- As a side note, imagine what would it mean if the metric. In case of the cross product, it is antisymmetric cross product machine is naked, while it is fully dressed so that A~ B~ = B~ A~; therefore, when the two arms in Eq. 14, which is ijkCiAjBk in the index notation. of the first× diagram− in× Eq. 13 are swapped—which is As some readers might already noticed, another name the third diagram—and yanked, a minus sign pops out, for the cross product machine is the Levi-Civita symbol, as depicted in Fig. 1. Associating a kinesthetic imagery ijk. It is a three-terminal machine (three-index tensor), that the lines of the cross product machine are elastic and antisymmetric in every pair of its arms (indices). but particularly stiff near the core might be helpful to intuitively remember this. Do not forget the minus sign. k Yanking a twist is a discontinuous “clank” process. ijk = (16) Note that in case of a general object (tensor), the value i j after swap-then-yanking its two arms is by no means re- lated to the original value, unless it bears symmetry or antisymmetry with respect to permutation of the two in- Next is the vector triple product. The BAC-CAB for- dices. mula translates into the graphical language as the follow- 4 ing. easily write down the equations equivalent to the BAC- CAB rule or the Jacobi identity.10 Knowing what fun- damental rules that identities are rooted in with being able to generate equivalent identities will effectively pro- = mote concrete understandings of the structure of vector A A − A (17) algebra. B C B C B C

l A~ B~ C~ ) = B~ (A~ C~ ) C~ (A~ B~ ) III. GRAPHICAL VECTOR CALCULUS × × · − · This holds for arbitrary A~, B~ , and C~ ; thus, one can ex- Now is the time for graphical vector “calculus.” Here, tract the “bones” only: we are considering not just scalars and vectors, but “scalar fields” f(~r), g(~r), and “vector fields” A~(~r), ··· B~ (~r), ; they depend on spatial coordinates, or equiv- = . (18) ··· − alently, the position vector ~r. In this section, “(~r)” is omitted unless there is an ambiguity whether it depends on ~r or not. Until now, all graphical equations followed from defining rules of graphical representation. However, Eq. 18 is the first—and indeed the only—nontrivial formula relating cross product machines and Kronecker deltas. This is the A. The Basics most important identity that serves as a basic “syntax” of our calculations. The first mission would be graphically representing Equation 18 is by no means “new.” With the index ∂ th = ~ei := ~ei∂i, where ~ei and xi are the i Carte- markers, it turns out that it is the well-known formula ∇ ∂xi about contracted two  ’s. sian basis vector and coordinate, respectively. is a ijk “vector” (that is, it carries an index), but also a differen-∇ j i j i j i tial operator at the same time. Therefore, to accomplish the mission, a notation that has one terminal and is ca- k = (19) pable of representing the Leibniz property (the product − rule of derivatives) should be devised. The later can be l m l m l m achieved by an empty circle, which reminds of a balloon. Things inside the balloon are subjected to differentiation. l   = δ δ δ δ (20) The balloon “eats” fg by first biting f only then g only: ijk klm jm il − jl im f g = f g + f g (fg)0 = f 0g + fg0. To However, the graphical way has multiple appealing ↔ points. First, it naturally serves as a quick visual “vectorize” this, we simply attach a single tail to it. mnemonic for Eq. 20. Also, in practical circumstances, the graphical form avoids the bulkiness of dummy in- f g f g f g dices and significantly simplifies the procedure of index = + replacement by δij’s. One does not have to say “i to l, j to m” over and over in one’s mind organizing the ex- (21) panded terms. This makes a greater difference in calcu- i i i lation time as the equation involves more operations and dummy indices (proof of the Jacobi identity,10 for exam- l ∂ (fg) = ∂ (f) g + f ∂ (g) ple). On the other hand, classification of vector algebraic i i i identities is immediate if they are written in the graphi- cal notation, because it shows the (tensorial) structure of This “differentiation hook” design was previously sug- equations explicitly. One can recognize identical struc- gested by Penrose.14,15 However, he has not published tures within a single glance, as comprehension of visuals how to do the Euclidean vector calculus in three dimen- is much faster than that of texts. Some may take a criti- sions using it. As you will see soon, it is powerful to dis- cal stance to this, because mere counting of the symbols tinguish vector algebraic manipulations from the range of “ ” and “ ” would also reveal the structure of equa- differentiation when an index-free format is kept, while tions,× albeit· for simple cases. However, with the graph- both are denoted without distinction by parentheses in ical notation, generating different identities of the same the ordinary notation. structure is also straightforward; it is accomplished by The Leibniz rule, Eq. 21, can be applied regardless of just attaching “flesh pieces” (vectors or arbitrary multi- the operand type.16 For instance, a vector can be fed to terminal objects13) to the “bone.” For instance, one can . ∇ 5

