Problem

Many people have asked this question, and many will continue to do so. It is the natural question of someone first learning the subject of calculus: what is “\(\mathrm{d}x\)”, and why is it everywhere in calculus?

Frankly, it’s mostly Leibniz’s fault. Leibniz, a brilliant philosopher and mathematician who may or may not have invented calculus depending on whom you ask, introduced the notation. His view of derivatives was as the ratio of related infinitesimals. In slightly more modern terms, $$\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}=\frac{dy}{dx}.$$ Unfortunately, people have carried this view for an unhealthily long amount of time. A branch of analysis known as “nonstandard analysis” found a clever and eponymously nonstandard way to make the idea of the infinitesimal rigorous. Still others have questioned the validity of the law of the excluded middle, which says that a proposition must be either true or false. By not accepting this law, some finagling and rather nonstandard logic can bring about the idea of an infinitesimal. The list goes on. In short, there is a myriad of “nonstandard” ways to realize \(\mathrm{d}x\).

This brings us to a much more refined question: is there a standard way to define \(\mathrm{d}x\)?

We will defend the claim that the answer is a resounding “yes.” What’s more, we will attempt to demonstrate that the concept is intuitive and natural, even to those relatively new to the subject of analysis.

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Intuition-based Introduction

In high school physics and precalculus, we are told that a vector is a magnitude accompanied by a direction. Although this is wonderful for the purposes of introduction, we aren’t fools. Position is not mentioned anywhere within this description of a vector. Thus, conceivably, we could push a large crate over a frictionless surface and have the force vector end up around the Mars rover. We need position to realistically use vectors.

With the additional datum of position, vectors become tangent vectors. If \(v\) is a vector and \(p\) is a point, then we will use \(v_p\) will denote the tangent vector corresponding to \(v\) at the point \(p\). The reason they are called tangent vectors comes from their use in differential geometry, where a vast, powerful generalization of lines, planes, and other “Euclidean” spaces, called a “(smooth) manifold,” is used. A circle is an example of such a space, and the tangent line to a point on the circle can be thought of as the set of all tangent vectors to the circle at that point. For the sake of intuition, we will stay within Euclidean space (\(\mathbb{R}^n\) for some nonnegative integer \(n\)) for this section.

Suppose \(f:\mathbb{R}^2\to\mathbb{R}\) is a smooth function from the real plane to the real line. At each point \(p\in\mathbb{R}^2\), we can take the directional derivative of \(f\) in whatever direction we like. For example, if we have coordinates \((x,y)\), then \(\left.\frac{\partial f}{\partial x}\right|_p\) is the directional derivative at \(p\) of \(f\) in the direction of increasing \(x\). If we wanted to take the directional derivative at \(p\) of \(f\) in the direction of decreasing \(x\), then we would just negate to obtain  \(-\left.\frac{\partial f}{\partial x}\right|_p\).

In fact, if we pick any tangent vector at \(p\), it will correspond to a directional derivative at \(p\) in the direction and with the corresponding magnitude of that tangent vector. Following the above example, taking the directional derivative at \(p\) in the direction of increasing \(x\) is the same as applying \(\left.\frac{\partial}{\partial x}\right|_p\), and going the opposite direction is the same as applying \(-\left.\frac{\partial}{\partial x}\right|_p\). Taking this a step further, we claim that tangent vectors and directional derivatives are the same thing! This may seem shocking, but all it says is that every tangent vector corresponds to a directional derivative with the same magnitude and direction. If we are working in the plane, then we can therefore identify \(\left.\frac{\partial}{\partial x}\right|_p\) with \(\begin{bmatrix}1 & 0\end{bmatrix}^T_p\) and \(\left.\frac{\partial}{\partial y}\right|_p\) with \(\begin{bmatrix} 0 & 1\end{bmatrix}^T_p\), where \(v^T\) denotes the transpose of a vector \(v\).

