# Usage of $\cdot$ in calculus

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!

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I have never seen the dot product operator used this way. Actually, let me ask you this. How do you interpret the dot operator here? –  Nameless Mar 18 '14 at 5:33
Generally I would remove symbols if doing so does not introduce ambiguity. –  copper.hat Mar 18 '14 at 5:41
The dot operator is simply shorthand for '*' in most cases. In the case of both it should be acceptable to forgo use. –  Display Name Mar 18 '14 at 5:49
I use $\cdot$ for multiplication when it improves readability or provides meaning. For example, I find $n!\cdot 2^n$ to be more readable than $n!2^n$, even though it’s not needed, and for a product of integers, $2\cdot11\cdot13$ is unambiguous, while juxtaposition isn’t, and other options, like $(2)(11)(13)$ or $2\times11\times13$, are clunkier. –  Steve Kass Mar 18 '14 at 5:49
I agree with the others. Not using it is acceptable. –  aricha1940 Mar 18 '14 at 6:33

## 2 Answers

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 $(-3, 4)\pmatrix{5\\-1}$ $\color{green}{\text{is correct}}$.

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You don't know because it doesn't make sense two multiply two row vectors :) . OTOH, $(3\;2)\begin{pmatrix}4\\1\end{pmatrix}$ makes perfect sense. –  Ruslan Mar 18 '14 at 7:10
@Ruslan Thank you for your input –  JChau Mar 18 '14 at 7:26
${(-3, 4)\pmatrix{5\\-1} }$ is correct because one assumes that two matrices are beeing multiplied: the $1\times 2$-matrix ${(-3, 4)}$ and the $2\times 1$-matrix ${\pmatrix{5\\-1} }$. –  Américo Tavares Mar 18 '14 at 7:33

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.

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