. f A A
B = ∂iAj = “ A~” (22) ∇ ij F g

i j E G D Here, visual reasoning comes earlier, naturally suggest- C ing the concept “ A~” without reference to coordinates (before we attach∇ index markers). This is one of the instances where the graphical notation intuitively hints FIG. 2. The “ecosystem” of the graphical vector calculus. students, who do not have abstract and rigorous mathe- matical understanding, to enter the world of tensors with its coordinate-free nature unspoiled. Thus, we obtain B~ ( A~) A~ ( B~ ). We do The expression Eq. 22 can be physically or geometri- not need to memorize· the∇ × tricky− minus· ∇ sign × or look up a cally meaningful, but it frequently appears in a particular 17 vector identity list all the time. All we need to do is just encoding: divergence and curl. They are obtained when to doodle the diagrams and see what happens. we let the two tails of Eq. 22 “interact” with each other with the machines we have seen in Section II. 2. ∇ × A~ × B~
A A = A,~ = A.~ (23)
∇ · ∇ × A~ B~ can readily be written in a graphical form from∇× the× diagrams for the cross product (Eq. 13) and a curl of a vector field (Eq. 23). The formula is rather A final note: the differentiation apply only on boxes, complex-looking: A~ B~
= ( B~ )A~ + (B~ )A~ not lines. It is because δ ’s and  ’s are all constants. ∇ × × ∇ · · ∇ − ij ijk ( A~)B~ (A~ )B~ . While proving this in the index So, one can freely rearrange the balloons (differentiation) notation,∇ · you− may· ∇ frown at equations to recognize which relative to connecting lines and cross product machines indices corresponds to which epsilon and delta; however, regardless of how they are entangled with each other. it is much neater in the graphical notation. The proof An imagery that the balloon membrane is impermeable proceeds by applying the Leibniz rule Eq. 21 and the to boxes but do not care whether lines or cross product “ = ” identity Eq. 18. machines pass through can be helpful. −

B A B A B A = (25) B. First Derivative Identities −

Finally, we will now show how easy deriving vector cal- culus identities is with the graphical notation! Essential B A B A B A B A = + examples are demonstrated; the remaining identities are − − worked on the supplementary material10 as exercises. Translating back to the ordinary notation gives the de- 1. ∇ · A~ × B~
sired result. Note that the second term in the bottom line translates into (B~ )A~, since B ( ) is the derivative · ∇ ··· From the diagrams for the cross product (Eq. 13) and “modified” by B~ : it “B~ -likely” differentiates ( ), that the divergence of a vector field (Eq. 23), A~ B~
··· is, the directional derivative with respect to B~ , B ∂ ( ). can be easily represented graphically. Then,∇ · apply× the i i ··· Leibniz rule Eq. 21. 3. ∇A~ · B~
B A = B A + B A (24) Lastly, we will demonstrate a graphical reasoning on the notorious vector calculus identity: A~ B~
. The formula is given by Eq. 28. It is perhaps∇ the· most com- The second term is a contraction of and A , B plicated among all vector calculus identities. However, which is B~ ( A~). The first term is a contraction of a bigger problem is that it is not clear how to massage · ∇ × A~ B~
into smaller expressions. In the graphical nota- B = B and A , which is ( B~ ) A~. ∇ · − −∇ × · tion, one can see the motivation of each step more trans- 6 parently. Start from the diagram for A~ B~
: where anything smooth that the derivatives commute can ∇ · be go inside the balloons. This is translated into the ordinary notation as ∂j∂i = ∂i∂j as an operator identity. A A B One of the most immediate results in second order B = + . (26) derivatives is the following. B A

