Functions of Several Variables

Functions of Two Variables #

Two inputs give one height.

A function of two variables gives a height for every point on a plane

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One input and one output give a curve: each point on it is fixed by a single number.

With two inputs, every point (x, y) of a plane is a possible input.

The output is a height above each point, so the graph is a surface, not a curve.

Surfaces in Three Dimensions #

A two-variable function graphs as a surface.

The graph of a two-variable function is a surface hanging above the plane

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A third axis comes up out of the page, and z measures height above the xy-plane.

$z=x^2+y^2$z = x² + y² is lowest at the origin and rises in every direction: a bowl.

Change one sign to $z=x^2-y^2$z = x² − y² and the surface rises along x while falling along y. That is a saddle.

Level Curves #

Slice at a height and you have a contour.

Slicing a surface at one height gives a contour, exactly as on a map

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Cut the bowl $z=x^2+y^2$z = x² + y² at height 4 and the slice is a circle.

Cut it again every 4 units up, and the cuts stack all the way to the rim.

Look straight down and each cut is an equation in x and y: $x^2+y^2=4$x² + y² = 4 has radius 2.

That is a contour map. Each step is 4 units, so closely spaced rings mean steep ground.

Limits of Functions of Two Variables #

Every path in must agree on the answer.

A limit on a plane must give the same answer along every path in

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On a line there are only two ways to approach a point: from the left or the right.

On a plane there are infinitely many ways to approach a point, along any path you like.

So two paths that give different answers are enough to prove there is no limit.

Continuity in Two Variables #

No tears, holes or cliffs in the sheet.

A surface is continuous where it has no tears, holes or sudden cliffs

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On a curve, continuity meant no lifting the pencil.

On a surface it means the same: the limit and the value must agree.

Picture a sheet with a rip or a step in it. Those are the points that fail.