We use the transpose to use column vectors for tangent vectors. By doing this, we can let row vectors act on them by left multiplication. Note that left multiplying a column vector by a row vector is the same as taking the dot product between those two vectors. Thus, row vectors can be viewed as linear functions from column vectors to the space of real numbers. Attaching these linear functionals to a point as we did with tangent vectors, we obtain cotangent vectors, or covectors. Hinting at the direction of conversation, in the plane we will denote the covector \(\alpha_p\) that satisfies \(\alpha_p\left(\left.\frac{\partial}{\partial x}\right|_p\right)=1\) and \(\alpha_p\left(\left.\frac{\partial}{\partial y}\right|_p\right)=0\) by \(\mathrm{d}x_p\).

The concept of a vector field from multivariate calculus is best seen in the context of tangent vectors. Using our terminology, a vector field is simply a function that takes in a point in \(\mathbb{R}^n\) and outputs a tangent vector at that point. Similarly, we define a differential \(1\)-form as a function that takes in a point in \(\mathbb{R}^n\) and outputs a covector at that point. For example, the map $$\mathrm{d}x:p\mapsto\mathrm{d}x_p$$ is a differential \(1\)-form. Thus, \(\mathrm{d}x\) is simply the differential \(1\)-form that takes each point \(p\) of the space to the cotangent vector \(\mathrm{d}x_p\) at \(p\) that satisfies \(\mathrm{d}x_p\left(\left.\frac{\partial}{\partial x}\right|_p\right)=1\) and \(\mathrm{d}x_p(v_p)=0\) for all \(v_p\not\in\left<\left.\frac{\partial}{\partial x}\right|_p\right>\).

How could these possibly be interpreted as infinitesimals? I think Spivak explains this best:

“Classical differential geometers (and classical analysts) did not hesitate to talk about ‘infinitely small’ changes \(\mathrm{d}x^i\) or the coordinates \(x^i\), just as Leibnitz (sic) had. No one wanted to admit that this was nonsense, because true results were obtained when these infinitely small quantities were divided into each other (provided one did it in the right way).

“Eventually it was realized that the closest one can come to describing an infinitely small change is to describe a direction in which this change is supposed to occur, i.e., a tangent vector. Since \(\mathrm{d}f\) is supposed to be an infinitesimal change of \(f\) under an infinitesimal change of the point, \(\mathrm{d}f\) must be a function of this change, which means that \(\mathrm{d}f\) should be a function on tangent vectors. The \(\mathrm{d}x^i\) themselves then metamorphosed into functions, and it became clear that they must be distinguished from the tangent vectors \(\partial/\partial x^i\).” [1]

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Interlude: Differential k-forms

The term differential \(1\)-form ominously implies it is part of a larger structure. In this section, we will provide blatantly formal definitions, then provide intuition. The formal subsection is designed to show that differential geometry is fundamentally different from your calculus homework (i.e. Please stop using the (differential-geometry) tag for questions that don’t apply to that level of generality). The formal subsection should, therefore, be skipped at the first hint of confusion.

Formal

Let \(M\) be a smooth manifold. We shall denote the space of smooth sections on the vector bundle \(\xi\) by \(\Gamma(\xi)\). Define \(\Gamma(\Lambda\,T^\vee M)\) to be the quotient algebra of the free algebra on \(\Gamma(T^\vee M)\) by the ideal generated by elements of the form \(\alpha\otimes\alpha\). In other words, $$\Gamma(\Lambda\,T^\vee M)=\left(\bigoplus_{k=0}^\infty(\Gamma(T^\vee M))^{\otimes k}\right)/\left<\alpha\otimes\alpha\vert \alpha\in\Gamma(T^\vee M)\right>.$$ We shall denote the product in \(\Gamma(\Lambda\,T^\vee M)\) by \(\wedge\).

