# 8: Calculus in More than One Variable

• • Contributed by Marcia Levitus

Chapter Objectives

• Review the concept of partial derivative.
• Review the properties of partial derivatives.
• Be able to use the properties of partial derivatives in the context of physical chemistry problems.
• Review the concept of double and triple integrals.
• Learn the concept of equation of state. Understand the concept of a van der Waals gas from the molecular point of view.
• Learn about phase transitions and critical phenomena.

• 8.1: Functions of Two Independent Variables
A function of two independent variables, z=f(x,y) , defines a surface in three-dimensional space. For a function of two or more variables, there are as many independent first derivatives as there are independent variables.
• 8.2: The Equation of State
An equation of state is an expression relating the density of a fluid with its temperature and pressure. Note that the density is related to the number of moles and the volume, so it takes care of these two variables together. There is no single equation of state that predicts the behavior of all substances under all conditions.
• 8.3: The Chain Rule
The chain rule allow us to create these ‘universal ’ relationships between the derivatives of different coordinate systems.
• 8.4: Double and Triple Integrals
We can extend the idea of a definite integral to more dimensions.
• 8.5: Real Gases
We have already mentioned some thermodynamic variables, but in order to make more connections between chemistry and math we need to introduce some concepts that we need to start discussing real gases.
• 8.6: Problems

Thumbnail: Surface $$Σ$$ with closed boundary $$∂Σ$$. $$\vec{F}$$ could be the $$\vec{E}$$ or $$\vec{B}$$ fields. $$n$$ is the unit normal. Image used with permission (Public Domain; Maschen).