Tell me and I’ll forget; show me and I may remember; involve me and I’ll understand, Chinese Proverb.

A function of two variables assigns a unique real number to each ordered pair in its domain.

Typically, we express such functions as z = f(x, y), where **x and y are the independent variables, and z is the dependent variable**.

The domain of a function of two variables is the set of all possible input values for both variables x and y.

The graph of a two-variable function is typically a surface in a three-dimensional Cartesian space . This surface represents all points (x, y, z) such that z = f(x, y).

- $x^2+4y^2+z^2 = 4 ↭ \frac{x^2}{4}+ \frac{y^2}{1}+ \frac{z^2}{4} = 1$ is an ellipsoid with its center at the origin (0, 0, 0) where a = 2, b = 1, and c = 1 are the length of the semi-axes (Figure iv).

Traces: In the yz-plane (x = 0), $\frac{y^2}{1}+ \frac{z^2}{4} = 1$ we see an ellipse with a vertical axis of length 2 (along the z-axis) and a horizontal axis of length 1 (along the y-axis). In the xz-plane (y = 0), we see a circle centered at the origin with radius 2, $\frac{x^2}{4}+ \frac{z^2}{4} = 1$. In the xy-plane (z = 0), we see an ellipse with a horizontal axis of length 2 (along the x-axis) and a vertical axis of length 1 (along the y-axis), $\frac{x^2}{4}+ \frac{y^2}{1} = 1$.

The general ellipsoid is a quadratic surface which is defines in Cartesian coordinates as $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1$ where a, b, and c are the length of the semi-axes of the ellipsoid. The points (a, 0, 0), (0, b, 0), and (0, 0, c) lie on the surface.

- $x = y^2 + z^2$. To any fixed x-value, y
^{2}+ x^{2}represent a circle in the yz-plane with radius $\sqrt{x}$. As x increases, the radius of the circle increases. It is a paraboloid. It opens along the x-axis because x is expressed as a sum of the squares of y and z e.g., x = 0, y^{2}+z^{2}= 0 is just a point, the origin (0, 0, 0). x = 1, y^{2}+z^{2}= 1 is a circle in the yz-plane with radius 1. y^{2}+z^{2}= 4 is a circle in the yz-plane with radius 2 (Figure v).

The vertex of the paraboloid is at the origin (0, 0, 0). y = 0, x = z^{2} is a parabola. z = 0, x = y^{2} is a parabola, too.

- z = y
^{2}-x^{2}is known as a hyperbolic paraboloid. It is a saddle surface because it curves upwards in one direction (along the y-axis) and downwards in another direction (along the x-axis) (Figure i).

A hyperbolic paraboloid is a type of quadratic surface in three-dimensional space. It is often referred to as a “saddle surface” because of its distinctive shape, which curves upwards in one direction and downwards in the perpendicular direction. The general equation for a hyperbolic paraboloid is $z = \frac{y^2}{b^2}-\frac{x^2}{a^2}$

- xy-Plane Trace: Set z = 0, y
^{2}= x^{2}⇒y = ± x. In the xy-plane, the trace is a pair of lines y = x and y = −x (This trace is not specially helpful). - xz-Plane Trace: Set y = 0. z=−x
^{2}. This is a**downward-opening parabola**in the xz-plane. - yz-Plane Trace: Set x = 0. z = y
^{2}. This is an**upward-opening parabola**in the yz-plane.

The surface is symmetric with respect to both the xz-plane and the yz-plane. At the origin (0, 0, 0), the function equals zero. Along the y-axis (x = 0), z = y^{2} describes an upward-opening parabola. For both positive or negatives values of y, the z-values increase as y^{2}, forming a parabolic shape opening upwards. For fixed x-values, such as x_{1}, x_{2}, ··· the equation $z=y^2-x_i^2$ represents a series of upward-opening parabolas shifted downward by the constant $x_i^2$. These parabolas have their vertices at $z = -x_i^2$ on the z-axis.

Along the x-axis (y = 0), z = −x^{2} describes an downward-opening parabola. For both positive and negative values of x, the
z-values decrease as -x^{2}, forming a parabolic shape opening downward. For fixed y-values, such as y = y_{1}, y_{2},···, the equation $z = y_i^2-x^2$ represents a series of downward-opening parabolas shifted upward by the constant $y_i^2$. These parabolas have their vertices at $z =y_i^2$ on the z-axis. This combination of upward and downward curvatures gives the surface its characteristic saddle shape.

