The relationship between math and physics is deep. When studying physics, one sometimes feels physics and math become one and the same. However, they are *not *the same. In fact, physicists warn against emphasizing the math side of physics too much. To paraphrase one of the greatest post–World War II scientists, Richard Feynman: “It is *not *because you understand the Maxwell equations mathematically inside out, that you understand physics inside out.”

Indeed, while vector equations and fields and all those other mathematical constructs do represent physical realities, one needs to develop a ‘physical’ – as opposed to a ‘mathematical’ – understanding of the equations. Now you’ll ask: what’s a ‘physical’ understanding? Well… Let me quote Feynman once again on that: “A physical understanding is a completely unmathematical, imprecise, and inexact thing, but absolutely necessary for a physicist.”

That’s surprising, isn’t it? Especially taking into account Feynman’s eminence. Judging from what he writes in his *Lectures,* Feynman doesn’t like philosophers, and he’d surely say that there’s no need for *meta*physics (i.e. the branch of philosophy that deals with first principles). That being said, I think that’s a matter of definition and interpretation. If metaphysics is defined as the branch of philosophy that deals with first principles, I’d say it’s rather obvious that physics is also based on some kind of ‘metaphysical’ model, and I wouldn’t hesitate to equate it with the Standard Model ! Indeed, from what I’ve learned so far, quantum mechanics has a lot in common with Pythagoras’ belief that mathematical concepts – and numbers in particular – might have greater ‘actuality’ than the reality they are supposed to describe.

But 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 get back to basics.

**Vectors in math and physics**

It may surprise you, but the term ‘vector’, in physics and in math, refers to more than a dozen *different *concepts. Just check it out on Wikipedia, if you don’t believe me. In fact, as an autodidact, I’d say this is probably one of the key sources of confusion for people like us. A vector means many things indeed. The most common definitions are:

- Some one-dimensional array of numbers (i.e. as an element of
**R**^{n}), or of anything really: numerical or alphanumerical values, blob files,… Whatever ! - A vector can also be a
*point*vector. In that case, it represents the*position*of a point in space in two, three or even four dimensions (if we include time), in relation to some*arbitrarily chosen*origin (i.e. the zero point). - A vector can also be a displacement vector: in that case, it will specify the
*change*in position of a point relative to its previous position. Again, such displacement vectors may be two-, three- or four-dimensional. - A vector may also refer to a so-called four-vector: a
*four-vector*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). But there’s many more four-vectors, one of them being**P**= (E/c,**p**), which relates energy and momentum in*spacetime*. - We also have vector
*operators*, like the gradient vector**∇**, and that’s what I want to write about in this post.

**Vectors in physics**

In physics, the term ‘vector’ may refer to any of the above but, usually, it will mean yet another thing: a *field *vector. Now, funnily enough, the term ‘field vector’, while being the most obvious description of what it is, is not widely used: what I call a ‘field vector’ is usually referred to as a *gradient*, and the vectors **E **and **B **are usually referred to as the electric or magnetic *field*. Period.

**E** and **B **behave like ** h**, which is the symbol used for the heat flow in some body or block of material: they are

*vector fields derived from a scalar field.*

* Huh? Scalar field? *I thought we were talking vectors. We are. And I will also need to qualify the statement above: the relation between A (the magnetic potential – i.e. the ‘magnetic scalar field’) and Φ (the electrostatic potential, or ‘electric scalar field’) and

**B**and

**E**and the magnetic and electrostatic potential A and Φ is a bit more complicated than the relationship between temperature (T) – i.e. the scalar field determining the heat flow – and

**, but the math that is involved is the same. But I am getting ahead of myself. Let’s look at**

*h***and T**

*h**only*for the moment.

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 but – whatever unit we use – *any observer using the same unit will measure the same at any given point*. That’s what distinguishes a ‘scalar’ from a ‘real number': a scalar field is something *real. *

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, that is]. 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 that *any observer using the same unit will measure the same at any given point*.

**Basics**

In physics, we can 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 view but, because they do represent a physical reality, these ‘numbers’ respect certain mathematical conditions: in practice, they will be a continuous or continuously differentiable*function*of position.*Heat flow*(), the*h**velocity*() 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:*any observer using the same unit should measure the same 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 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 say anything about them, and so I won’t.

