**Pre-script** (dated 26 June 2020): This post has become less relevant (even irrelevant, perhaps) because my views on all things quantum-mechanical have evolved significantly as a result of my progression towards a more complete *realist *(classical) interpretation of quantum physics. In addition, some of the material was removed by a dark force (that also created problems with the layout, I see now). In any case, we recommend you read our recent papers. I keep blog posts like these mainly because I want to keep track of where I came from. I might review them one day, but I currently don’t have the time or energy for it. đ

**Original post**:

The relationship between math and physics is deep.Â When studying physics, one sometimes feels physics and math become one and the same. But they are *not*. In fact, eminent physicists such asÂ Richard FeynmanÂ warn against emphasizing the math side of physics too much: âIt is *notÂ *because you understand the Maxwell equations *mathematically* inside out, that you understand *physics* inside out.â

We should never lose sight of the fact that all these equations and mathematical constructs represent *physical* realities. So the math is nothing but the ‘language’ in which we express physical reality and, as Feynman puts it, one (also) needs to develop a âphysicalâ â as opposed to a ‘mathematical’ âÂ understanding of the equations. Now you’ll ask: whatâs a âphysicalâ understanding? WellâŚ Let me quote Feynman once again on that: â*A physical understanding is a completely unmathematical, imprecise, and inexact thing, but absolutely necessary for a physicist.*â

It’s rather surprising to hear that from him: this is a rather philosophical statement, indeed, and Feynman doesn’t like philosophyÂ (see, for example, what he writes on the philosophical implications of the Uncertainty Principle). Indeed, while most physicists â or scientists in general, I’d say âÂ will admit there is some value in a philosophy of science (that’s the branch of philosophy concerned with the foundations and methods of science), they will usually smile derisively when hearing someone talk about *meta*physics.Â However, ifÂ metaphysics is theÂ branch of philosophy that deals with ‘first principles’, then itâs obvious that the Standard Model (SM) in physics is, in fact, also some kind of âmetaphysicalâ model! Indeed, what everything is said and done, physicists *assume* those complex-valued wave functions are, somehow, ‘real’, but all they can ‘see’ (i.e. measure or verify by experiment) are (real-valued) probabilities: we can’t ‘see’ the probability *amplitudes*.

The only reason why we accept the SM theory is because its predictions agree so well with experiment.Â *VeryÂ *well indeed. The agreement between theory and experiment is most perfect in the so-called electromagnetic sector of the SM, but the results for the weak force (which I referred to as the ‘weird force’ in some of my posts) are very good too.Â For example, using CERN data, researchers could finally, recently, observe an extremely rare decay mode which, once again, confirms that the Standard Model, as complicated as it is, is the best we’ve got: just click on the link if you want to hear more about it. [And please do: stuff like this is quite readable and, hence, interesting.]

As this blog makes abundantly clear, it’s not easy to ‘summarize’ the Standard Model in a couple of sentences or in one simple diagram. In fact, I’d say that’s impossible. If there’s one or two diagrams sort of ‘covering’ it all, then it’s the two diagrams that you’ve seen *ad nauseam *already: (a) the overview of the three generations of matter, with the gauge bosons for the electromagnetic, strong and weak force respectively, as well as the Higgs boson, next to it, and (b) the overview of the various interactions between them. [And, yes, these two diagrams come from Wikipedia.]

I’ve said it before: the complexity of the Standard Model (it has not less than 61 ‘elementary’ particles taking into account that quarks and gluons come in various ‘colors’, and also including all antiparticles â which we have to include them in out count because they are justÂ as ‘real’ as the particles), and the ‘weirdness’ of the weak force, plus a astonishing range of other ‘particularities’ (these ‘quantum numbers’ or ‘charges’ are really not easy to ‘understand’), do not make for a aesthetically pleasing theory but, let me repeat it again, it’s the best we’ve got. Hence, we may not ‘like’ it but, as Feynman puts it: “Whether we like or don’t like a theory is *notÂ *the essential question. It is whether or not the theory gives predictions that agree with experiment.” (Feynman, QED â *The Strange Theory of Light and Matter*, p. 10)

It would be foolish to try to reduce the complexity of the Standard Model to a couple of sentences. That being said, when digging into the subject-matter of quantum mechanics over the past year, I actually got the feeling that, when everything is said and done, modern physics has quite a lot in common with Pythagorasâ ‘simple’ belief that mathematical conceptsÂ â and numbers in particularÂ â might have greater âactualityâ than the reality they are supposed to describe. To put it crudely, the only ‘update’ to the Pythagorean model that’s needed is to replace Pythagorasâ numerological ideas by the equipartition theorem and quantum-mechanical wave functions, describing probability amplitudes that are represented by *complexÂ numbers*. Indeed, complex numbers are numbers too, and Pythagoras would have reveled in their beauty. In fact, I can’t help thinking that, if he could have imagined them, he would surely have created a âreligionâ around Euler’s formula, rather than around the tetrad. đ

In any case… Let’s leave the jokes and the silly comparisons aside, as that’s not what I want to write about in this post (if you want to read more about this, I’ll refer you another blog of mine). In this post, I want to present the basics of vector calculus, an understanding of which is *absolutelyÂ *essential in order to gain both a mathematical as well as a ‘physical’ understanding of what *fields* really are. So that’s classical mechanics once again. However, as I found out, one can’t study quantum mechanics without going through the required prerequisites.Â So let’s go for it.

**Vectors in math and physics**

What’s a vector? It may surprise you, but the term ‘vector’, in physics and in math, refers to more than a dozenÂ *differentÂ *concepts, and that’s a major source of confusion for people like usâautodidacts. The term ‘vector’ refers to many different things indeed. The most common definitions are:

- The term ‘vector’ often refers to a (one-dimensional)
*array of numbers*. In that case, a vector is, quite simply, an element ofÂ**R**^{n}, while the array will be referred to as an. This definition can be generalized to also include arrays of alphanumerical values, or blob files, or any type of object really, but that’s a definition that’s more relevant for other sciences â most notably computer science. In math and physics, we usually limit ourselves to arrays of*n*-tuple*numbers*. However, you should note that a ‘number’ may also be a complex number, and so we have real as well as complex vector spaces. The most straightforward example of a complex vector space is the set of complex numbers itself:Â**C**. In that case, the*n*-tuple is a ‘1-tuple’, aka as a singleton, but the element in it (i.e. a complex number) will have ‘two dimensions’, so to speak. [Just like the term ‘vector’, the term ‘dimension’ has various definitions in math and physics too, and so it may be quite confusing.] However, we can also have 2-tuples, 3-tuples or, more in general, n-tuples of complex numbers. In that case, the vector space is denoted by**C**^{n}. I’ve written about vector spaces before and so I won’t say too much about this. - A vector can also be aÂ
. In that case, it represents the*pointÂ*vector*position*of a point in physical space â in one, two or three dimensions âÂ in relation to someÂ*arbitrarily chosen*origin (i.e. the zero point). As such, we’ll usually write it as x (in one dimension) or, in three dimensions, as (x, y, z). More generally, a point vector is often denoted by the bold-face symbolÂ**R**. This definition is obviously ‘related’ to the definition above, but it’s not the same: we’re talking*physicalÂ*space here indeed, not some ‘mathematical’ space. Physical space can be*curved*, as you obviously know when you’re reading this blog, and I also wrote about that in the above-mentioned post, so you can re-visit that topic too if you want. Here, I should just mention one point which may or may not confuse you: while (two-dimensional) point vectors and complex numbers have a lot in common,Â*they are not the same*, and it’s important to understand both the similarities as well as the differences between both. For example, multiplying two vectors and multiplying two complex numbers is definitely*notÂ*the same.Â I’ll come back to this. - A vector can also be a
: in that case, it will specify the*displacement vector**change*in position of a point relative to its previous position. Again, such displacement vectors may be one-, two-, or three-dimensional, depending on the space we’re envisaging, which may be one-dimensional (a simple line), two-dimensional (i.e. the plane), three-dimensional (i.e. three-dimensional space), or four-dimensional (i.e. space-time). A displacement vector is often denoted by**s**orÂ Î**R**, with the delta indicating we’re talking a a*distanceÂ*or aÂ*difference**Â*indeed: s = ÎR =**R**_{2}Â âÂ**R**_{1}Â = (x_{2}Â â x_{1}, y_{2}Â â y_{1}, z_{2}Â â z_{1}). That’s kids’ stuff, isn’t it? - A vector may also refer to a so-called
: aÂ*four-vector**four-vectorÂ*obeys very specific transformation rules, referred to as the Lorentz transformation. In this regard, you’ve surely heard of space-time vectors, referred to as*events*, and noted as**X**= (*c*t,**r**), with r the spatial vector**r**= (x, y, z) and*c*the speed of light (which, in this case, is nothing but a proportionality constant ensuring that space and time are measured in compatible units). So we can also write**X**asÂ**X**= (*c*t, x, y, z). However,Â there is another four-vector which you’ve probably also seen already (see, for example, my post on (special)Â RelativityÂ Theory):**P**= (E/c,**p**), which relates energy and momentum in spacetime. Of course, spacetime can also be*curved*. In fact, Einstein’s (general) Relativity Theory is about the curvature of spacetime, not of ordinary space. But I should not write more about this here, as it’s about time I get back to the main story line of this post. - Finally, we also have
**vector**, like the gradient vector*operators***â**.**Now**VectorÂ*thatÂ*isÂ what I want to write about in this post.*operatorsÂ*are also considered to be ‘vectors’ â to some extent, at least: we use them in a ‘vector products’, for example, as I will show below â but, because they are operators and, as such, “hungry for something to operate on”, they are obviously quite different from any of the ‘vectors’ I defined in point (1) to (4) above. [Feynman attributes this ‘hungry for something to operate on’ expression to theÂ British mathematician Sir James Hopwood Jeans, who’s best known from the infamous Rayleigh-Jeans law, whose inconsistency with observations is known as the ultraviolet catastrophe or ‘black-body radiation problem’. But that’s a fairly useless digression so let me got in with it.]

