- 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.
