• Notation: Make sure your students understand that $f$ is now a function of two (or more) variables $f=f(x,y)$, not a function of one variable $f=f(x)$. Therefore, in the graph, $f$ is plotted along the “$z$”-axis, not the “$y$”-axis.
• Students may not understand why they only need two partial derivatives to specify the tangent plane to a function of two variables (and three partial derivatives to specify the tangent space to a function of three variables, etc.). You can show the geometry of the two variable case by taking an individual white board or a stiff piece of cardboard to represent the tangent plane and tilting it to represent the two partial derivatives.
• Students have almost certainly never seen Taylor series for multiple variables. This would be a natural place to introduce this topic if they have seen one variable Taylor series and if you have time.

In their mathematics classes, students will almost always have seen functions written only in terms of the rectangular variables $x$, $y$, etc. In this case, it is always clear that the other rectangular variables are held constant in the partial derivative. $$\frac{\partial f}{\partial x} \qquad\hbox{$y=$ constant, etc.}$$ In many physics applications, particularly in thermodynamics (which is NOT the subject of this book), it is important to distinguish between partial derivatives with different variables held fixed. Make sure to discuss this, drawing lots of pictures, and introduce the physicists' notation of parentheses, with a subscript indicating the fixed variable. $$\left(\frac{\partial P}{\partial V}\right)_T \ne \left(\frac{\partial P}{\partial V}\right)_S $$