In a text on physics, the term ‘vector’ may refer to any of the above but it’s often the second and third definition (point and/or displacement vectors) that will be implicit. As mentioned above, I want to write about the fifth ‘type’ of vector: vector operators. Now, the title of this post is ‘vector calculus’, and so you’ll immediately wonder why I say these vector operators may or may not be defined as vectors. Moreover, the fact of the matter is that these operators operate on yet another type of ‘vector’ â so that’s a ** sixth** definition I need to introduce here:Â

**.**

*fieldÂ*vectorsNow, funnily enough, the term ‘field vector’, while being the most obvious description of what it is, is actually not widely used: what I call a ‘field vector’ is often referred to as a *gradient vector*, and the vectors **EÂ **andÂ **B **are usually referred to as the electric or magnetic *field*, *tout court*. Indeed, if you *googleÂ *the terms ‘electromagnetic vector’ (or electric or magnetic vector), you will usually be redirected. However, when everything is said and done,Â **E** and **B**Â *areÂ *vectors: they have a magnitude, and they have a direction. To be even more precise, while they depend on both space and time â so we can write **E** as **E** = **E**(x, y, z, t) and we have *four*Â independent variables hereÂ â they have three components: one of each direction in space, so we can write **E** as:

Â **E** = **E**(x, y, z, t) = [E_{x}, E_{y}, E_{z}] = [E_{x}(x, y, z, t), E_{y}(x, y, z, t), E_{z}(x, y, z, t)]

So, truth be told, **vector calculus** (aka vector analysis) in physics is about (vector) ** fields** and (vector)

**,. While the ‘scene’ for these fields and operators is, obviously, physical space (or spacetime) and, hence, aÂ**

*operators**vector space*, it’s good to be clear on terminology and remind oneself that, in physics, vector calculus is not about mathematical vectors: it’s about real stuff. That’s why Feynman prefers a much longer term than vector calculus or vector analysis: he calls itÂ

*differential calculus of vector fields*Â which, indeed, is what it is â but I am sure you would not have bothered starting reading this post if I would have used that term too. đ

Now, this has probably become the longest introduction *ever* to a blog post, and so I had better get on with it. đ

**Vector fields and scalar fields**

Let’s dive straight into it. Vector fields like**Â E** and **B **behave like ** h**, which is the symbol used in a number of textbooks for the heat flow in some body or block of material:

**E**,

**B**and

**are all**

*h**vector fields derived from a scalar field.*

* Huh? Scalar field?Â *Aren’t we talking vectors? We are.Â If I say we can derive the vector field

**(i.e. the heat flow) from a scalar field, I am basically saying that the relationship between**

*h***and the temperature T (i.e. the scalar field) is direct and very straightforward. Likewise, the relationship between**

*h***E**and the scalar field ÎŚ is also direct and very straightforward.

[To be fully precise and complete, IÂ should qualify the latter statement: it’s only true in *electrostatics*, i.e. when we’re considering charges that don’t move. When we have moving charges, magnetic effects come into play, and then we have a more complicated relationship between (i) *twoÂ *scalar fields, namelyÂ A (the magnetic potential – i.e. the ‘magnetic scalar field’) and ÎŚ (the electrostatic potential, or ‘electric scalar field’), and (ii) *two* vector fields, namelyÂ **B** and **E**. The relationships between the two are then a bit more complicated than the relationship between T and ** h**. However, the math involved is the same. In fact, the complication arises from the fact that magnetism is actually a relativistic effect. However, at this stage, this statement will only confuse you, and so I will write more about that in my next post.]

Let’s look at ** h** and T.Â As you know, the temperature is a measure for energy. In a block of material, the temperature T will be a

*scalar*: some real number that we can measure in Kelvin, Fahrenheit or Celsius butÂ â whatever unit we useÂ â

*any observer using the same unit willÂ measure the sameÂ at any given point*. That’s what distinguishes a ‘scalar’ quantity from ‘real numbers’ in general: a scalar field is something

*real.*It represents something

*physical*.Â A real number is just… Well… A real number, i.e. a mathematical concept only.

*Â Â*

The same is true for a vector field: it is something *real*. As Feynman puts it: “It is not true that any three numbers form a vector [in physics]. It is true only if, when we rotate the coordinate system, the components of the vector transform among themselves in the correct way.” What’s the ‘correct way’? It’s a way that ensures thatÂ *any observer using the same unit willÂ measure the sameÂ at any given point*.

**How does it work?**

In physics, we associate a point in space with *physical realities*, such as:

*Temperature*, the ‘*height*‘ of a body in a gravitational field, or the*pressure*distribution in a gas or a fluid, are all examples of**scalar fields**: they are just (real) numbers from a math point of viewÂ but, because they do represent a physical reality, these ‘numbers’ respect certain mathematical conditions: in practice, they will be a continuous or continuously differentiableÂ*function*Â of position.*Heat flowÂ*(), theÂ*h**velocity*()Â of the molecules/atoms in a rotating object, or the electric field (*v***E**), are examples of**vector fields**. As mentioned above, the same condition applies:Â*any observer using the same unit should measure the sameÂ at any given point*.*Tensors*, which represent, for example, stress or strain at some point in space (in various directions), or the curvature of space (or spacetime, to be fully correct) in the general theory of relativity.- Finally, there are also
*spinors*, which are often defined as a “generalization of tensors using complex numbers instead of real numbers.” They are very relevant in quantum mechanics, it is said, but I don’t know enough about them to write about them, and so I won’t.

How do we derive a vector field, like ** h**, from a scalar field (i.e. T in this case)? The two illustrations below (taken from Feynman’s

*Lectures*) illustrate the ‘mechanics’ behind it: heat flows, obviously, from the hotter to the colder places. At this point, we need some definitions. Let’s start with the definition of the heat flow:

**the (magnitude of the)**

*Â*

**heat****Â**

*flow*Â (*h*) is the amount ofÂ thermal*energy*(ÎJ) that passes, per unit time and per unit area, through an infinitesimal surface element at right angles to the direction of flow.A vector has both a magnitude and a direction, as defined above, and, hence, if we define *e** _{f}*Â as the unit vector in the direction of flow, we can write:

** h** =

*h*Âˇ

*e**Â = (ÎJ/Îa)Âˇ*

_{f}

*e*

_{f}ÎJ stands for the thermal energy flowing through an area marked as Îa in the diagram above *per unit time*. So, if we incorporate the idea that the aspect of time is already taken care of, we can simplify the definition above, and just say that **the heat flow is the flow of thermal energy per unit area**.Â Simple trigonometry will then yield an equally simple formula for the heat flow through

*any*surface Îa

_{2}Â (i.e. any surface that isÂ

*not*at right angles to the heat flow

**):**

*h*ÎJ/Îa_{2}Â = (ÎJ/Îa_{1})*cos*Î¸ =Â ** h**Âˇ

*n*When I say ‘simple’, I must add that all is relative, of course, Frankly, I myself did *notÂ *immediately understand why the *heat flow* through the Îa_{1}Â and Îa_{2}Â areas below must be the same. That’s why I added the **blue** square in the illustration above (which I took from Feynman’s *Lectures*): it’s the same area as Îa_{1}, but it shows more clearly â I hope! âÂ why the heat flow through the two areas is the same indeed, especially in light of the fact that we are looking at infinitesimally small areas here (so we’re taking a limit here).

