I am beginning to learn vector calculus and over the last few days I have been looking at integrals.
An example the lecturer used was fluid flow. He said that the volume of fluid flowing through a small area $\delta S$ in time $\delta t$ is $(\mathbf{u} \cdot \delta\mathbf{S})\delta t$, where $\mathbf{u}$ is the velocity field of the fluid. He followed it by saying the total amount of fluid flowing through the whole surface $S$ in time $\delta t$ is $$\delta t \int_S \mathbf{u} \cdot d \mathbf{S}.$$
For whatever reason, he wrote the expression in the form above, not as an equation (i.e. something with an equals sign). So is the correct equation $$\delta \Phi = \delta t \int_S \mathbf{u} \cdot d \mathbf{S},$$ or $$\Phi = \delta t \int_S \mathbf{u} \cdot d \mathbf{S}?$$
The use of $\delta t$ rather than $dt$ is confusing me because it is unclear if he means to say $\delta t \to dt$ or something else.
Thanks for the help.