SUNY Geneseo Department of Mathematics

Introduction to Multivariable Functions

Monday, March 6

Math 223
Spring 2023
Prof. Doug Baldwin

Anything You Want to Talk About?

(No.)

PRISM (Not Quite) Pi Day

Since Pi Day (March 14, i.e., 3.14) is during spring break, PRISM is celebrating it this Thursday (3.09) from 4:00 - 5:00 in South 309. Everyone is welcome to drop by for pie and conversation.

Osculating Circles

Formally, an osculating circle for a curve at point P, with unit tangent T and principal unit normal N at P, is a circle that…

Lies in the same plane as T and N
Lies on the concave side of the curve
Is tangent to the curve
Has the same curvature as the curve has at P.

Graph of a curve with a circle nestled inside one of its bends; circle is tangent to curve in bend

Multivariable Functions

Based on section 3.1 in the textbook.

Key Ideas

A multivariable function is one that depends on x, y, etc. It maps R² (or R³, etc.) to a scalar result.

You can plot 2-variable functions, although using technology to do it is far better than doing it by hand.

Level curves (or surfaces for 3-variable functions) are curves (or surfaces) in the xy plane (or xyz space) along which the function has a constant value:

3 D plot with curve in X Y plane and curve of constant height above it

Multivariable functions can have domain restrictions and limited ranges, just as single-variable functions can.

Examples

Some examples of multivariable functions that occur to me include…

The formula for the length of the hypotenuse of a right triangle: h(a,b) = √(a² + b²). This has no domain restriction, since a² + b² is always non-negative, but has a range that is only the non-negative reals, not all reals.
The distance a moving object travels as a function of its speed and time: D(s,t) = st. This has no domain or range restrictions.

See if you can think of some other examples.

Speed from distance and time: s(D,t) = D/t. This has a domain that is all pairs of reals, except those in which t = 0.

Consider the function z = x² - y², and find an equation for the level curve z = 3.

Level curve equations are as simple as setting the result of the function equal to a constant. Once you do that, some simplification may be possible, but not always. So in this case, the level curve is the hyperbola 3 = x² - y².

Plotting

Mathematica can plot 2-variable functions with the Plot3D function (which we briefly saw in connection with plotting certain quadric surfaces).

The ContourPlot function plots level curves for 2-variable functions. The command typically looks like

ContourPlot[ function, {x, lowX, highX}, {y, lowY, highY} ]

where function is the function to plot level curves of, and lowX, highX, lowY, and highY are the lower and upper bounds on x and y. A Contours option lets you specify the exact level curves you want, or the total number of curves.

You can download this notebook to see some examples of multivariable function and level curve plots, and experiment with them for yourself.

Limits of multivariable functions.

Please read section 3.2 of the textbook. But I plan to discuss it over 2 class days, probably continuity and limits that do exist tomorrow and limits that don’t exist Wednesday, if you want to break up the reading accordingly.

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