As for the cosine factor in the formula above, you should note that, in thatÂ ÎJ/Îa_{2}Â = (ÎJ/Îa_{1})*cos*Î¸ =Â ** h**Âˇ

**equation, we have a**

*nÂ**dot*product (aka as a

*scalar*product)Â of

*two*vectors: (1)Â

**, the heat flow and (2)**

*h***, the unit vector that is normal (orthogonal) to the surface Îa**

*n*_{2}. So let meÂ remind you of the definition of theÂ

*scalar*(or

*dot*)Â product of two vectors. It yields a (real) number:

**A**Âˇ**B** =Â |**A**||**B**|*cos*Î¸, withÂ Î¸ the angle between **A** and **B**

In this case, * h*Âˇ

*Â =Â |*

**n***||*

**h****|**

*n**cos*Î¸ = |

*|Âˇ1Âˇ*

**h***cos*Î¸ = |

*|*

**h***cos*Î¸ =

*hÂˇcos*Î¸. What we are saying here is that

*we get the component of the heat flow that’s perpendicular*(or

*normal*, as physicists and mathematicians seem to prefer to say)

*to the surface*Îa

_{2}Â

*by taking the dot product of the heat flow*Â

*h**and the unit normal*Â

**. We’ll use this formula later, and so it’s good to take note of it here.**

*n*OK. Let’s get back to the lesson. The only thing that we need to do to prove thatÂ ÎJ/Îa_{2}Â = (ÎJ/Îa_{1})*cos*Î¸ formula is show that Îa_{2}Â = Îa_{1}/*cos*Î¸ or, what amounts to the same, that Îa_{1}Â = Îa_{2}*cos*Î¸. Now *that* is something you should be able to figure out yourself: it’s quite easy to show that the angle between ** h** and

**is equal to the angle between the surfaces Îa**

*n*_{1}Â and Îa

_{2}. The rest is just plain triangle trigonometry.

For example, when the surfaces coincide, the angleÂ Î¸ will be zero and thenÂ * h*Âˇ

*Â is just equal to |*

**n***|*

**h***cos*Î¸ = |

*| =*

**h***h*Âˇ1 =

*h*=Â ÎJ/Îa

_{1}. The other extreme is that orthogonal surfaces: in that case, the angleÂ Î¸ will be 90Â° and, hence,

*Âˇ*

**h***Â =Â |*

**n***||*

**h****|**

*n**cos*(Ď/2)Â = |

*|Âˇ1Âˇ0Â = 0: there is no heat flow normal to the direction of heat flow.*

**h**OK. That’s clear enough.Â The point to note is that the vectors ** h** and

**Â represent physical entities and, therefore, they do not depend on our reference frame (except for the units we use to measure things). That allows us to defineÂ**

*n**Â vector equations*.

**The â (del) operator and the gradient**

Let’s continue our example of temperature and heat flow. In a block of material, the temperature (T) will vary in the x, y and z direction and, hence, the partial derivatives âT/âx, âT/ây and âT/âz make sense: they measure how the temperature varies with respect to position. Now, the remarkable thing is that the 3-tuple (âT/âx, âT/ây, âT/âz) is a physical vector itself: it is independent, indeed, of the reference frame (provided we measure stuff in the same unit) â so we can do a translation and/or a rotation of the coordinate axes and we get the same value. This means this set of three numbers is aÂ *vectorÂ *indeed:

(âT/âx, âT/ây, âT/âz) = a vector

If you like to see a formal proof of this, I’ll refer you to Feynman once again â but I think the intuitive argument will do: if temperature and space are *real*, then the derivatives of temperature in regard to the x-, y- and z-directions should be equally real, isn’t it? Let’s go for the more intricate stuff now.

If we go from one point to another, in the x-, y- or z-direction, then we can define some (infinitesimally small)Â *displacement vector* Î**R** = (Îx, Îy, Îz), and theÂ *difference* in temperature between two nearby points (ÎT) will tend to the (total)* differential*Â of T â which we denote by ÎT â as the two point get closer and closer. Hence, we write:

ÎT = (âT/âx)Îx + (âT/ây)Îy + (âT/âz)Îz

The two equations aboveÂ combine to yield:

ÎT = (âT/âx, âT/ây, âT/âz)(Îx, Îy, Îz) = **â**TÂˇÎ**R**

In this equation, we used the **â **(*del*)Â operator, i.e. the *vectorÂ *differential operator. It’s an operator like the differential operator â/âx (i.e. the *derivative*) but, unlike the derivative, it returns not one but three values, i.e. a *vector*, which is usually referred to as the gradient, i.e.Â **â**T in this case. More in general, we can writeÂ **â***f*(x, y, z), **â**ĎÂ or **â **followed by whatever symbol for the function we’re looking at.

In other words, the **â **operator *acts on a scalar-valued function* (T), aka a *scalar field*, and yields a *vector*:

**â**T** = **(âT/âx, âT/ây, âT/âz)

That’s why we write **â**Â inÂ **bold-type**Â too, just like the vector **R**. Indeed, using bold-type (instead of an arrow or so) is a convenient way to mark a vector, and the difference (in print) betweenÂ **â** andÂ â is subtle, but it’s thereÂ â and for a good reason as you can see!

[To be precise, I should add that we do not write all of the operators that return three components in bold-type. The most simple example is the common derivativeÂ â**E**/ât = [âE_{x}/ât, âE_{y}/ât, âE_{z}/ât]. We have a lot of other possible combinations. Some make sense, and some don’t, like â* h*/ây =Â [â

*h*

_{x}/ây, â

*h*

_{y}/ây, â

*h*

_{z}/ây], for example.]

If **â**T is a vector, what’s its *direction*? Think about it. […] The *rate of change* of T in the x-, y- and z-direction are theÂ x-, y- and z-component of ourÂ **â**T vector respectively. In fact, the rate of change of T in *any* direction will be the component of theÂ **â**T vector *in that direction*. Now, the magnitude of a vector *component* will always be smaller than the magnitude of the vector itself,Â except if it’s the component in the same direction as the vector, in which case the component *isÂ *the vector. [If you have difficulty understanding this, read what I write once again, but *veryÂ slowly and attentively*.]Â Therefore, the direction of **â**T will be the direction in which the (instantaneous) rate of change of T is largest. In Feynman’s words: “**The gradient of T has the direction of the steepest uphill slope in T**.”Â Now, it should be quite obvious what direction that really is: it is theÂ *opposite *direction of the heat flow ** h**.

That’s all you need to know to understand our first realÂ ** vector equation**:

** h** = â

*Îş*

**â**T

Indeed, you don’t need too much math to understand this equation in the way we want to understand it, and that’s in some kind of ‘*physical*‘ way (as opposed to just the math side of it). Let me spell it out:

- The
*direction*of heat flow is opposite to the direction of the*gradient*vectorÂ**â**T. Hence, heat flows from*higher*to*lower*temperature (i.e. ‘downhill’), as we would expect, of course!). So that’s the minus sign. - The
*magnitude*ofis proportional to the magnitude of the gradientÂ*h***â**T, with the constant of proportionality equal to*Îş*Â (kappa), which is called the*thermal conductivity*. Now, in case you wonder what this means (again: do go beyond the math,*please!*), remember that the heat flow is the flow of thermal energy per unit area (and per unit time, of course): || =**h***h*=Â ÎJ/Îa.