= = = = 0 (30) We aim to express Eq. 26 in tractable terms; we must − transform it into vectorial terms that can be written in a coordinate-free manner in the ordinary notation (such as divergence, curl, or directional derivatives).18 The second At the first equality, the inner balloon is rearranged to term in the right hand side is identical to the first term if be the outer one according to Eq. 29; the second equal- A is substituted to B and B is substituted to A; therefore, ity comes from the dancing rule; at the third equality, we may work on the first term first then simply do the the “clank” process is used. One can easily see that substitution to obtain the result for the second term. ( f) = 0 and ( A~) = 0 are all the conse- The central observation that guides us is that if the quences∇ × ∇ of this property.∇ · The∇ × details are contained in the 10 A supplementary material with the proof of other second first term was , it can be written as (B~ )A~. Then, and higher order identities. B ·∇ interchanging two lines is readily possible by = . − IV. PRACTICAL EXAMPLES

A A A So far, this is the story of the graphical notation, a beginners’ companion to vector calculus. In this section, = − we provide practical examples in the physics context. B B B

(27) A. The Economy of the Graphical Notation: The A A Same Diagram, Different Readings = + Remember the economy of the graphical notation in B B Section II C? In music, there are musical objects that have multiple names in ordinary notation. For exam- = (B~ )A~ + B~ ( A~) · ∇ × ∇ × ple, D] and E[ are the same when they are aurally rep- resented. Likewise, there are situations that different In the second line, the upper cross product machine is plaintext equations are represented as a single graphical “clanked.” Finally, expression so that one can easily recognize their equiva- A~ B~
= (B~ )A~ + B~ ( A~) + (A~ B~ ), (28) lence. The following two, which appears when one deals ∇ · · ∇ × ∇ × ↔ with the equations of motion of a rigid electric dipole translating and rotating in a magnetic field,19 are equal where “+(A~ B~ )” means adding the same expression ~ ~↔ in their values but spelled differently in the ordinary no- with A and B interchanged. This trick of interchanging tation. two lines, = , is often useful. With the graphical − notation, utilizing it and recognizing when to use it is ~v (~ω ~p) B~
, ~p (~v B~ )
~ω (31) achieved without difficulty. · × × − × × · To see the equivalence of them, one should spend time on permuting the vectors according to properties of the C. Second and Higher-order Derivative Identities triple products. However, it is strikingly easy if one draws a diagram corresponding to them. Graphical proofs of second and higher order iden- tities can be easily proceeded analogously. Second- ω v order derivatives are depicted as double-balloon dia- (32) grams. There are no new graphical rules introduced ex- p B cept the following “commutativity of derivatives,” Two expressions in Eq. 31 are just different readings (groupings) of Eq. 32. It is the matter of grouping the = , (29) left branch (~ω ~p) first or the right branch (~v B~ ) first in Eq. 32. Permuting× the vectors in the ordinary× notation 7 and in the graphical notation are just two different ways which is ∂ixj = δij in the ordinary notation. If the two of manipulating an identical tensor structure, but it is terminals are connected by Kronecker delta, a “vacuum much easier in the graphical notation. Then, why not bubble” is obtained: use the graphical notation, at least as a mnemonic? r i = = δijδij = 3 . (36) B. Cross Your Fingers j

