Usage of $\cdot$ in calculus

Question

I often find myself caught in the dilemma of whether or not to use the symbol $\cdot$ in calculus. Take for example, the chain rule:

$$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$$

Is the $\cdot$ there really necessary? Can we write it simply instead as:

$$\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$$

How about the case when one considers axillary substitutions during integration, for example:

$$x = \sin\theta \implies dx = \cos\theta\cdot d\theta$$

Is the $\cdot$ here again necessary? Will $dx = \cos\theta d\theta$ be the same (I've seen a mixture of both quite often), by the strict standards of Mathematics? Up to date, all I've heard from my lecturers is that omitting the $\cdot$ is fine, although strictly speaking, it is necessary. But why so? What does the $\cdot$ mean?

Or is this all just a matter of preferences? If so, what best practices would you advise?

Thanks!

Answer 1

From what I have seen in textbooks and the like, I have never seen the $\cdot$ symbol being used. The chain rule would be: $$\dfrac{dy}{dx}=\color{green}{\frac{dy}{du}\frac{du}{dx}}$$ A particular integral solvable by u-substitution would be written like: $$\int \color{green}{\sin x \cos x \ dx}$$ $$u=\sin x$$ $$du = \color{green}{\cos x \ dx}$$

As for using the actual notation $\cdot$ is acceptable, I think it is allowed. Some other textbooks may have that notation, but I have not seen $\cdot$ to represent multiplication and the like. Now I do not want you to think you do not use $\cdot$ when dealing with dot products. I will stop here because I actually do not really know what a dot product is, other than the fact that it is related to vectors. You do not need to use the dot symbol (as pointed out by user Américo Tavares) $$\color{green}{(-3, 4)\cdot \pmatrix{5\\-1} \ \text{is definitely correct.}}$$ $$\color{green}{(-3, 4)\pmatrix{5\\-1} \ \text{is also correct.}}$$

Edit: User Américo Tavares pointed out in a comment that $(-3, 4)\pmatrix{5\\-1}$ $\color{green}{\text{is correct}}$.

Answer 2

When dealing with analysis vs linear algebra the concepts of $\cdot$ vs $\times$ vs * are all the same in real analysis and are just multiplication. Multiplication of vectors is different. Multiplying two vectors by $\cdot$ produces a number (called a scalar) and we refer to that operation as a dot product. When you multiply two vectors using the $\times$ symbol this refers to a cross product and produces another vector. These concepts are not in a beginning calculus or real analysis course so you should view the use of $\cdot$ as a purely aesthetic way of denoting multiplication.