*But…* Yes? *Why would it be proportional? Why don’t we have some exponential relation or something?Â *Good question, but the answer is simple, and it’s rooted in physical realityÂ â of course! The heat flow between two places â let’s call them 1 and 2 âÂ is proportional to the temperature *difference* between those two places, so we have: ÎJ âźÂ T_{2}Â âÂ T_{1}. In fact, that’s where the factor of proportionality comes in. If we imagine a very small slab of material (infinitesimally small, in fact) with two faces, parallel to the isothermals, with a surface area ÎA and a tiny distanceÂ Îs between them, we can write:

ÎJ =Â *Îş*(T_{2}Â âÂ T_{1})ÎA/Îs = ÎJ =Â *ÎşÂˇ*ÎTÂˇÎA/Îs â ÎJ/ÎA = *Îş*ÎT/Îs

Now, we definedÂ ÎJ/ÎA as the *magnitude* of ** h**. As for its direction, it’s obviously perpendicular (not parallel) to the isothermals. Now, as Îs tends to zero, ÎT/Îs is nothing but the

*rate of changeÂ*of T with position. We know it’s the

*maximum*rate of change, because the position change is also perpendicular to the isotherms (if the faces are parallel, that tiny distance Îs is perpendicular). Hence, ÎT/Îs

*must*be the

*magnitude*of the gradient vector (

**â**T). As its direction is opposite to that of

**, we can simply pop in a minus sign and switch from magnitudes to vectors to write what we wrote already:**

*h***= â**

*h**Îş*

**â**T.

But let’s get back to the lesson. I think you ‘get’ all of the above. In fact, I should probably *not* have introduced that extra equation above (the ÎJ expression) and all the extra stuff (i.e. the ‘infinitesimally small slab’ explanation), as it probably only confuses you. So…Â *What’s the point really?* Well… Let’s look, once again, at that equationÂ ** h** = â

*Îş*

**â**T and Â let us generalize the result:

- We have a
here,Â the temperature T â but it could be any scalar field really!*scalar field* - When we have the ‘formula’ for the scalar field â it’s obviously some function T(x, y, z) â we can derive theÂ heat flowÂ
Â from it, i.e. a vector quantity, which has a property which we can vaguely refer to as ‘flow’.*h* - We do so using this brand-new operator
**â**. That’s a so-called vector differential operator aka theÂ. We just apply it to the scalar field and we’re done! The only thing left is to add some proportionality factor, but so that’s just because of our units. [In case you wonder about the symbol it self, â is the so-called*del*Â operator*nabla*symbol: the name comes from the Hebrew word for a harp, which has a similar shape indeed.]*Â*

This truly *is* a most remarkable result, and we’ll encounter the same equation elsewhere. For example, if the electric potential isÂ ÎŚ, then we can immediately calculate the electric field using the following formula:

**E** =Â â**â**ÎŚ

Indeed, the situation is entirely analogous from a mathematical point of view. For example, we have the same minus sign, so **E** also ‘flows’ from higher to lower potential. Where’s the factor of proportionality? Well… We don’t have one, as we assume that the units in which we measure **E** and ÎŚ are ‘fully compatible’ (so don’t worry about them now). Of course, as mentioned above, this formula for **E**Â is only valid in electrostatics, i.e. when there are no *moving* charges. When moving charges are involved, we also have the magnetic force coming into play, and then equations become a bit more complicated. However, this extra complication does not fundamentally alter the logic involved, and I’ll come back to this so you see how it all nicely fits together.

**Note**: In case you feel I’ve skipped some of the ‘explanation’ of that vector equationÂ ** h** = â

*Îş*

**â**T… Well… You may be right.

*I*feel that it’s enough to simply point out thatÂ

**â**T is a vector with opposite direction to

**, so that explains the minus sign in front of the**

*h***â**T factor. The only thing left to ‘explain’ then is the

*magnitudeÂ*of

**, but so that’s why we pop in that**

*h**kappa*factor (

*Îş*), and so we’re done, I think, in terms of ‘understanding’ this equation. But so that’s whatÂ

*I*think indeed. Feynman offers a much more elaborate ‘explanation‘, and so you can check that out if you think my approach to it is a bit too much of a shortcut.

**Interim summary**

So far, we have only have shown two things really:

**[I]**Â The first thing to *alwaysÂ *remember is that ** h**Âˇ

**product: it gives us the component of ‘flow’ (per unit time and per unit area) of**

*n***perpendicular through**

*hÂ**any*surface element Îa. Of course, this result is valid for any other vector field, or any vector for that matter: the scalar product of a vector and a

*unit*vector will

*always*yieldÂ the component of that vector in the direction of that unit vector. [But note the second vector needs to be aÂ

*unitÂ*vector: it isÂ

*notÂ*generally true that the dot product of one vector with another yields the component of the first vector in the direction of the second: there’s a scale factor that comes into play.]

Now, you should note that the term ‘component’ (of a vector) usually refers to a *numberÂ *(not to a vector) âÂ and surely in this case,Â because we calculate it using a scalar product!Â I am just highlighting this because it did confuse *me* for quite a while. Why? Well… The concept of a ‘component’ of a vector is, obviously, intimately *associated* with the idea of ‘direction’: we always talk about the component *in some direction*, e.g. in the x-, y- or z-direction, or in the direction of any combination of x, y and z. Hence, I think it’s only natural to think of a ‘component’ as a vector in its own right. However, as I note here, we shouldn’t do that: a ‘component’ is just a *magnitude*, i.e. a number only. If we’d want to include the idea of direction, it’s simple: we can just multiply the component with the normal vectorÂ * n*Â once again, and then we have a

*vector*quantity once again, instead of just a scalar. So then we just write (

**Âˇ**

*h***)Âˇ**

*n***Â = (**

*n***Âˇ**

*h***)**

*n*

*n**.Â*Simple, isn’t it? đ

[As I am smiling here, I should quickly say something about this dot (Âˇ) symbol: we use the same symbol here for (i) a product between scalars (i.e. real or complex numbers), like 3Âˇ4; (ii) a product between a scalar and a vector, like 3Âˇ** vÂ **â but then I often omit the dot to simply write 3

**; and, finally, (iii) a scalar product of two vectors, like**

*v***Âˇ**

*h***indeed. We should, perhaps, introduce some new symbol for multiplying numbers, like â for example, but then I hope you’re smart enough to see from the context what’s going on really.]**

*nÂ*Back to the lesson. Let me jot down the formula once again: ** h**Âˇ

**Â = |**

*n**||*

**h****|**

*n**cos*Î¸ =Â

*hÂˇcos*Î¸. Hence, the

*number*we get here isÂ

*hÂ*(i.e. theÂ

*amountÂ*of heat flow in the

*direction*of flow) multiplied by

*cos*Î¸, with Î¸ the angle between (i) the surface we’re looking at (which, as mentioned above, is any surface really) and (ii) the surface that’s perpendicular to the direction of flow.

Hmm… […]Â The direction of flow? Let’s take a moment to think about what we’re saying here. Is there any particular or unique direction really? Heat flows in all directions from warmer to colder areas, and not just in one direction, doesn’t it? You’re right. Once again, the terminology may confuse you â which is yet another reason why math is so much better as a language to express physical ideas đ â and so we should be precise: the direction of ** h **is the direction in which the amount of heat flow (i.e.

*hÂˇcos*Î¸)Â is largest (hence, the angleÂ Î¸ is zero). As we pointed out above, that’s the direction in which the temperature T changes the fastest. In fact, as Feynman notes: “We can, if we wish, consider that this statement

*defines*

**.”**

*h*That brings me to the second thing you should âÂ *always*Â and* immediately*Â â remember from all of that I’ve written above.

**[II]**Â If we write the infinitesimal (i.e. the *differential*) change in temperature (in whatever direction) as ÎT, then we know that

ÎT = (âT/âx, âT/ây, âT/âz)(Îx, Îy, Îz) = **â**TÂˇÎ**R**

Now, what does this say really? Î**RÂ **is an (infinitesimal)Â *displacement *vector: Î**R** = (Îx, Îy, Îz). Hence, it has someÂ *direction*. To be clear: that can be *anyÂ *direction in space really. So that’s simple.Â What about the second factor in this dot product, i.e. that *gradient *vectorÂ **â**T?Â

The direction of the gradient (i.e.Â **â**T) is *notÂ *just ‘any direction’: **it’s theÂ direction in which the rate of change of T is largest**, and we know what direction that is: it’s the *opposite *direction of the heat flow ** h**, as evidenced by the minus sign in our vector equationsÂ

**= â**

*h**Îş*

**â**T or

**E**=Â â

**â**ÎŚ.Â So, once again, we have a (scalar) product of two vectors here,Â

**â**TÂˇÎ

**R**, which yields…

**Good question. ThatÂ**

*Hmm…Â***â**TÂˇÎ

**RÂ**expression is

*veryÂ*similar to that

**Âˇ**

*h***Â expression above, but it’s not quite the same. It’s also a vector dot product â or a scalar product, in other words, but, unlike that**

*n***vector, theÂ Î**

*n***R**vector is

*not*a unit vector: it’s an infinitesimally small displacement vector. So we do

*notÂ*get some ‘component’ of

**â**T.Â More in general, you should note that theÂ dot product of two vectors

**A**and

**B**doesÂ

*not*, in general, yield the component of

**A**in the direction of

**B**, unless

**B**is a unit vector â which, again, is

*not*the case here. So if we don’t have that here, what

*do*we have?

