## Functions of Several Variables## IntroductionIt is a good idea to review the basic notion of functions. This website contains an introductory text on functions with a quiz that you may want to solve to solidify your knowledge. Recall that the definition of Cartesian product of sets (from the text Introduction to sets) allows us to write \( \mathbb R^2 \) for the set of all ordered pairs \( (x,y) \) where \( x \) and \( y \) are real numbers. Elements of \( \mathbb R^2 \) are \( (1,3) \), \( (-6,-3) \), and many others. It is very important to be clear on the fact that \( 1\not\in \mathbb R^2 \) as \( 1 \) itself is not an ordered pair. In multivariable calculus we will often consider functions \( f:\mathbb R^2\to\mathbb R \) and call them An example of a function of two variables is \( f(x,y)=9+\sin(x+y^2)-\cos(xy)+e^{x\cdot(\sin x+y)} \). This is one of the function we’ll probably never study as it is too complicated, but you got the idea. ## Graphs of functionsIf \( f:\mathbb R^2\to\mathbb R \) is a function of two variables, its graph is the set in \( \mathbb R^3 \) consisting of all points \( (x,y,z) \) such that \( z=f(x,y) \). Consider, for example, the function \[ f(x,y)=\frac{9}{1+\sin^2 x+y^2}.\] The points on its graph are colored and shown in the picture below:## Level curves
The concept of level curves comes from geography where people use two-dimensional maps to draw the three dimensional terrain. This is done through a series of optical illusions. Geographers plot contour lines that connect points of the same altitude. These contours are called In mathematics, the concept of level curves has to be formally defined. We start by assuming that we are given a surface. We assume that the surface is defined as the graph of a function \( z=f(x,y) \). For each constant \( c\in\mathbb R \), the intersection of the plane \( z=1 \) with the surface \( z=f(x,y) \) is a curve whose \( x \) and \( y \) coordinates satisfy the equation \( f(x,y)=c \). For the function \( \displaystyle f(x,y)=\frac{9}{1+\sin^2 x+y^2} \) the level sets are shown in the picture below. |

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