In many situations, a variable is a function of more than one other variable. For example, the volume of a circular cylinder, $V$, depends on the radius of its ends, $r$, and its height, $h$, through $V=\pi r^2 h$ (see Fig. 1). We may say the volume of a circular cylinder is a function of two variables, $r$ and $h$, and write $V=f(r,h)$. To describe the temperature, $T$, of different points in a place we need three variables $x, y$, and $z$, which are the coordinates of the points. If the temperature also varies with time, $t$, we need four variables. In this case, we deal with a function of four variables $T=f(x,y,z,t)$.


Figure 1: The volume of a circular cylinder $V$ depends on $r$ and $h$.


In this chapter, we will extend some fundamental concepts of calculus including limits, continuity, and derivatives, to functions of two or more than two variables. As the independent variables may vary in different ways, these concepts are more complicated than their counterparts in the calculus of a single variable.

In this chapter, we learn about

  1. Functions of Several Variables
  2. Graph of a Function
  3. Level Curves and Level Surfaces
  4. Open and Closed Sets
  5. Limits and Continuity
  6. Partial Derivatives
  7. Higher Partial Derivatives
  8. Tangent Planes
  9. Linear (or Affine) Approximation
  10. Differentiability
  11. The Differential of a Function
  12. The Chain Rule
  13. Directional Derivatives and Gradients
  14. The Direction of the Fastest Increase
  15. Tangent Planes — Revisited
  16. Implicit Functions
  17. Taylor’s Theorem (Optional)
  18. The Maxima and Minima