Let’sÂ look at the (physical) geometry of the situation once again. Heat obviously flows in one direction only: from warmer to colder places â *notÂ *in the opposite direction. Therefore, the Î¸ in the * h*Âˇ

*Â =Â*

**n***hÂˇcos*Î¸ expression varies from â90Â° to +90Â° only. Hence, the cosine factor (

*cos*Î¸) is always positive.

**Â Indeed, we do not have any right- or left-hand rule here to distinguish between the ‘front’ side and the ‘back’ side of our surface area. So when we’re looking at that**

*Always.***Âˇ**

*h***product, we should remember that that normal unit vector**

*n***is a unit vector that’s normal to the surface but which is oriented, generally,Â**

*n**towards*the direction of flow. Therefore, thatÂ

**Âˇ**

*h***product will**

*n**alwaysÂ*yield some positive value, because Î¸ varies from â90Â° to +90Â° only indeed.

When looking at that ÎT =Â **â**TÂˇÎ**R**Â product, the situation is quite different: while **â**T has a very *specific* direction (I really mean *unique*) Â âÂ which, as mentioned above is opposite to that ofÂ ** h**Â â thatÂ Î

**R**vector can point in

*any*direction â and then I mean literally

*anyÂ*direction,

*including directions ‘uphill’*.Â Likewise, it’s obvious that the temperature difference ÎT can be both positive orÂ negative (or zero, when we’re moving on a isotherm itself).Â In fact, it’s rather obvious that,

*if we go in the direction of flow*, we go from higher to lower temperatures and, hence, ÎT will, effectively, be negative: ÎT = T

_{2}Â âÂ T

_{1}Â < 0, as shown below.

Now, because |**â**T| andÂ |Î**R**| are absolute values (or magnitudes) of vectors, they are always positive (*always*!). Therefore, ifÂ ÎT has a minus sign, it will have to come from the cosine factor in the ÎT =Â **â**TÂˇÎ**R**Â = |**â**T|Âˇ|Î**R**|Âˇ*cos*Î¸ expression. [Again, if you wonder where this expression comes from: it’s just the definition of a vector dot product.]Â Therefore,Â ÎT and *cos*Î¸ will have the same sign, *always*, and Î¸ can have any value between â180Â° to +180Â°. In other words, we’re effectively looking at the full circle here. To make a long story short, we can write the following:

ÎT = |**â**T|Âˇ|Î**R**|Âˇ*cos*Î¸ = |**â**T|ÂˇÎRÂˇ*cos*Î¸ â ÎT/ÎR = |**â**T|*cos*Î¸

As you can see, Î¸ is the angle between **â**T and Î**R **here and, as mentioned above, it can take on any value â well… Any value between â180Â° to +180Â°, that is. ÎT/ÎR is, obviously, theÂ rate of change of T in the direction of Î**R**Â and, from the expression above, we can see it is equal to the component of **â**T in the direction of Î**R**:

ÎT/ÎR = |**â**T|*cos*Î¸

So we have a *negativeÂ *component here? Yes. The rate of change is negative and, therefore, we have a negative component. Indeed, any vector has components inÂ *allÂ *directions, including directions that point *away* from it. However, in the directions that point away from it, the component will be *negative*. More in general, we have the following interesting result: **the rate of change of a scalar fieldÂ Ď in the direction of a (small) displacement ÎRÂ is the component of the gradient âĎ along that displacement**. We write that result as:

ÎĎ/ÎR = |**â**Ď|*cos*Î¸

[Note the (not so) subtle difference betweenÂ Î**R **(i.e. a *vector*) andÂ ÎR (some real number). It’s quite important. ]

We’ve said a lot of (not so) interesting things here, but we still haven’t answered the original question: what’s **â**TÂˇÎ**R**? Well… We can’t say more than what we said already: it’s equal to ÎT, which is a *differential*: ÎT = (âT/âx)Îx + (âT/ây)Îy + (âT/âz)Îz. A differential and a derivative (i.e. a *rate of change*) are not the same, but they are obviously closely related, as evidenced from the equations above: the *rate* of change is the change *per unit distance*. [Likewise, note thatÂ |**â**Ď|*cos*Î¸ is just a product of two real numbers, whileÂ **â**TÂˇÎ**RÂ **is a vector dot product, i.e. a (scalar) product of two vectors!]

In any case, this is enough of a recapitulation. In fact, this ‘interim summary’ has become longer than the preceding text! We’re now ready to discuss what I’ll call the First Theorem of vector calculus in physics. Of course, never mind the term: what’s first or second or third doesn’t matter really: you’ll needÂ *allÂ *of the theorems below to understand vector calculus.

**The First Theorem**

Let’s assume we have some scalar field Ď in space: Ď might be the temperature, but it could be any scalar field really. Now, if we go from one point (1) to another (2) in space, as shown below, we’ll follow some arbitrary path, which is denoted by the curveÂ Î in the illustrations below. Each point along the curve can then be associated with a *gradient*Â **â**Ď (think of the ** h** = â

*Îş*

**â**T and

**E**=Â â

**â**ÎŚ expressions above if you’d want examples). Its tangential component is obviously equal to (

**â**Ď)

*ÂˇÎs*

_{t}_{Â }=

**â**ĎÂˇÎ

**s**.Â [Please note, once again, the subtle difference between Î

**s**

*(with the*

_{Â }**s**in bold-face) and Îs: Î

**s**

*is a vector, and Îs*

_{Â }*is its magnitude.]*

_{Â }_{Â }

As shown in the illustrations above, we can mark off the curve at a number of points (a, b, c, etcetera) and join these points by straight-line segments Î**s*** _{i}*. Now let’sÂ consider the first line segment, i.e. Î

**s**

_{1}. It’s obvious that the

*changeÂ*in Ď from point 1 to point a is equal to ÎĎ

_{1Â }= Ď(a) â Ď(1). Now, we have that generalÂ ÎĎ = (âĎ/âx, âĎ/ây, âĎ/âz)(Îx, Îy, Îz) =

**â**ĎÂˇÎ

**s**equation. [If you find it difficult to interpret what I am writing here, just substitute ĎÂ for T and Î

**s**Â forÂ Î

**R**.] So we can write:

ÎĎ_{1Â }= Ď(a) â Ď(1) = (**â**Ď)_{1}ÂˇÎ**s**_{1}

Likewise, we can write:

Ď(b) â Ď(a) = (**â**Ď)_{2}ÂˇÎ**s**_{1}

In these expressions, (**â**Ď)_{1Â }and (**â**Ď)_{2Â }mean the gradient evaluated at segment Î**s**_{1}Â and pointÂ Î**s**_{2}Â respectively, *notÂ *at point 1 and 2 â obviously. Now, if we add the two equations above, we get:

Ď(b) â Ď(1) = (**â**Ď)_{1}ÂˇÎ**s**_{1Â }+ (**â**Ď)_{2}ÂˇÎ**s**_{1}

To make a long story short, we can keep adding such terms to get:

Ď(2) â Ď(1) = â(**â**Ď)* _{i}*ÂˇÎ

**s**

_{i}

We can add more and more segments and, hence, take a limit: as Îs* _{i}*Â tends to zero, â becomes a sum of an infinite number of terms â which we denote using the integral sign âŤ â in which d

**s**isÂ â quite simply âÂ just the infinitesimally smallÂ

*displacement*Â vector. In other words,Â we get the following

*line integral*Â along that curveÂ Î:

*Â*

This is a gem, and our First Theorem indeed. It’s a remarkable result, especially taking into account the fact that the path doesn’t matter: we could have chosen any curveÂ Î indeed, and the result would be the same. So we have:

You’ll say: so what? What do we do with this? Well… Nothing much for the moment, but we’ll need this result later. So I’d say: just hang in there, and note this is the first significant use of our del operator in a mathematical expression that you’ll encounterÂ *veryÂ *often in physics. So just let it sink in, and allow me to proceed with the rest of the story.