We may now define differential forms as elements of \(\Gamma(\Lambda\,T^\vee M)\), and a differential \(k\)-form is simply an element of the subalgebra $$\Gamma(\Lambda^k\,T^\vee M)=(\Gamma(T^\vee M))^{\otimes k}/(\left<\alpha\otimes\alpha\vert \alpha\in\Gamma(T^\vee M)\right>\cap (\Gamma(T^\vee M))^{\otimes k}).$$ It should be noted that \(\Gamma(\Lambda^0\,T^\vee M)=C^\infty(M)\).

We define the exterior derivative \(\mathrm{d}\) as the unique \(\mathbb{R}\)-linear map \(\mathrm{d}:\Gamma(\Lambda\,T^\vee M)\to\Gamma(\Lambda\,T^\vee M)\) such that

• If \(\alpha\) is a \(k\)-form, then \(\mathrm{d}\alpha\) is a \((k+1)\)-form.
• If \(f\in C^\infty(M)\), then \(\mathrm{d}f_p(X_p)=X_p(f)\) for all tangent vectors \(X_p\in T_pM\).
• If \(\alpha\) is a \(k\)-form and \(\beta\) is another differential form, then \(\mathrm{d}(\alpha\wedge\beta)=\mathrm{d}\alpha\wedge\beta+(-1)^k\alpha\wedge\mathrm{d}\beta\).
• For all differential forms \(\alpha\), \(\mathrm{d}(\mathrm{d}\alpha)=0\).

Informal

Let’s take this step by step, and stay in \(\mathbb{R}^n\). We introduce a “wedge product” \(\wedge\) to extend the space of differential \(1\)-forms. A \(0\)-form is just a smooth function, and for \(f\in C^\infty(\mathbb{R}^n)\) and a differential form \(\alpha\), \(f\wedge\alpha=f\alpha\). If \(\alpha\) and \(\beta\) are distinct \(1\)-forms, then \(\alpha\wedge\beta\) is a \(2\)-form. If \(\gamma\) is yet another distinct \(1\)-form, then \(\alpha\wedge\beta\wedge\gamma\) is a \(3\)-form. If {\({\alpha^i}\)} is a set of \(k\) distinct \(1\)-forms, then \(\alpha^1\wedge\alpha^2\wedge\cdots\wedge\alpha^k\) is a \(k\)-form. However, \(0\wedge\alpha=0\) and \(\alpha\wedge\alpha=0\) for all \(1\)-forms \(\alpha\).

Differential \(k\)-forms generalize \(1\)-forms by taking in \(k\) tangent vectors instead of only \(1\). For example, in the plane with coordinates \((x,y)\), \((\mathrm{d}x\wedge\mathrm{d}y)_p(v_p,w_p)=\frac{1}{2}(\mathrm{d}x_p(v_p)\mathrm{d}y_p(w_p)-\mathrm{d}x_p(w_p)\mathrm{d}y_p(v_p))\).

In addition to this, there is a linear map \(\mathrm{d}\) that takes \(k\)-forms to \((k+1)\)-forms. If the coordinates are given by \((x^1,\,\dots\,,x^n)\) and $$\alpha=\sum_{1\leq i_1<\cdots<i_k\leq n}f_{i_1\cdots i_k}\mathrm{d}x^{i_1}\wedge\cdots\wedge\mathrm{d}x^{i_k}=\sum_If_I\mathrm{d}x^I$$ is a \(k\)-form, then $$\mathrm{d}\alpha=\sum_{j=1}^n\sum_I\frac{\partial f_I}{\partial x^j}\mathrm{d}x^j\wedge\mathrm{d}x^I.$$

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Why would this be the natural choice?

We have claimed that this interpretation is incredibly natural to our approach to calculus. While it is implausible to mathematically prove this subjective viewpoint, providing a few examples of how it works will invariably display the beauty of the concept.

