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On page 162 of Anderson's, Fundamentals of Aerodynamics the equation of a streamline of a flow is developed. In this section, $\boldsymbol {ds}$ is said to be "a directed element of the streamline". Then it is stated that in cartesian coordinates, $$\boldsymbol {ds} = dx \hat{\imath} + dy \hat{\jmath} + dz \hat{k}$$ In this answer to What is the rigorous definition of dy and dx?, $dx$, $dy$, and $dz$ are said to be linear functionals which form the standard basis of $(\mathbb R^3)'$. If this is the case then are there any meaningful definitions for the multiplication of $dx$ and $\hat{\imath}$?

Justin H
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    Regardless of your interpretation, $dx \hat \imath$ is something like multiplication, not function application, so it doesn't reduce to $1$. – Mark S. Sep 18 '21 at 02:08
  • @MarkS. Oh I see thanks I'm not sure why I jumped to that conclusion my bad. – Justin H Sep 18 '21 at 02:22
  • Intuitively, you can just think of $dx, dy$, and $dz$ as very tiny numbers. When physics and engineering authors write about $dx$ etc they're usually not trying to be rigorous. Calculus is all about using linear approximation to estimate the change in the value of the output of the function when the inputs are changed by tiny amounts. – littleO Sep 18 '21 at 02:55
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    This expression is a vector-valued $1$-form. – Ted Shifrin Sep 18 '21 at 03:26
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    take it as $ds=(dx,dy,dz)$ – janmarqz Sep 18 '21 at 03:54

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