The capacity of the graphical notations is more than If a cross product machine is used, a mnemonic. It is a calculation tool equipped with its own syntax so that one can proceed the entire process r of vector calculus in the graphical notation without ref- = = = = 0 , (37) erence to indices. Let us demonstrate such calculational − − advantages. The trick of interchanging lines introduced in Sec- as you know that ~r = 0. The second and the third tion III B 3 has an objective to reassign contractions be- equality proceed by∇ “swap-then-yanking” × the cross prod- tween indices to obtain a more convenient form. For an uct machine and the Kronecker delta part, respectively. example of its practical usage, consider the electrostatic force formula for a point electric dipole ~p in an electric Lastly, note that r = n , where ~n := ~r/r field E~ (~r). It is given by ~p E~ (~r)
, but also (~p )E~ (~r). (r := ~r ) is the unit radial vector. It would be an overkill to∇ look· up the vector calculus·∇ iden- With| | these basic graphical equations, one can graph- tity table and apply the general formula Eq. 28, because ically prove identities involving r and ~r such as the fol- ~p is not differentiated by . Simply, the following graph- lowing. ical equations completes∇ the proof of the equivalence of the two. (A~ ) ~r = A~ A r = A (38) ∇ ↔ p p p 2~r = 0 r = = 0 (39) p = = (33) ∇ ↔ − E E E k 20 E Here, the fact that ∂ δ = 0 j = 0 is used. k ij ↔ i Also, expressions such as (r sin θ~e ˆ)(~e ˆ := φ/ φ , ∇× φ φ ∇ |∇ | where φ is the azimuthal angle) can be calculated by Note that = E~ (~r) = 0. This shows the inten- recasting it into a coordinate-free expression: (~e ~r). E ∇ × ∇× z × tion of the calculation evidently, without memorizing the whole formula. In case of a point magnetic dipole ~m in ~ = = = = 2 (40) a magnetic field B(~r), − ez r e r z ez ez ez m m m m The last step is due to = 2 , which can be proved m = = + µ0 , (34) by the following. − − J B B B B = = 3 = 2 (41) − − − so the force exerted on the dipole is ~m B~ (~r)
= (~m 1∇ · · Rather than using coordinate expressions of gradient, )B~ (~r)+µ0 ~m J~(~r), where J~(~r) = B~ (~r) is current ∇ × µ0 ∇× curl, and divergence in particular coordinates, working in density at ~r. a coordinate-free manner has several benefits. In complex cases, it can be faster and has a lower possibility to make mistakes.21 Also, it offers an algebraic way to find the C. Identities Involving ~r δ(3)(~r) term in the divergence or curl of a vector field.10 It is notable that such advantages are doubled with the As a specific and important example, consider the vec- graphical notation that significantly lowers the difficulty tor calculus with the position vector, ~r. First, note that of handling higher-rank index manipulations. For var- ious physical examples such as dipolar electromagnetic r = , (35) fields and flow configurations in fluid dynamics, refer to the supplementary material.10 8

D. A First Look on Tensors in an unambiguous and less-bulky form, in comparison to ordinary notations. Moreover, as one finds in the 10 Lastly, we want to comment about tensors, since they supplementary material, one can wisely calculate enor- occasionally appear in undergraduate physics. Students mous tensor expressions in a shortcut with the guidance are likely to develop the ideas of tensors by themselves of the graphical notation. Lastly, the graphical notation while utilizing the graphical vector calculus; the exten- is considerably useful in denoting and explaining the in- sion from zero and one-terminal objects to multi-terminal variance property of tensorial expressions. As elaborated 10 objects is straightforward, and the graphical notation in the supplementary material, one can easily exam- naturally involves the manipulation of multiple termi- ine how the terminals of a tensor expression transforms nals. Also, graphical representations are useful to ex- under rotation intuitively by “arrow pushing”—the pair plain the concept of tensors to students, utilizing the creation/annihilation and propagation of arrowheads. “machine view.” For example, think about the inertia tensor, Iij = i I j . It is simply a two-terminal device that “modulates” a one-terminal input (angular velocity, V. CONCLUSIONS ω ) into a one-terminal output (angular momentum, L = I ω ). Imagine as if a “signal” generated Graphical notations of tensor algebra have a history from the ω box propagates from right to left. Swapping spanning over a century.2 The basic idea can be traced the two arms of the inertia tensor does not affects the back to the late 19th century works on invariant theory value), because it is symmetric: i I j = Iij = Iji = that related invariants to graphs.23–26 In the mid-20th i I = I . However, this is not the case for century, diagrammatic methods such as Levinson and i j j Yutsis’ diagrams for 3n-j symbols27,28 and Cvitanovi´c’s a general multi-terminal object unless it is symmetric, as birdtracks2,29,30 are devised to conduct group-theoretic we have already discussed in Section II C. For the details calculations and applied to quantum theory.3,31–33 Ac- of graphical representations of such general objects, refer cording to Levinson,27 one of the major motivations to to the supplementary material.10 Here, we restrict our develop such apparatus was “the extreme inconvenience attention to symmetric rank-2 tensors. due to the bulkiness” of the ordinary plaintext notation. At least there are three of the practical benefits of us- On the other hand, Penrose1,34 devised a graphical no- ing graphical notation for tensor equations. First, it is tation for tensor algebra and utilized it in tensors and convenient to calculate the trace22 and related quantities spinors in general relativity, theory of angular momen- of a tensor.10 Next, the graphical notation provides a tum and spin networks, and twistor theory.14,15,35 Simi- transparent and unambiguous way to denote contraction lar to Levinson,27 one of his motivation was also to sim- structures. For example, consider the two expressions be- plify the complicated equations and to effectively grasp low denoting K = 1 ω I ω and  ω L =  ω I ω 2 i ij j ijk j k ijk j kl l the various interrelations they have by visual reasoning;36 respectively, however, he was also intended to introduce the concept of “abstract tensor system” by a coordinate-free notation that transparently retains the full syntactic structures of 1 ω ω 1,14,15 K = 2 I , L = I ω , (42) tensor equations. The concept of the abstract ten- ω ω sor system and the Penrose graphical notation motivated the study of category theory and its graphical language or the following more complex example that appears in in algebraic geometry,37–40 and served as a background40 the formula for the angular profile of electric quadrupole to “language engineering” works to physics,41–43 such as radiation power. diagrams in tensor network of states44–49 or quantum in- 40,41,50 " # formation and computing. n Q n So, why is the three-dimensional Euclidean vector cal- n n (43) Q Q∗ culus so quiet with such “graphicalism?” Perhaps it has − n Q∗ n been already being used as a private calculation tech-

