Scalars and vectors

In Year I Mathematics, we saw that in applications in physics and engineering, it is common to define two quantities: scalars and vectors. A scalar is a quantity that is determined by its magnitude; examples include length and temperature. A vector is a quantity that is determined by both its magnitude and its direction. Examples include force or velocity. See notes from Year I Mathematics for a review of the basic definitions of vector algebra. Recall that we denote vectors by bold, lower case letters, e.g. $\boldsymbol{v}, \boldsymbol{a}, \boldsymbol{b}$ or, with an arrow notation, $\vec{v}, \vec{a}, \vec{b}$. In these notes, we will use the bold notation to denote vectors.

There are four kinds of functions involving scalars and vectors:

1. Scalar functions of a scalar
   • e.g. the temperature of a body as a function of time $T(t)$. The calculus here is ordinary differential calculus.
2. Vector functions of a scalar
   • e.g. the position, velocity or acceleration as a function of time, $\boldsymbol{r}(t)$. The calculus here is very similar to ordinary differential calculus.
3. Scalar functions of a vector
   • e.g. density or temperature as a function of position, $\rho(\boldsymbol{r})$ or $T(\boldsymbol{r})$. These functions define scalar fields. A new type of calculus is needed here.
4. Vector functions of a vector
   • e.g. velocity in a fluid as a function of position, $\boldsymbol{u}(\boldsymbol{r})$. These functions define vector fields and, again, a new type of calculus is needed here.