Back to basics indeed. How do we derive a vector field from a scalar field? Let’s study temperature and heat flow. The two illustrations below (taken from Feynman’s *Lectures*) illustrate the ‘mechanics’ behind it: the (magnitude of the)* heat*

**(which is, obviously, from the hotter to the colder places).**

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

** h** =

*h*·

*e**= (ΔJ/Δa)·*

_{f}

*e*

_{f}ΔJ stands for the thermal energy flowing through an area marked as Δa in the diagram above. It is measured per unit time, obviously. Hence, the heat flow is the flow of thermal energy *per unit area*.

Using simple trigonometry (but all is relative: it took me a while to figure out that the heat flow through the Δa_{1} and Δa_{2} areas below, are, in effect, the same :-( and then I also needed some time to figure out the cosine factor in the formula below) yield an equally simple formula for the heat flow through *any *surface Δa_{2} (i.e. any surface that is *not *at right angles to the heat flow ** h**):

ΔJ/Δa_{2} = (ΔJ/Δa_{1})·*cos*θ = ** h**·

*n*In this equation, we have the *scalar *product of *two* vectors: (1) ** h**, the heat flow and (2)

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

*n*_{2}. At this point, I need to remind you of the definition of the

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

**A**·**B** = |**A**||**B**|*cos*(θ), with θ the angle between **A** and **B**

In this case, * h*·

*= |*

**n***||*

**h****|**

*n**cos*θ = |

*|·1·*

**h***cos*θ = |

*|*

**h***cos*θ. For example, when the surfaces coincide, the angle θ will be zero and then

*·*

**h***is just equal to |*

**n***|*

**h***cos*θ = |

*| =*

**h***h*·1 =

*h*= ΔJ/Δa

_{1}. The other extreme is that orthogonal surfaces: in that case, the angle θ will be 90° and, hence,

*·*

**h***= |*

**n***||*

**h****|**

*n**cos*(π/2) = |

*|·1·0 = 0: there is no heat flow normal to the direction of heat flow.*

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

**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 define**

*n**vector equations*.

**The ∇ (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 ∂T/∂x, ∂T/∂y and ∂T/∂z make sense: they measure how the temperature varies with respect to position. Now, the remarkable thing is that the 3-tuple (∂T/∂x, ∂T/∂y, ∂T/∂z) is a physical vector itself: it is independent, indeed, of the reference frame (provided we measure stuff in the same unit) – 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 a *vector *indeed:

(∂T/∂x, ∂T/∂y, ∂T/∂z) = a vector (1)

If you like to see a formal proof of this, I’ll refer you to Feynman once again – 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 *displacement vector* Δ**R** = (Δx, Δy, Δz), and the *difference* in temperature between two nearby points (ΔT) will tend to the total differential:

ΔT = (∂T/∂x)Δx + (∂T/∂y)Δy + (∂T/∂z)Δz (2)

Equation (1) and (2) combine to:

ΔT = (∂T/∂x, ∂T/∂y, ∂T/∂z)(Δx, Δy, Δz) = **∇**T·Δ**R**

In this equation, we used the **∇ **(*del*) operator, i.e. the *vector *differential operator. It’s an operator like the differential operator ∂/∂x but, unlike the derivative, it returns a *vector*, referred to as the gradient: **∇**T, in this case. In other words, the **∇ **operator *acts on a scalar* (T) and yields a *vector*:

**∇**T** = **(∂T/∂x, ∂T/∂y, ∂T/∂z)

That’s why we write **∇** in **bold-type** too, just like the vector **R**. [As you know, using bold-type (instead of an arrow or so) is a convenient way to mark a vector.]

If **∇**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 the x-, y- and z-component of our **∇**T vector respectively. In fact, the rate of change of T in *any* direction will be the component of the **∇**T vector *in that direction*. Now, the magnitude of a vector *component* will always be smaller than the magnitude of the vector itself, except if it’s the component in the same direction as the vector, in which case the component *is *the vector. Therefore, the direction of **∇**T will be the direction in which the 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.” But it is obvious what direction that is: it is the *opposite *direction of the heat flow ** h**.

Now we’re ready to write our first vector equation:

** h** = –

*κ*

**∇**T

This is simple enough to understand: the direction of heat flow is opposite to the direction of **∇**T (so its flows from higher to lower temperature, as we would expect, of course!), and its magnitude is proportional, with the constant of proportionality equal to *κ* (kappa), which is called the *thermal conductivity*. In case you wonder what this means (don’t get lost in the math, indeed!), remember that the heat flow is the flow of thermal energy per unit area (and per unit time, of course): |* h*| =

*h*= ΔJ/Δa.