Here, i Q j = Qij is the electric quadrupole moment nique, but its intractability to be printed due to graphi- 37,51 which is also a symmetric tensor. The asterisk stands for cal format might hindered its publication. However, complex conjugation. For a calculus example, consider regarding the popularity of Feynman diagrams that is the divergence of the stress tensor σ, σ. Which index also a graphical notation, it is worth casting light on the of σ is in charge of the inner product∇ in · the expression graphical tensor notation, as graphical vector calculus “ σ?’—find the answer in the following diagram. has its own pedagogical benefits. (Moreover, it conceptu- ∇ · ally precedes to Feynman diagrams.) On the other hand, educators, already well-acquainted with the index nota- σ (44) tion and less sensible to beginners’ difficulties, might have not tried to employ a graphical machinery to do vector The contraction structures and their symmetry are calculus. However, there are introductory materials for clearly evident at a glance and can be quickly denoted graphical vector algebra and linear algebra,5–8,52 where 9 differentiation does not comes into play. Therefore, pub- simpler expression of the same symmetry up to a propor- lishing an educator’s manual for the application of the tionality constant”10 is also notable.57 graphical notation in vector calculus would be a useful Finally, it serves as an excellent primer to the graphical thing to do. languages of advanced physics for undergraduates. After What is newly proposed in this work is the graphical learning the graphical vector algebra, one can easily learn derivations and tricks of the vector differential calculus. the birdtracks notation that is capable of group-theoretic No previous publications have dealt with the differentia- calculations in quantum theory. Also, the graphical vec- tion and integration of vector fields, while the graphical tor calculus provides exercises of “diagrammatics,” trans- vector algebra introduced in this paper can be found also lating equations into graphics and vice versa that is an in other publications.2,3,5,6,53 Also, pedagogical values of everyday task when one learns quantum field theory. En- the graphical notation are demonstrated, and sufficient thusiastic undergraduates who have always been curious exercises containing both mathematical and physical cal- about the working principles of Feynman diagrams will culations are provided. Overall, this paper will serve as quench their thirst by learning the graphical tensor alge- a self-contained educational material. bra. In essence, graphs for tensorial expressions of var- ious symmetry groups, birdtracks, is a group-theoretic The graphical notation has a lot of advantages. First, portion of Feynman diagrams. It is easy to learn Feyn- it provides a quick mnemonic or derivation for identities man diagrams after learning birdtracks or graphical ten- (e.g. Eq. 18 or the vector calculus identities). It also en- sor algebra and vice versa because the way they denote hances the calculation speed,54 giving a bird's eye view mathematical structures is alike: loop diagrams for trace to calculation scenario. The strategy of reducing compli- (“vacuum bubbles,” Eq. 36) or etc. Meanwhile, bird- cated expressions can be wisely decided. Although they tracks may leave a more concrete impression because it are best performed in the graphical environment, such has graphical “progression rules”58 that enables to jump techniques on index gymnastics gained from graphical from an expression to another via equality unlike Feyn- representations are inherited altogether into the index man diagrams. Furthermore, when one considers a se- notation environment. An index notation user also will ries expansion of a tensorial expression, one encounters benefit from association of a tensorial expression with a the exact parallel with diagrammatic perturbation in sta- graphical image. tistical mechanics or quantum field theory. Pedagogical Next, it has advantages in denoting and comprehend- examples can be found in the supplementary material.10 ing tensors. If it is unambiguous, an index-free no- The core characteristic that provides a background to tation is preferred, that is, “ A~” is preferred over ∇ × all these advantages is the “physically