Indefinite Integration

It is tempting to simply write that \(\int\mathrm{d}\alpha=\alpha\), but this is not well defined. Denote the space of differential \(k\)-forms on \(\mathbb{R}^n\) by \(\Omega^k(\mathbb{R}^n)\). If \(\alpha\in\Omega^{k+1}(\mathbb{R}^n)\) and \(\alpha=\mathrm{d}\beta\) for some \(\beta\in\Omega^k(\mathbb{R}^n)\), then \(\alpha=\mathrm{d}(\beta+\mathrm{d}\gamma)\) as well. Thus, there is no unique \(k\)-form mapping to \(\alpha\).

Define a \(k\)-form \(\alpha\) to be exact if and only if \(\alpha=\mathrm{d}\beta\) for some \(\beta\), and say \(\alpha\in\mathrm{d}(\Omega^{k-1}(\mathbb{R}^n))\). Similarly, define a \(k\)-form \(\alpha\) to be closed if and only if \(\mathrm{d}\alpha=0\), and say \(\alpha\in\Omega^k_\text{Closed}(\mathbb{R}^n)\). Using these, we may now define $$\int:\mathrm{d}(\Omega^{k}(\mathbb{R}^n))\to\Omega^k(\mathbb{R}^n)/\Omega^{k}_\text{Closed}(\mathbb{R}^n),$$

$$\mathrm{d}\alpha\mapsto\alpha+\Omega^k_\text{Closed}(\mathbb{R}^n).$$ Thus, \(\int\) maps exact forms \(\mathrm{d}\alpha\) back to the coset of \(\Omega^k_\text{Closed}(\mathbb{R}^n)\) determined by \(\alpha\). In less formal terms, this means that \(\int\) takes the exact form \(\mathrm{d}\alpha\) to the collection \(\begin{Bmatrix}\alpha + \sigma : \mathrm{d}\sigma=0\end{Bmatrix}\). (For you sheafites, this can easily be seen as a map between presheaves.)

Does this work with the indefinite integral we already use? Let \(k=0\) and \(n=1\). For \(\mathrm{d}f=f’\,\mathrm{d}x\), $$\int\mathrm{d}f=\int f’\,\mathrm{d}x=f+\Omega^0_\text{Closed}(\mathbb{R})\ni f+c,$$ where \(c\) is an “arbitrary constant of integration.” Thus, the indefinite integral arises naturally from this choice of perspective.

Differentiation

With this interpretation, we can define differentiation as simply the application of a tangent vector or vector field to a smooth function. Though this looks rather snazzy at first glance, it really just says that differentiation is defined as usual; we have only changed the context. For the sake of entertainment, we shall introduce a generalization.

Suppose \(\phi\) is a smooth bijection with a smooth inverse, \(\psi\) is a smooth map, and \(f\) be a smooth, real-valued function. We define the pushforward \(\psi _ \ast X_p\) of a tangent vector \(X_p\) at \(p\) by \((\psi _ \ast X_p)(f)=X_{\psi(p)}(f\circ\psi)\) and the pushforward \(\phi_\ast X\) of a vector field \(X\) by \(\phi\) by $$(\phi_\ast X)_p(f)=X _ {\phi^{-1}(p)}(f\circ\phi).$$ For a differential \(k\)-form \(\alpha\), we define the pullback \(\psi^\ast\alpha\) of \(\alpha\) by \(\psi\) by $$(\psi ^ \ast \alpha)_p(X_1(p),\,\dots\,,X_k(p))=\alpha _ {\psi(p)}(\psi _ \ast (X_1(p)),\,\dots\,,\psi _ \ast (X_k(p))).$$

If \(X\) is a vector field defined on \(\mathbb{R}^n\) and \(p\in\mathbb{R}^n\), then there exists an open neighborhood \(U\subseteq\mathbb{R}^n\) of \(p\) and an \(\varepsilon>0\) such that there is a unique \(\phi^X:(-\varepsilon,\varepsilon)\times U\to\mathbb{R}^n,~(t,q)\mapsto\phi^X_t(q)\) such that