So what’s the point? Well… We have a scalar field here, the temperature T, from which we can derive the heat flow ** h**, i.e. a vector quantity, using this new operator

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

**E** = –**∇**Φ

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.

**Note**: The formula for E above 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 more complicated.

**Operations with ∇: divergence and curl**

You may think we’ve covered a lot of ground already but, in fact, we only just got started. In what I wrote above, I emphasized the physics. Let me now turn to the math involved. Let’s start by dissociating the operator from the scalar field, so we just write:

**∇**** = **(∂/∂x, ∂/∂y, ∂/∂z)

This doesn’t *mean *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 with something else. We can do a scalar vector product, for example:

**∇**·(a vector)

We can do this product because **∇** has three components, so it’s a ‘vector’ too, and then we need to make sure that the vector we’re operating on also has three components. To continue with our examples, we can write

**∇**·** h **= (∂/∂x, ∂/∂y, ∂/∂z)·(

*h*

_{x},

*h*

_{y},

*h*

_{z}) = ∂

*h*

_{x}/∂x + ∂

*h*

_{y}/∂y, ∂

*h*

_{z}/∂z

What we have here, in fact, is a *new *operator. It’s a *vector* operator, because it acts on a vector, but note the operator yields a scalar as a result. [Remember that our del operator acted on a scalar to yield a vector, so it was the other way around.] It’s an important operator in physics, and so it has a name and a symbol of its own:

**∇**·** h **= div

**= the divergence of**

*h*

*h*The physical significance of the divergence is related to the so-called *flux *of a vector field: it measures the magnitude of a field’s *source *or *sink *at a given point. Continuing our example with temperature, consider *air* as it is heated or cooled. The relevant vector field is now the *velocity *of the 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 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. 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:

**∇**·E = ρ/ε_{0} and **∇**·B = 0

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

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

The first equation basically says that electric charges cause an electric field. The second equation basically says there is no such thing as a *magnetic *charge: the magnetic force only appears when charges are *moving* and/or when electric fields are *changing*.

Of course, you’ll anticipate the second new operator now, because that’s the one that appears in the *other *two equations in Maxwell’s *set* of equations. It’s the cross product:

**∇×E **= (∂/∂x, ∂/∂y, ∂/∂z)×(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:

(**∇×E**)_{z} = ∇_{x}E_{y }– ∇_{y}E_{x }= ∂E_{y}/∂x – ∂E_{x}/∂y

(**∇×E**)_{x} = ∇_{y}E_{z }– ∇_{z}E_{y }= ∂E_{z}/∂y – ∂E_{y}/∂z

(**∇×E**)_{y} = ∇_{z}E_{x }– ∇_{x}E_{z }= ∂E_{x}/∂z – ∂E_{z}/∂x

I know this looks pretty monstrous, but so that’s how cross products work. I gave the geometric formula for a dot product above, so I should also give you the same for a cross product:

**A**×**B** = |**A**||**B**|*sin*(θ)**n**

In this formula, we once again have θ, the angle between **A** and **B**, but note we take its sine this time. In addition, we have **n**: a unit vector at right angles to both **A** and **B**. It’s what makes the cross product a vector, and its direction is given by that right-hand rule which we encountered a couple of times already.

Just like we did when using the del operator in a dot product, we also have a special name and symbol for using the del operator in a cross product:

**∇×**** h **= curl

**= the curl of**

*h*

*h*The **curl** is, just like the divergence, a vector operator, because its acts on a vector, but so its result is a vector too, *not* a scalar. 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* of a vector field, so the length and direction of the curl vector characterize the rotation at that point:

- The direction of the curl is the axis of rotation, as determined by the right-hand rule. [By now, you know that, in physics, there’s not much use for a left hand. :-)]
- 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. As for now, just note we defined *three *new operators in this ‘introduction’ to vector analysis:

**∇**T**=**grad**T**= a vector**∇·**= div*h*= a scalar*h***∇**×= 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 equation—that is, in more than a strictly mathematical sense—was described by Dirac. He said: “I understand what an equation means if I have a way of figuring out the characteristics of its solution without actually solving it.” So if we have a way of knowing what should happen in given circumstances without actually solving the equations, then we “understand” the equations, as applied to these circumstances.”

Well… It’s obvious this short post isn’t enough to reach such understanding. For that, we’ll need *much *more. But, for the moment, I’ll leave it at this. :-)