- z = 3 -x
^{2}-y^{2}. xy-trace: 0 = 3 -x^{2}-y^{2}, it is a circle of radius $\sqrt{3}$. For any fixed z-value, the equation x^{2}+ y^{2}= 3 -z represents a circle in the xy-plane with radius $\sqrt{3-z}$. For z = 3, x^{2}+ y^{2}= 3 -3 = 0. This is just a point at (0, 0, 3). For z = 2, x^{2}+ y^{2}= 3 -2. This is a circle in the xy-plane with radius 1. For z = 0, x^{2}+ y^{2}= 3 -0 = 2. This is a circle in the xy-plane with radius $\sqrt{3}$ (Figure vi).

It is a paraboloid. It opens downwards along the z-axis because z is expressed as a decreasing function of the sum of the squares of x and y. The vertex of the paraboloid is (0, 0, 3). For x = 0, z = 3 -y^{2} is a parabola. For y = 0, z = 3 -x^{2} is a parabola, too.

- x
^{2}+y^{2}= z^{2}. It is a double cone centered at the origin. It is symmetric with respect to the x-, y-, and z-axes (Figure A).Double cone is two cones placed apex to apex (the vertex at the top or bottom). The double cone is given by the algebraic equation $\frac{z^2}{c^2}= \frac{x^2+y^2}{a^2}$.

- xy-Plane, z = 0 x
^{2}+y^{2}= 0 is just the origin (0, 0, 0). z = 1, x^{2}+y^{2}= 1 represents a circle of radius 1. z = 2, x^{2}+ y^{2}= 2 is a circle of radius $\sqrt{2}$. In other words, moving away from the origin,**in planes parallel to the xy-plane, the surface form circles of increasing radius.** - yz-Plane. Set x = 0, y
^{2}= z^{2}, z = ±y, these are two lines passing through the origin with slopes of 1 and -1. For different values of x, say x = x_{1}, x = x_{2},… the equation becomes $y^2 = z^2 -x_i^2$. These represent hyperbolas opening in the y- and z-directions (Figure B). - xz-Plane. Set y = 0, x
^{2}= z^{2}, z = ±x, which are two lines passing through the origin with slopes of 1 and -1. For different values of y, say y = y_{1}, y = y_{2},… the equation becomes $x^2 = z^2 -y_i^2$. These represent hyperbolas opening in the x- and z-directions.

In the xz- and yz-planes, the surface forms intersecting lines that pass through the origin and create a cone shape. Combining these traces, we see that the surface forms two identical cones, one opening upwards and the other downwards, meeting at the origin.

- x
^{2}+ y^{2}= z^{2}-1. This equation is a specific case of a more general equation for a two-sheeted circular hyperboloid along the z-axis $\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2} = -1$. In our case a = b = c = 1 (Figure C).

The surface consists of **two separate parts, one for z ≥ 1 and one for z ≤ −1, which are symmetrical about the xy-plane**.

- xy-Plane. Set z = 0, x
^{2}+y^{2}= -1. This equation has no real solutions. Therefore, there are no intersections with the xy-plane. For z = ±1, x^{2}+ y^{2}= 0. The only solutions are the points (0, 0, 1) and (by symmetry) (0, 0, -1). For z = ±2, x^{2}+y^{2}= 2^{2}-1 = 3. These are two circles with radius $\sqrt{3}$ centered at (0, 0, 2) and (0, 0, -2). Similarity, for any z = k where |z| ≥ 1, x^{2}+ y^{2}= k^{2}-1, i.e., circles with radii $\sqrt{k^2-1}$. - yz-Plane. Set x = 0, y
^{2}- z^{2}= -1. Rearranging gives: z^{2}-y^{2}= 1. This is a hyperbola opening upwards in the y- and z-directions. - xz-Plane. Set y = 0, x
^{2}= z^{2}-1. Rearranging gives: z^{2}-x^{2}= 1. This is also a hyperbola opening upwards in the x- and z-directions.

This surface forms two separate, symmetrical sheets opening along the z-axis, creating the shape known as a two-sheeted hyperboloid.

This content is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License and is based on MIT OpenCourseWare [18.01 Single Variable Calculus, Fall 2007].

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