Before doing so, however, I should note that even Feynman sins when trying to explain this theorem in a more ‘intuitive’ way. Indeed, in his *LectureÂ *on the topic, he writes the following: “Since the gradient represents the rate of change of a field quantity, if we integrate that rate of change, we should get the total change.” Now, from that ÎĎ/ÎR = |**â**Ď|*cos*Î¸ formula, it’s obvious that the gradient is the rate of change in a specific direction only. To be precise, in *this* particular caseÂ â with the field quantityÂ Ď equal to the temperature T â it’s the direction in which T changes the fastest.

You should also note that the integral above is *not *the type of integral you known from high school. Indeed, it’sÂ *not *ofÂ the rather straightforwardÂ âŤf(x)dx type, with f(x) the integrand and dx the variable of integration. *ThatÂ *type of integral, we knew how to solve. A line integral is quite different.Â Look at it carefully: we have a vector dot product after the âŤ sign. So, unlike what Feynman suggests, it’s *notÂ *justÂ a matter of “integrating the rate of change.”

Now, I’ll refer you to Wikipedia for a good discussion of what a *line* integral really is, but I can’t resist the temptation to copy the animation in that article, because it’s very well made indeed. While it shows that we can think of a line integral as the two- or three-dimensional equivalent of the standard type of integral we learned to solve in high school (you’ll remember the solution was also the area under the graph of the function that had to be integrated), the way to go about it is quite different. Solving them will, in general, involve some so-called parametrization of the curve C.

However, this post is becoming way too long and, hence, I really need to move on now.

**Operations withÂ â: Â divergence and curl**

You may think we’ve covered a lot of ground already, and we did. At the same time, everything I wrote above is actually just the start of it. I emphasized the *physics* of the situation so far. Let me now turn to the math involved. Let’s start by dissociating the del operator from the scalar field, so we just write:

**â****Â = **(â/âx, â/ây, â/âz)

This doesn’tÂ *meanÂ *anything, you’ll say, because the operator has nothing to operate on. And, yes, you’re right. However, in math, it doesn’t matter: we can *combine* this ‘meaningless’ operator (which looks *like* a vector, because it has three components)Â *with something else*. For example, we can do a vector dot product:

**â**Âˇ(a vector)

As mentioned above, we can ‘do’ this product because **â** has three components, so it’s a ‘vector’ too (although I find such name-giving quite confusing), and so we just need to make sure that the vector we’re operating on has three components too. To continue with our heat flow example, we can write, for example:

**â**Âˇ** h **=Â (â/âx, â/ây, â/âz)Âˇ(

*h*

_{x},Â

*h*

_{y},Â

*h*

_{z}) =Â â

*h*

_{x}/âx + â

*h*

_{y}/ây, â

*h*

_{z}/âz

This del operator followed by a dot, and acting on a vectorÂ â i.e.Â **â**Âˇ(*vector*) â is, in fact, aÂ *newÂ *operator. Note that we use *two*Â existing symbols, the del (**â**) and the dot (Âˇ), but it’s one operator really. [Inventing a new symbol for it would not be wise, because we’d forget where it comes from and, hence, probably scratch our head when we’d see it.] It’s referred to as a *vector* operator, just like the del operator, but don’t worry about the terminology here because, once again, the terminology here might confuse you. Indeed, our del operator acted on a *scalar* to yield a *vector*, and now it’s the other way around: we have an operator acting on a *vector* to return a *scalar*. In a few minutes, we’ll define yet another operator acting on a *vector* to return a *vector*. Now, all of these operators are so-called vector operators, *notÂ *because there’s some vector involved, butÂ because they all involve the del operator. It’s that simple. So there’s no such thing as a scalar operator. đ But let me get back to the main line of the story. ThisÂ **â**Âˇ Â operator is quite important in physics, and so it has a name (and an abbreviated notation) of its own:

**â**Âˇ** h **= div

**= the**

*h**divergence*of

*h*The physical significance of the divergence is related to the so-called *fluxÂ *of a vector field: itÂ measures the magnitude of a field’s *sourceÂ *orÂ *sinkÂ *at a given point. Continuing our example with temperature, consider *air* as it is heated or cooled. The relevant vector field is now the *velocityÂ *of theÂ moving air at a point. If air is heated in a particular region, it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a *source*. If the air cools and contracts, the divergence has a negative value, as the region is a *sink*.

A less intuitive but more accurate definition is the following: **the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point**.

*Phew!* That sounds more serious, doesn’t it? We’ll come back to this definition when we’re ready to define the concept of flux somewhat accurately. For now, just note two of Maxwell’s famous equations involve the divergence operator:

**â**ÂˇE = Ď/Îľ_{0}Â andÂ **â**ÂˇB = 0

In my previous post, I gave a verbal description of those two equations:

- The flux of
**E**Â through a*closed*Â surface = (the net charge inside)/Îľ_{0} - The flux of
**B**Â through a*closedÂ*surface = zero

The first equation basically says that electric charges cause an electric field. The second equation basically says there is no such thing as aÂ *magneticÂ *charge: the magnetic force only appears when charges areÂ *moving*Â and/or when electric fields areÂ *changing*. Note that we’re talking closed surface here, so they define a volume indeed. We can also look at the flux through a non-closed surface (and we’ll do that shortly) but, in the context of Maxwell’s equations, we’re talking volumes and, hence, closed surfaces.

Let me quickly throw in some remarks on the units in which we measure stuff. Electric field strength (so the unit we use to measure the magnitude of **E**) is measured in Newton per Coulomb, so force divided by charge. That makes sense, because **E **is defined as the force on the unit charge: **E** = **F**/q, and so the unit is N/C. Please do think about why we have q in the denominator: if we’d have the same force on an electric charge that isÂ *twiceÂ *as big, then we’d have a field strength that’s only half, so we have an *inverse* proportionality here. Conversely, if we’d have twice the force on the same electric charge, the field strength would also double.

Now, flux and field strength are obviously related, but not the same. The flux is obviously proportional to the field strength (expressed in N/C), but then we also know it’s some number expressed *per unit area*. Hence, you might think that the unit of flux is field strength *per square meter*, i.e. N/C/m^{2}. It’s not. It’s a stupid mistake, but one that is commonly made. Flux is expressed in N/C *timesÂ *m^{2}, so that’s the *product* (N/C)Âˇm^{2Â }= (NÂˇm/C)Âˇm^{Â }= (J/C)Âˇm.Â Why is that? Think about the common graphical representation of a field: we just draw lines, all tangent to the direction of the field vector at every point, and the *density* of the lines (i.e.Â the number of lines *per unit area*) represents the magnitude of our electric field vector. Now, the flux through some area is the number of lines we count in that area. Hence, if you double the area, you should get twice the flux. Halve the area, and you should get half the flux. So we have aÂ *directÂ *proportionality here. In fact, assuming the electric field is uniform, we can write the (electric) flux as the product of the field strength **E** and the (vector) area **S**, so we write ÎŚ_{E}Â = **E**Âˇ**S** = EÂˇSÂˇcosÎ¸.

*Huh? *Yes. The origin of the mistake is that we, somehow, think the ‘per unit area’ qualification comes with the flux. It doesn’t: it’s in the idea of field strength itself. Indeed, an alternative to the presentation above is just to draw arrows representing the same field strength, as illustrated below. However, instead of drawingÂ *moreÂ *arrows (of some standard length) to represent increasing field strength, we’d just drawÂ *longerÂ *arrowsânot *more* of them. So then the idea of the number of lines per unit area is no longer valid.

