# What are equations such as $dz = dx + dy$ called?

These look like differential equations:

$$dz = dx + dy,$$ $$dU = TdS - PdV,$$

and many other equations in physics, but they aren't ordinary differential equations or partial differential equations. What are they then?

• In physics courses these expressions are sometimes referred to as calculus of/with infinitesimals or the variational approach, indicating that you are considering infinitesimal changes to the system. More formally, these are equations with differential forms and connect to quite complicated subjects as symplectic geometry and contact geometry. Jul 5, 2019 at 21:39
• Yeah they're differential forms, from differential geometry. Jul 5, 2019 at 21:41
• They can be thought of as a simple and a very compact way of expressing how different variables are related to each other. From them you can derive differential/integral equations for evolution of the variables depending on the specific system you are dealing with, for example from the last one can write down the definition of temperature $T \equiv \left(\frac{\partial U}{\partial S}\right)_{V~\text{constant}}$ and get the formula for the change of energy when the temperature is constant $\Delta U = -\int P{\rm d}V$ and so on and so on. Jul 5, 2019 at 21:47
• I would probably call these "equations of differentials". Jul 5, 2019 at 22:03

I don't know what physicists call them. What they are (in physics) are equations that relate small (infinitesimal) changes in physical quantities. For example, $$ds = v \ dt$$ says that a small change in time causes a small change in position that's proportional to the velocity.

Your second example looks like one from thermodynamics. $$T$$ is temperature, $$P$$ pressure and $$V$$ volume. (I don't remember enough physics to identify $$S$$ and $$U$$.)

Often these equations lead to differential equations - ordinary or partial.

In some formal mathematical physics they are equations that relate quantities defined for manifolds.

In mathematics -- especially multivariable calculus and differential geometry -- quantities like $$y\,dx$$ or $$T\,dS - P\,dV$$, etc., are called differential $$1$$-forms.

An equation of the form $$df = 0$$, where $$f$$ is a function, is called an exact equation.

More generally, an equation of the form $$\alpha = 0$$, where $$\alpha$$ is any differential $$1$$-form, is called a Pfaffian equation.