Quadric surfaces, multivariable functions and level curves

The introductory week of multivariable calculus. We do two things. First you will learn to recognize quadric surfaces in space from their equations: it comes in the exam as an “identify and sketch” question, and you also need it to see the region and the constraint in double integral and Lagrange questions. Then we move on to functions of two variables of the form $z = f(x,y)$: domain, range, level curves and limits. When we move on to partial derivatives next week, this language will be ready.

Golden rules:

1. To recognize a surface, look at its traces: the plane curves you get by holding $x$, $y$ or $z$ fixed.
2. Count the signs: how many squares have a minus sign, is the right side $0$ or $1$, is there a first-degree variable?
3. In the domain, inside a root $\ge0$, inside a logarithm $>0$, denominator $\neq0$.
4. A level curve is $f(x,y) = c$; if the one through a point is asked, compute $c$ from that point.
5. In a two-variable limit, if two different paths give different results, the limit does not exist. But a few paths giving the same result does NOT PROVE that the limit exists.

Quadric surfaces

The general form: a second-degree equation in $x$, $y$, $z$. The six types you will meet in this course:

If $a = b$, the ellipses turn into circles and the surface is a surface of revolution about the $z$ axis (like a circular paraboloid, a circular cone).

Recognition template

1) Is there a first-degree variable?  (z = ... ) → paraboloid
     squares with the same sign → elliptic,   opposite signs → hyperbolic (saddle)
2) All squares. Is the constant term 0?  → cone
3) All squares, right side 1 (a positive constant):
     no minus → ellipsoid
     one minus → hyperboloid of one sheet
     two minus → hyperboloid of two sheets
4) Axis: the axis of the "different" variable (the one with the minus sign, or the first-degree one)
5) If a variable is missing altogether → cylinder

Example 1. $4x^2+y^2+4z^2 = 16$. Divide by $16$: $\dfrac{x^2}{4}+\dfrac{y^2}{16}+\dfrac{z^2}{4} = 1$. An ellipsoid; the semi-axis is $2$ in the $x$ and $z$ directions and $4$ in the $y$ direction. A “football” elongated along the $y$ axis.

Example 2. $y = x^2+z^2$. $y$ is first degree, the squares have the same sign: a circular paraboloid, its axis is the $y$ axis, and it opens toward the positive direction of $y$. The trace $y = 4$ is the circle $x^2+z^2 = 4$.

Example 3. $x^2-y^2-z^2 = 1$. Two minus signs: a hyperboloid of two sheets, the axis is the $x$ axis (the only positive one). There is no surface for $\lvert x\rvert<1$.

PITFALL · which variable is the axis? In hyperboloids, the axis is the axis of the variable whose sign is different from the others. In $x^2+y^2-z^2 = 1$ this is $z$; in $x^2-y^2-z^2 = 1$ it is $x$. If you memorize it as “the one with the minus sign”, you will go wrong in the second example.

The sphere and completing the square

$(x-h)^2+(y-k)^2+(z-l)^2 = r^2$ is the sphere with center $(h,k,l)$ and radius $r$. In an equation written out in full, complete the square for each variable.

Example 4. $x^2+y^2+z^2-2x+4y-6z = 2$.

$(x-1)^2-1+(y+2)^2-4+(z-3)^2-9 = 2\;\Rightarrow\;(x-1)^2+(y+2)^2+(z-3)^2 = 16.$

Center $(1,-2,3)$, radius $4$. Check: $(5,-2,3)$ is $4$ units from the center and must satisfy the equation: $25+4+9-10-8-18 = 2$ ✓.

Multivariable functions

$f(x,y)$ assigns a number to each point $(x,y)$ in the plane. Its graph is the surface $z = f(x,y)$ in space. Domain: the points where the formula makes sense. Range: the values taken.

When drawing the domain, state whether the boundary is included: a solid line if $\ge$ (closed), a dashed line if $>$ (open).

Level curves

The set of points that satisfy the equation $f(x,y) = c$ is a level curve. All the level curves drawn for different values of $c$ are called a contour map; the contour lines on maps are exactly this.

Example 5. $f(x,y) = x^2+y^2$: the level curves are $x^2+y^2 = c$, centered circles ($c>0$). When $c$ increases in equal steps, the circles get closer to each other, because the paraboloid gets steeper outward.

Example 6. $f(x,y) = y-x^2$: the level curves are $y = x^2+c$, parabolas shifted up.

Example 7. $f(x,y) = y^2-x^2$: hyperbolas for $c\neq0$, the lines $y = \pm x$ for $c=0$. The contour map of the saddle surface.

For $f(x,y,z)$ of three variables, the same idea gives a level surface: $f(x,y,z) = c$. $x^2+y^2+z^2 = c$ gives concentric spheres.

TIP · contour density. Where the contour lines are close together the function changes quickly (the surface is steep), where they are far apart it changes slowly. The gradient vector in week 10 gives exactly this information and is always perpendicular to the level curve.

Limits and continuity

$\displaystyle\lim_{(x,y)\to(a,b)}f(x,y) = L$ means that $f$ goes to $L$ no matter which path the point $(x,y)$ approaches $(a,b)$ along. In one variable there were only two directions (right, left); in the plane there are infinitely many paths.

Continuity. Polynomials, rational functions on their domains, and their compositions with $\sin$, $e^x$, $\ln$, $\sqrt{\ }$ are continuous. For a continuous function, limit = plugging in: $\displaystyle\lim_{(x,y)\to(2,1)}\frac{x^2+3y}{x+y} = \frac{7}{3}$.

The two-path test

If different limits come out along two different paths, the limit does not exist.

Example 8. $\dfrac{x^2-y^2}{x^2+y^2}$ at $(0,0)$. The $x$ axis ($y=0$): $\dfrac{x^2}{x^2} = 1$. The $y$ axis ($x=0$): $\dfrac{-y^2}{y^2} = -1$. No limit.

Example 9 (lines are not enough). $\dfrac{x^2y}{x^4+y^2}$. Along every line $y = mx$, $\dfrac{mx^3}{x^4+m^2x^2} = \dfrac{mx}{x^2+m^2}\to0$. But along the parabola $y = x^2$, $\dfrac{x^4}{2x^4} = \dfrac12$. No limit. If lines give you no result, try a curve that equalizes the degrees of the numerator and denominator.

Showing that a limit exists

If the paths give the same result, prove the limit with squeeze: $\lvert f(x,y)-L\rvert\le g(x,y)$ and $g\to0$. Polar coordinates also work: put $x = r\cos\theta$, $y = r\sin\theta$; if the expression goes to $0$ as $r\to0$ independently of $\theta$, the limit is $0$.

Common mistakes