[…] OK. I realize I am probably just confusing you here. Just one more thing, perhaps. We also have magnetic flux, denoted as ÎŚ_{B}, and it’s defined in the same way: ÎŚ_{B}Â = **B**Âˇ**S** = BÂˇSÂˇcosÎ¸. However, because the unit of *magneticÂ *field strength is different, the unit of magnetic flux is different too. It’s theÂ *weber*, and I’ll let you look up its definition yourself. đ Note that it’s a bit of a different beast, because the magnetic force is a bit of a different beast. đ

So let’sÂ get back to our operators.Â You’ll anticipate the second new operator now, because that’s the one that appears in theÂ *otherÂ *two equations in Maxwell’s *set* of equations. It’s the cross product:

**âĂEÂ **= (â/âx, â/ây, â/âz)Ă(E_{x}, E_{y}, E_{z}) = … *What?*

Well… The cross product is not as straightforward to write down as the dot product. We get a *vector* indeed, not a scalar, and its three components are:

(**âĂE**)_{z}Â = â_{x}E_{yÂ }â â_{y}E_{xÂ }= âE_{y}/âx â âE_{x}/ây

(**âĂE**)_{x}Â = â_{y}E_{zÂ }â â_{z}E_{yÂ }= âE_{z}/ây â âE_{y}/âz

(**âĂE**)_{y}Â = â_{z}E_{xÂ }â â_{x}E_{zÂ }= âE_{x}/âz â âE_{z}/âx

I know this looks pretty monstrous, but so that’s how cross products work. Please do check it out: you have to play with the order of the x, y and z *subscripts*. I gave the geometric formula for a dot product above, so I should also give you the same for a cross product:

**A**Ă**B** =Â |**A**||**B**|*sin*(Î¸)**n**

In this formula, we once again have Î¸, the angle between **A** and **B**, but note that, this time around, it’s the sine, not the cosine, that pops up when calculating the magnitude of this vector. In addition, we have ** n** at the end:

**is a unit vectorÂ at right angles to both**

*n***A**and

**B**. It’s what makes the cross product a vector. Indeed, as you can see, multiplying by n will not alter the magnitude (|

**A**||

**B**|

*sin*Î¸) of this product, but it gives the whole thing a

*direction*, so we get a new vector indeed.Â Of course, we have

*twoÂ*unit vectors at right angles to

**A**and

**B**, and so we need a rule to choose one of these: theÂ direction of the

**vector we want is given by that right-hand rule which we encountered a couple of times already.**

*n*Again, it’s two symbols but one operator really, and we also have a special name (and notation) for it:

**âĂ**** h **= curl

**= the**

*h**curl*of

*h*The **curl** is, just like the divergence, a so-called vector operator but, as mentioned above, that’s just because it involves the del operator. Just note that it acts on a vector and that its result is a vector too. What’s the geometric interpretation of the curl? Well… It’s a bit hard to describe that but let’s try. The curl describes the ‘rotation’ or ‘circulation’Â of a vector field:

- The direction of the curl is the axis of rotation, as determined by the right-hand rule.
- The magnitude of the curl is the magnitude of rotation.

I know. This is pretty abstract, and I’ll probably have to come back to it in another post. Let’s first ask some basic question: should we associate some unit with the curl? In fact, when youÂ *google*, you’ll find lots of units used in electromagnetic theory (like the *weber*, for example), but nothing for circulation. I am not sure why, because if flux is related to some density, the idea of curl (or circulation) is pretty much the same. It’s just that it isn’t used much in actual engineering problems, and surely not those you may have encountered in your high school physics course!

In any case, just note we defined *threeÂ *new operators in this ‘introduction’ to vector calculus:

**â**T**=**gradÂ**T**= a vector**âÂˇ**= div*h*= a scalar*h***â**Ă= curl*h*= a vector*h*

That’s all. It’s all we need to understand Maxwell’s famous equations:

** Huh? **Hmm… You’re right: understanding the symbols, to some extent, doesn’t mean we ‘understand’ these equations. What does it mean to ‘understand’ an equation? Let me quote Feynman on that: “What it means really to understand an equationâthat is, in more than a strictly mathematical senseâwas described by Dirac. He said: âI understand what an equation means if I have a way of figuring out the characteristics of its solution without actually solving it.â So if we have a way of knowing what should happen in given circumstances without actually solving the equations, then we âunderstandâ the equations, as applied to these circumstances.”

We’re surely not there yet. In fact, I doubt we’ll ever reach Dirac’s understanding of Maxwell’s equations. But let’s do what we can.

In order to ‘understand’ the equations above in a more ‘physical’ way, let’s explore the concepts of divergence and curl somewhat more. We said the divergence was related to the ‘flux’ of a vector field, and the curl was related to its ‘circulation’. In my previous post, I had already illustrated those two concepts copying the following diagrams from Feynman’s *Lectures*:

flux = (average normal component)Âˇ(surface area)

So that’s the flux (through a non-closed surface).

To illustrate the concept of circulation, we have not one but three diagrams, shown below. Diagram (a) gives us the vector field, such as the velocity field in a liquid. In diagram (b), we imagine a tube (of uniform cross section) that follows some arbitrary closed curve. Finally, in diagram (c), we imagine weâd suddenly freeze the liquid everywhere except inside the tube. Then the liquid in the tube would circulate as shown inÂ (c), and so that’s the concept of circulation.

We have a similar formula as for the flux:

circulation = (the average tangential component)Âˇ(the distance around)

In both formulas (flux and circulation), we have a product of two scalars: (i) the average normal component and the average tangential component (for the flux and circulation respectively) and (ii) the surface area and the distance around (again, for the flux and circulation respectively). So we get a scalar as a result. Does that make sense? When we related the concept of flux to the divergence of a vector field, we said that the flux would have a positive value if the region is a *source*, and a negative value if the region is a *sink*. So we have a number here (otherwise we wouldn’t be talking ‘positive’ or ‘negative’ values). So that’s OK. But are we talking about the same number? Yes. I’ll show they are the same in a few minutes.

But what about circulation? When we related the concept of circulation of the curl of a vector field, we introduced a vectorÂ cross product, so that yields a vector, not a scalar. So what’s the relation between that vector and the number we get when multiplying the ‘average tangential component’ and the ‘distance around’. The answer requires some more mathematical analysis, and I’ll give you what you need in a minute. Let me first make a remark about conventions here.

From what I write above, you see that we use a plus or minus sign for the flux to indicate the *directionÂ *of flow:Â the flux has a positive value if the region is a *source*, and a negative value if the region is a *sink*. Now, why don’t we do the same for circulation? We said the curl is a vector, and its direction isÂ the *axis* of rotation as determined by the right-hand rule. Why do we need a vector here? Why can’t we have a scalar taking on positive or negative values, just like we do for the flux?

The intuitive answer to this question (i.e. the ‘non-mathematical’ or ‘physical’ explanation, I’d say) is the following. Although we can calculate the flux through aÂ *non-closedÂ *surface, from a mathematical point of view, flux is effectively being defined by referring to the infinitesimalÂ *volume* around some point and, therefore, we can easily, andÂ *unambiguously*, determine whether we’re inside *or* outside of that volume. Therefore, the concepts of positive and negative values make sense, as we can define them referring to some *unique* reference point, which is *either* inside *or* outside of the region.

When talking circulation, however, we’re talking about some *curve* in space. Now it’s *notÂ *so easy to find some unique reference point. *We* may say that *weÂ *are looking at some curve from some point ‘in front of’ that curve, but some other person whose position, from our point of view, would be ‘behind’ the curve, would not agree with our definition of ‘in front of’: in fact, his definition would be exactly the opposite of ours. In short, because of the geometry of the situation involved, our convention in regard to the ‘sign’ of circulation (positive or negative) becomes somewhat more complicated. It’s no longer a simple matter of ‘inward’ or ‘outward’ flow: we need something like a ‘right-hand rule’ indeed. [We could, of course, also adopt a left-hand rule but, byÂ now, you know that, in physics, there’s not much use for a left hand. :-)]

That also ‘explains’ why the vector cross product is non-commutative: **A**Ă**B** â **B**Ă**A**. To be fully precise, **A**Ă**B**Â andÂ **B**Ă**AÂ **have the same magnitude but opposite direction: **A**Ă**B** =Â |**A**||**B**|*sin*(Î¸)* n* = â|

**A**||

**B**|

*sin*(Î¸)(â

*)*

**n***= â*(

**B**Ă

**A**) =Â

*â*

**B**Ă

**A**. The dot product, on the other hand, is fully commutative:

**A**Âˇ

**B**Â =

**B**Âˇ

**A**.

In fact, the concept of circulation is very much related to the concept of angular momentum which, as you’ll remember from a previous post, also involves a vector cross product.

[…]

I’ve confused you too much already. The only way out is the full mathematical treatment. So let’s go for that.

