Functions of Two and Three Variables

Department of Mathematics and Statistics

October 3, 2012

Calculus III (James Madison University) Math 237 October 3, 2012 1 / 5 Functions of Two Variables

Review: Deﬁnition Let A and B be sets. A function f : A B is a rule which assigns an → element of the set B to each element of the set A. The set A is called the domain of the function. The set B is called the codomain or target of the function.

In a function of two variables, the domain will be a subset of R2 and the codomain will be a subset of R.

Calculus III (James Madison University) Math 237 October 3, 2012 2 / 5 Graphs of Functions of Two Variables

Deﬁnition The graph of a function f of two variables is the set of ordered triples:

Graph(f )= (x, y, f (x, y)) (x, y) Domain(f ) . { | ∈ } Most of the functions of two variables that we will discuss this semester will be two-dimensional surfaces which exist in three-dimensional space. One technique for understanding these surfaces is to understand their level curves. Deﬁnition Let f be a function of two variables and let c be a point in the range of f . The level curve for f at height c is the curve in the plane with equation f (x, y)=c.

Calculus III (James Madison University) Math 237 October 3, 2012 3 / 5 Functions of Three Variables

In a function of three variables, the domain will be a subset of R3 and the codomain will be a subset of R. Deﬁnition The graph of a function f of three variables is the set of ordered quadruples:

Graph(f )= (x, y, z, f (x, y, z)) (x, y, z) Domain(f ) . { | ∈ } Most of the functions of three variables that we will discuss this semester will be three-dimensional objects which exist in four-dimensional space.

Calculus III (James Madison University) Math 237 October 3, 2012 4 / 5 Level Surfaces

One technique for understanding these objects is to understand their level surfaces. Deﬁnition Let f be a function of three variables and let c be a point in the range of f . The level surface for f at height c is the surface in R3 with equation f (x, y, z)=c.

Calculus III (James Madison University) Math 237 October 3, 2012 5 / 5