implemented syn- “~eiijk∂jAk,” probably because it is more simple and tax” of the notation. It is believed that Feynman dia- easy to read off the tensorial structure in groups of se- grams work because it is indeed a faithful representation mantic units (such as parsing B~ A~ into “B~ dot A~,” ·∇× ∇× of the physical reality (to the best of our knowledge)— not “(B~ cross ) dot A~”). Particularly, the graphical no- the nature is implemented by worldlines of particles that ∇ tation is preferable to other index-free notations, because are isomorphic to Feynman diagrams. In the graphi- it can flexibly represent tensor equations which become cal notation of tensors, the grammar of tensors is “em- bulky in the ordinary index-free notation and transpar- bodied” in the wires, 3-junctions, nodes, beads, and all ently displays the contraction structure. The symmetry that: the symbols behave as its physical appearance of a tensorial expression also can be grasped at a single (self-explanatory design of symbols in Section II B and glance. Moreover, students will automatically discover Section II C). Consequently, the language is highly in- the concept of tensors as an invariant n-terminal object tuitive and automatically simplifies tensorial expressions and develop essential ideas of tensors in a coordinate-free (the economy of the graphical notation). The association setting using the graphical notation. For example, stu- of a kinesthetic imagery further simplifies the perception dents will realize themselves interpreting the first term in and manipulation of the elements (the dancing rule and the right hand side of Eq. 26 as Eq. 22 contracted with B~ the “clank” in Section II C). As Feynman diagrams are at its second terminal (“input slot”). As a result, the idea the most natural language to describe the microscopic of the tensor “ A~” can be understood without leaving process of elementary particles, the graphical notation is a vague impression,∇ as its graphical representation pro- the canonical language of the vector calculus system. vides a concrete comprehension of its functionality (as a Last but not least, the graphical notation will change “machine”). As parse trees (graphs) can promote under- a vector calculus class into an enjoyable game. As a child standing syntactic structures and generating sentences playing with educational toys such as Lego blocks or mag- of the same structure, the graphical representation can netic building sticks, it will be an entertaining experience do the same in tensor calculus and its education.55 Fur- to “doodle” with the dancing diagrams. Even a calcula- thermore, an unsupervised acquisition of tacit knowledge tion of complicated tensorial invariants can be a challeng- during graphical manipulation experiences such as “the ing task that thrills a person; one would feel as if he or equations are also valid after undressing test vectors from she is doing cat's cradle or literally “gymnastics” involv- them” (Section II D) or “a compound n-terminal object ing their visual, kinesthetic, or even multimodal neural that has a permutation symmetry can be reduced into a substrates. Such an amusing character can attract stu- 10 dents interest and offer a motivation to study vector cal- about boosting your education by the graphical notation? culus. Students would voluntarily build various tensorial structures, heuristically find the identities, and gain intu- itions. One possible “creative classroom” scenario can be VI. ACKNOWLEDGEMENT suggested is to present students only the basic grammar of the graphical notation and letting them spontaneously We thank Elisha Peterson for the provision of access and exploratively find the “sentences (identities),” per- to his research materials and clarification on the rea- haps in a group. The teacher can collect their results and son why he gave the name “trace diagrams” to his di- have a group presentation, then introduce missing iden- agrams via e-mail. The work of K.-Y. Kim was sup- tities if any. This will turn a formula-memorizing class ported by Basic Science Research Program through the into an amusing voluntary learning experience. So, how National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF2017R1A2B4004810) and GIST Research Institute (GRI) grant funded by the GIST in 2019.