**Flux**

Some of the confusion as to what flux actually means in electromagnetism is probably caused by the fact that the illustration above is *notÂ *a closed surface and, from my previous post, you should remember that Maxwell’s first and third equation define the flux of **E** and **B** through *closed* surfaces. It’s not that the formula above for the flux through a *non*-closed surface is wrong: it’s just that, in electromagnetism, we usually talk about the flux through a closed surface.

A closed surface has no boundary. In contrast, the surface *areaÂ *above does have a clear boundary and, hence, it’s *notÂ *a closed surface. A sphere is an example of a closed surface. A cube is an example as well. In fact, an infinitesimally small cube is what’s used to prove a very convenient theorem, referred to as *Gauss’ Theorem*. We will not prove it here, but just try to make sure you ‘understand’ what it says.

Suppose we have some vector field ** C**Â and that we have some closed surface S â a sphere, for example, but it may also be some very irregular volume. Its shape doesn’t matter: the only requirement is that it’s defined by a closed surface. Let’s then denote the

*volumeÂ*that’s enclosed by this surface by V. Now, the flux throughÂ someÂ (infinitesimal) surface element d

*a*Â will, effectively, be given by that formula above:

flux = (average normal component)Âˇ(surface area)

What’s the average normal component in this case? It’s given by that ÎJ/Îa_{2}Â = (ÎJ/Îa_{1})*cos*Î¸ =Â ** h**Âˇ

**formula, except that we just need to substitute**

*nÂ***for**

*h***here, so we have**

*C**Âˇ*

**C****Â instead of**

*n***Âˇ**

*h***. To get the flux through the**

*n**closed*surface S, we just need to add all the contributions. Adding those contributions amounts to taking the followingÂ

*surface integral*:

Now, I talked about Gauss’ Theorem above, and I said I would not prove it, but this is what **Gauss’ Theorem** says:

* Huh?Â *Don’t panic. Just try to ‘read’ what’s written here. From all that I’ve said so far, you should ‘understand’ the surface integral on the left-hand side. So that should be OK. Let’s now look at the right-hand side. The right-hand side uses theÂ

*divergenceÂ*operator which I introduced above:Â

**â**Âˇ(vector). In this case,

**â**Âˇ

**. That’s a scalar, as we know, and it represents the outward flux from an infinitesimally small cube inside the surface indeed. The**

*C**volumeÂ*

*integral*on the right-hand side adds all of the fluxes out of each part (think of it as zillions of infinitesimally small cubes) of the volume V that is enclosed by the (closed) surface S. So that’s what Gauss’ Theorem is all about. In words, we can state Gauss’ Theorem as follows:

**Gauss’ Theorem**: *The (surface) integral of the normal component of a vector (field) over a closed surface is the (volume) integral of the divergence of the vector over the volume enclosed by the surface.*

Again, I said I would not prove Gauss’ Theorem, but its proof is actually quite intuitive: to calculate the flux out of a large volume, we can ‘cut it up’ in smaller volumes, and then calculate the flux out of these volumes. If we add it up, we’ll get the total flux. In any case, I’ll refer you to Feynman in case you’d want to see how it goesÂ *exactly*. So far, IÂ did what I promised to do, and that’s to relate the formula for flux (i.e. thatÂ (average normal component)Âˇ(surface area) formula) to the divergence operator. Let’s now do the same for the curl.

**Curl**

For non-native English speakers (like me), it’s always good to have a look at the common-sense definition of ‘curl’: as a verb (to curl), it means ‘to form or cause to form into a curved or spiral shape’. As a noun (e.g. a curl of hair), it means ‘something having a spiral or inwardly curved form’. It’s clear that, while not the same, we can indeed relate this common-sense definition to the concept of circulation that we introduced above:

circulation = (the average tangential component)Âˇ(the distance around)

So that’s the (*scalar*) product we already mentioned above. How do we relate it to that curl operator?

*Patience, please !*Â The illustration below shows what we *actuallyÂ *have toÂ do to calculate the circulation around some loopÂ Î: we take an infinite number of vectorÂ *dot* products ** C**Âˇd

*. Take a careful look at the notation here: I use bold-face for*

**s****and, hence, d**

*s***is some little vector indeed. Going to the limit, d**

*s***Â becomes a differential indeed. The fact that we’re talking a vector dot product here ensures that only theÂ**

*s**tangentialÂ*component of

*C**‘*enters the equation’, so to speak. I’ll come back to that in a moment. Just have a good look at the illustration first.

Such infinite sum ofÂ vectorÂ *dot* products ** C**Âˇd

*is, once again, an integral. It’s another ‘special’ integral, in fact. To be precise, it’s a*

**sÂ***line integral*. Moreover, it’s not just any line integral: we have to go all around the (closed) loop to take it. We cannot stop somewhere halfway. That’s why Feynman writes it with a little loop (Îż) through the integral sign (âŤ):

Note the subtle difference between the two products in the integrands of the integrals above: *C*_{t}ds versus ** C**Âˇd

**. The first product is just a product of two scalars, while the second is a dot product of two vectors. Just check it out using the definition of a dot product (**

*s***A**Âˇ

**B**=Â |

**A**||

**B**|

*cos*Î¸) and substitute

**A**and

**B**by

**and d**

*C***respectively, noting that the tangential componentÂ**

*s**C*

_{t}Â equalsÂ

*C*timesÂ cosÎ¸ indeed.

Now, once again, we want to relate this integral with that dot product inside to one of those vector operators we introduced above. In this case, we’ll relate the circulation with the *curlÂ *operator. The analysis involves infinitesimal squares (as opposed to those infinitesimal cubes we introduced above), and the result is what is referred to as **Stokes’ Theorem**. I’ll just write it down:

Again, the integral on the left was explained above: it’s a line integral taking around the full loop Î. As for the integral on the right-hand side, that’s a *surface *integral once again but, instead of a *div* operator, we have the *curl* operator inside and, moreover, the integrand is the normal component of the curl only. Now, remembering that we can always find the normal component of a vector (i.e. the component that’s normal to the surface) by taking the dot product of that vector and the unit normal vector (** n**), we can write Stokes’s Theorem also as:

That doesn’t look any ‘nicer’, but it’s the form in which you’ll usually see it. Once again, I willÂ *not *give you any formal proof of this. Indeed, if you’d want to see how it goes, I’ll just refer you to Feynman’s* Lectures*. However, the philosophy behind is the same. The first step is to prove that we can break up the surface bounded byÂ the loop Î into a number of smaller areas, and that the circulation around Î will be equal to the sum of the circulations around the little loops. The idea is illustrated below:

Of course, we then go to the limit and cut up the surface into an infinite number of infinitesimally small squares. The next step in the proof then shows that the circulation of ** CÂ **around an infinitesimal square is, indeed, (i) the component of the curl of

**normal to the surface enclosed by that square multiplied by (ii) the area of that (infinitesimal) square. The diagram and formula below do not give you the proof but just illustrate the idea:**

*C*OK, you’ll say, ** so what?** Well… Nothing much. I think you have enough to digest as for now. It probably looks very daunting, but so that’s all we need to knowÂ â for the moment that isÂ â to arrive at a better ‘physical’ understanding of Maxwell’s famous equations. I’ll come back to them in my next post. Before proceeding to the summary of this whole post, let me just write down Stokes’ Theorem in words:

**Stokes’ Theorem**:Â *The line integral of the tangential component of a vector (field) around a closed loop is equal to the surface integral of the normal component of the curl of that vector over any surface which is bounded by the loop.*

**Summary**

We’ve defined three so-called vector operators, which we’ll use *veryÂ *often in physics:

**â**T**=**gradÂ**T**= a vector**âÂˇ**= div*h*= a scalar*h***â**Ă= curl*h*= a vector*h*

Moreover, we also explained three important theorems, which we’ll use as least as much:

**[1] The First Theorem**:

**[2] Gauss Theorem**:

**[3] Stokes Theorem:**

As said, we’ll come back to them in my next post. As for now, just try to familiarize yourself with these *div* and *curl* operators. Try to ‘understand’ them as good as you can. Don’t look at them as just some weird mathematical definition: try to understand them in a ‘physical’ way, i.e. in a ‘completely unmathematical, imprecise, and inexact way’, remembering that’s what it takes to understand to truly understand physics. đ

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## 6 thoughts on “Applied vector analysis (I)”