Where is the differentiation in the differential form

Question

I do understand what a differential 1-form does. But I don't understand where the differentiation is. Why are differential 1 forms (or even k -forms) named the way they are.

Also a general differentiation of the term 'form' would be useful

Answer 1

Differential forms are so named because they are built from differentials. The word differential is related to difference. As you probably knows, differentials, like $\mathrm{d}x$ and $\mathrm{d}y$ are often thought of as infinitessimally small differences. That's the origin of Leibniz's notation for derivatives: $$\frac{\mathrm{d}y}{\mathrm{d}x} = \lim \frac{\Delta y}{\Delta x}$$

Thus, the name differential form isn't directly related to differentiation. That said, a multivariable function $f$ is said to be differentiable at $x_0$ if there exists a linear map $J$ such that $$\lim_{h \to 0} \frac{\|f(x_0+h)-f(x_0)-Jh\|}{\|h\|} = 0$$ It can be shown that the action of $J$ on $h=(h^i)$ is given by $$Jh = \sum_i \frac{\partial f}{\partial x^i} h^i.$$

On a differential $p$-form $\omega = f \, \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_p}$ the exterior derivative $\mathrm{d}$ is defined by $$\mathrm{d}\omega = \frac{\partial f}{\partial x^j} \, dx^j \wedge \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_p}.$$

For a $0$-form $f$, which just is a function, we get $$\mathrm{d}f = \sum_i \frac{\partial f}{\partial x_i} \mathrm{d}x_i.$$ Comparing this with the expression for $Jh$ we see that we can identify $J$ with the exterior derivative or differential $\mathrm{d}f.$ Thus, differentiability and differentials are closely related concepts.

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Regarding the word form I haven't seen any general definition. But it seems to be used when some abstract expressions are constructed, perhaps without having a precise concrete definition. For example, a differential form is a construction of the form (pun intended) $f \, \mathrm{d}x^{i_1} \wedge \cdots \wedge \mathrm{d}x^{i_p}$. You can come a long way just working with them in this abstract form without having a formal definition as a quotient algebra of tensor products of linear maps.