• \(\phi^X\) is smooth.
• For fixed \(t\), \(\phi^X_t:U\to\phi^X_t(U)\) is a smooth bijection with a smooth inverse.
• If \(|a|<\varepsilon\), \(|b|<\varepsilon\), and \(|a+b|<\varepsilon\), then \(\phi^X_{a+b}=\phi^X_a\circ\phi^X_b\).
• Let \(\psi_q:t\mapsto\phi_t(q)\). For all \(q\in U\), \((\psi_q) _ \ast\!\left(\left.\frac{\partial}{\partial t}\right|_0\right)=X_q\).

We call \(\phi^X\) the local flow of \(X\).

For any differential form \(\alpha\), we can differentiate it with respect to a vector field \(X\) using the Lie derivative: $$\mathcal{L} _ X\alpha=\lim _ {\delta\rightarrow 0}\frac{1}{\delta}\hspace{-.5em}\left((\phi^X _ \delta) ^ \ast \alpha-\alpha\right).$$

For \(0\)-forms \(f\), this is simply \(\mathcal{L} _ Xf=X(f)\). Again, this setting is the natural playground of calculus.

Definite Integration

Define a singular \(k\)-cube to be a smooth map (as a map from a manifold with boundary) \(c:[0,1]^k\to\mathbb{R}^n\). For example, a path between two points is a singular \(1\)-cube.

There is plenty we can do with these singular cubes, but we need to be able to add them. Thus, we say define finite formal sums of \(k\)-cubes as \(k\)-chains. This lets us add cubes together. If we need to visualize this, we can just think of passing through one cube, then passing through the other.

This notion of addition gives us a bit more leeway with what we can do. For example, we can define the boundary \(\partial c\) of a \(k\)-cube \(c\) by $$\sum_{i=1}^k(-1)^i(c_{(i,0)}-c_{(i,1)}),$$ where \(c_{(i,j)}:(x^1,\,\dots\,,x^{k-1})\mapsto c(x^1,\,\dots\,,x^{i-1},j,x^{i},\,\dots\,,x^{k-1})\). We can also extend this to chains, using \(\partial\hspace{-.3em}\left(\sum_r a_rc_r\right)=\sum_ra_r\partial(c_r)\). This essentially gives the oriented boundary of the singular cube, and by intuition we can see that \(\partial(\partial c)=0\) for any chain \(c\).

For coordinates \((x^1,\,\dots\,,x^k)\) on \([0,1]^k\), a \(k\)-form \(\alpha\), a singular \(k\)-chain \(c=\sum_{r}a_rc_r\), a partition \(P=(t_{i_1},\,\dots\,, t_{i_k})\) of \([0,1]^k\), with \(0\leq t_{i_j}\leq 1\) and \(1\leq i_j\leq l_j\), and a collection \(\rho_P\) of \(\rho_{i_1\cdots i_k}\in[t_{i_1-1},t_{i_1}]\times\cdots\times[t_{i_k-1},t_{i_k}]\), we define $$S(c,\alpha,P,\rho _ P)=\sum_{r}\left(a_r\hspace{-.5em}\sum _ {\small{i _ 1,\dots, i _ k}}(c _ r ^ \ast \alpha) _ {\rho _ {i _ 1\cdots i _ k}}\hspace{-.5em}\left(\left.\frac{\partial}{\partial x^1}\right| _ {\rho _ {i _ 1\cdots i _ k}}\hspace{-1.5em},\,\dots\,,\left.\frac{\partial}{\partial x^k}\right| _ {\rho _ {i _ 1\cdots i _ k}}\right)(t _ {i _ 1}\hspace{-.5em}-t _ {i _ 1-1})\cdots(t _ {i _ k}\hspace{-.5em}-t _ {i _ k-1})\right).$$

With this, we define $$\int\limits_{c}\alpha=\lim_{\small{l_1,\dots,l_k\rightarrow\infty}}S(c,\alpha,P,\rho_P).$$