∗ [email protected] to imply such technical term when we say “objects.”. † [email protected] 14 R. Penrose and W. Rindler, Spinors and space-time, Cam- ‡ [email protected]; https://phys.gist.ac.kr/gctp/ bridge Monographs on Mathematical Physics, Vol. 1 (Cam- 1 R. Penrose, Combinatorial mathematics and its applica- bridge University Press Cambridge, 1987). tions , 221 (1971). 15 R. Penrose, The road to reality: A complete guide to the 2 P. Cvitanovi´c, Group theory: birdtracks, Lie’s, and excep- physical universe, vintage ed. (Vintage Books, New York, tional groups (Princeton University Press, 2008) p. 273. 2007). 3 G. E. Stedman, Diagram techniques in group theory (Cam- 16 However, if one considers covariant derivatives, its action bridge University Press, Cambridge, 2009). depends on the operand type. The graphical notation for 4 E. Peterson, Trace diagrams, representations, and low- it can be devised easily, also. dimensional topology, Ph.D. thesis, University of Maryland 17 “∇A~” is decomposed into three invariant combinations un- (2006). der SO(3) action: divergence, curl, and “shear.” However, 5 J. Blinn, IEEE Computer Graphics and Applications 22, as Romano and Price60 points out, shear is a rather un- 86 (2002). popular concept in usual undergraduate courses. 6 18 th E. Peterson, arXiv e-prints , arXiv:0910.1362 (2009), The i component of the first term in Eq. (26) is Bj ∂i(Aj ), arXiv:0910.1362 [math.HO]. which is inaccessible in the ordinary coordinate-free nota- 7 E. Peterson, arXiv e-prints , arXiv:0712.2058 (2007), tion. Some notations, such as Hestenes’ overdot notation arXiv:0712.2058 [math.HO]. and Feynman’s subscript notation, had been suggested to 8 E. Peterson, On A Diagrammatic Proof of the Cayley- avoid such componentwise description, while the graph- Hamilton Theorem, Tech. Rep. (United States Military ical notation being the most clear and transparent one. 61 Academy, 2009) arXiv:0907.2364v1. In Hestenes’ overdot notation, ~eiBj ∂i(Aj ) is denoted as 9 ˙ J. Richter-Gebert and P. Lebmeir, Discrete & Computa- ∇˙ (A~ · B~ ). The overdot specifies which quantity is subject tional Geometry 42, 305 (2009). 10 to differentiation; parentheses is for vector algebraic pars- J.-H. Kim, M. S. H. Oh, and K.-Y. Kim, “An Invitation ing. On the other hand, in Feynman’s subscript notation,62 to Graphical Tensor Methods: Exercises in Graphi- ~ ~ it is denoted as ∇A~ (A · B). Both can be used as long as cal Vector and Tensor Calculus and More,” https: they don’t arise confusion with preexisting notations (such //www.researchgate.net/publication/337831066_An_ as time derivatives and directional derivatives with respect Invitation_to_Graphical_Tensor_Methods_Exercises_ to vector fields). in_Graphical_Vector_and_Tensor_Calculus_and_More. 19 11 For example, see the theoretical problem 2 of the Asian Cf. the “machine” view of tensors, such as in Misner, Physics Olympiad in 2001,63 an undergraduate-level prob- Wheeler, and Thorne’s book.59. 12 lem that is interesting and physically meaningful, hav- When you translate a graphical equation with no index ing implications on special relativistic electrodynamics and markers specified, the locations of terminals are the refer- “Gilbert-Ampre duality.”64–67. ence for assigning indices. Imagine both sides of the equa- 20 As mentioned before, the coordinates we are considering tion are wrapped in a black box. Then, assign the same in- here is Cartesian. In curvilinear coordinates, one should dices for identical sites on the black box surface; i.e. “same prove ∇2~r = 0 by ∇2~r = ∇(∇ · ~r) − ∇ × (∇ × ~r) = 0. index for same terminal” of the black box. This rule must 21 This is also true for integral calculus of vector fields. be respected also when you write a graphical equation with 22 Our graphical notation for Kronecker delta and the no index markers specified. If the terminals of both sides cross product machine is identical to Peterson’s “trace of the equation do not match, such as “ ⊃ = | ”, then it is diagrams.”4,6–8 In fact, the name “trace diagrams” orig- invalid. We say, “types do not match.”. 13 inates from the fact that the trace of a matrix is one of the In category theory, what we are calling “objects” here has a simplest diagram and easily calculated in the notation. name “morphism.” “Objects” rather refer to indices in cat- 23 J. J. Sylvester, American Journal of Mathematics 1, 64 egory theorists’ terminology. However, we are not intended (1878). 11

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