This, admittedly, looks awful. However, it already agrees with our original definitions. If \(c:[0,1]\to\mathbb{R},~t\mapsto (b-a)t+a\) and \(\alpha=\mathrm{d}f\), then we can pick \(t_i=\frac{i}{n}\), \(0\leq i\leq n\), to get $$\int\limits _ c\mathrm{d}f=\lim _ {n\rightarrow\infty}\sum _ {i=0} ^ {n-1}(c ^ \ast\mathrm{d}f) _ {t _ i}\left(\left.\frac{\partial}{\partial x}\right| _ {t _ i}\right)(t _ {i+1}-t _ i)=\lim _ {n\rightarrow\infty}\sum _ {i=0} ^ {n-1}c'(t _ i)f'(c(t _ i))(t _ {i+1}-t _ i).$$ Since \(c'(t_i)=b-a\) and \(t_{i+1}-t_i=\frac{i+1}{n}-\frac{i}{n}=\frac{1}{n}\), this further simplifies to $$\int\limits_c\mathrm{d}f=\int_a^bf’\,\mathrm{d}x=\lim _ {n\rightarrow\infty}\sum _ {i=0} ^ {n-1}(b-a)f'(a+(b-a)\frac{i}{n})\frac{1}{n}=\lim _ {n\rightarrow\infty}\sum _ {i=0} ^ {n-1}f'(a+i\frac{b-a}{n})\frac{b-a}{n},$$ which is just the lower Riemann sum.

As hideous as this all is, though already natural in a kind of organic way, this all culminates into something euphorically beautiful.

Theorem: If \(\alpha\) is a \(k\)-form and \(c\) is a \((k+1)\)-chain, then $$\int\limits_{\partial c}\alpha=\int\limits_{c}\mathrm{d}\alpha.$$

See that theorem? It’s debatably the most important theorem in all of calculus and analysis.

The fundamental theorem of calculus states that \(\int_a^bf’\,\mathrm{d}x=f(b)-f(a)\). By our new theorem, called the generalized Stokes theorem, $$\int_a^bf’\,\mathrm{d}x=\int\limits_c\mathrm{d}f=\int\limits_{\partial c}f=f(b)-f(a).$$ Similar thinking leads to easy proofs of Green’s theorem and the divergence theorem from multivariate calculus. Even the residue theorem from complex analysis has a nice proof using this theorem.

Thus, the drudgery of all the new machinery we’ve developed is easily forgiven for this theorem, which is truly beautiful and natural.

Differentiation (again)

Occasionally, we encounter mathematicians that are unwilling to part with the idea of the derivative as a quotient of differentials. When I come across such people, I typically appease them using the following:

Define the division of two linearly dependent vectors \(v\) and \(w=\lambda v\) by \(w\div v=\lambda\). The exterior derivative of \(f:\mathbb{R}\to\mathbb{R}\), which is \(\mathrm{d}f=f’\,\mathrm{d}x\), and \(\mathrm{d}x\) are scalar multiples of each other at each point. That is, \(\mathrm{d}f_t=f'(t)\mathrm{d}x_t\). Thus, \(\mathrm{d}f_t\div\mathrm{d}x_t=f'(t)\).

I privately call this the Gooby derivative lemma, in reference to a series of webcomics depicting the Disney characters Donald Duck (Dolan) and Goofy (Gooby), where Donald crudely exploits Goofy’s lack of intelligence under the guise of friendship.

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 Conclusion

We’ve discussed the benefits of thinking of \(\mathrm{d}x\) as a differential form and demonstrated some of the beauty of differential geometry. Hopefully, this has convinced you of the depth and power of calculus, and taught you that infinitesimals are not the only explanation for it.

References:

[1] Michael Spivak, A Comprehensive Introduction to Differential Geometry, 3rd ed., Vol. 1, Publish or Perish, 1999.