Functions of several variables

Functions of several variables has a domain that is multi-dimensional. Hence, a function of $n$ variables takes a set of $n$ numbers as its input.

Table of contents

    Intro

    Only being able to work with single variable functions put a strict constrain on the type of issues we can investigate. Most phenomena around us depend not on only one, but multiple different factors.

    For instance, if we want to successfully complete a moon landing, we must first of all take three spatial coordinates , and into consideration.

    Multivariable functions isn't rocket science, but rocket science certainly depends on them

    In addition to those, variations in temperature and pressure impacts the rockets trajectory and needs to be accounted for as well.

    Weather conditions are actually interesting multivariable functions themselves, depending on a large number of different variables.

    Concept

    A function works like a predictable machine that takes in some input and spits out an output as per the form of the function. The most basic ones have only single in- and outputs.

    Now the only difference when it comes to multivariable functions is that they take in more than one input, which the function combines in some way to produce its output.

    Both functions of one and those of several variables can produce outputs consisting of any number of output components though.

    Math

    In earlier courses we have studied function with one dimensional input and output,

    we now extend this to several dimensions.

    Scalar functions only have a single output component, and in one variable they can be thought of as taking an and mapping it to a , resulting in a curve in the plane.

    The analogous extension to higher dimensions is a scalar function of several variables, producing some surface in space.

    Closed and bounded sets

    Here comes a lecture note with definitions. We will need them later, so just keep them in the back of your mind. Namely, the characteristics of a domain of a function determine whether we can do certain types of calculations on it. And domains are sets.

    A set is an ensemble of points, determined by coordinates . To each point in a set, there are as many coordinates as there are dimensions in the space where the set lives. For example, each subset of is made up of points with two coordinates.

    We'll look at two characteristics of sets: closedness and boundedness.

    Open and closed

    Recall how we define closed and open intervals:

    The first interval is open, the second one is closed.

    The boundary of an interval is just two points. The boundary of a set in is a curve, and in it's a surface. In , it's not so easy to think about how it looks, but it's okay, the qualities coming with a boundary stays the same.

    Analogously to the closed interval, a set that is closed has all its boundary points included in it. This is a closed set:

    If we take all the boundary points away, we end up with an open set, which can be illustrated like this:

    As an example, consider a solid sphere, centered at the point . An open solid sphere is:

    This, on the other hand is a closed solid sphere:

    A curious side note: is neither open nor closed, as it has no boundary points.

    That closes the topic on closedness for now.

    Bounded and unbounded

    Let be a point in . Then, is bounded if there is a number so that for all .

    The blobs above are bounded, as well as the solid spheres. is unbounded: there is no number such that for all points, as the points go all the way to infinity.

    For the same reason, the set is unbounded as well:

    Domains and co-domains

    For a function in one variable, the domain refers to the set of values we can plug in for , and the co-domain is a set defining what type of outputs to expect. We write:

    When looking at for all possible , we may find that not all values are mapped to. Therefore, we define the image, or range, as the set of possible outputs of , given the domain .

    The domain decides which and how many inputs a function takes. The co-domain sets the dimension of its output.

    The same terms are used for functions in several variables. The difference is now that rather than being sets of single values for and , the domain, co-domain, and image of a function can be sets of points in some multidimensional space.

    Functions of several variables

    Scalar functions

    The functions we focus on in this course are mostly ones that take in two real independent variables and , to spit out one dependent variable . These are called scalar functions of two variables, and give rise to curved surfaces in a three dimensional space.
    Usually, the co-domain is , all the reals, and we say it's a real valued function.

    As an example, look at the function above, where the domain is the set of two-dimensional points with and .

    A scalar function of three variables , can be thought of as taking a point in a 3-dimensional space and assigning a density of the material that makes up the space to that point.

    If we imagine the 3-dimensional space as being made up of poorly mixed pancake batter full of lumps, we can describe the varying flour content throughout the batter with a function .

    Functions of yet more variables are hard to describe intuitively, but the math works just the same.

    The general form of scalar functions is

    where signifies the dimension of any point in the domain.

    Vector valued functions

    In contrast to scalar functions, a vector valued function produces an output with variables, where defines the dimension of the elements in the co-domain. Each variable in the output depends on the input variables.

    The input can consist of one independent variable:

    We have already seen functions of this type when talking of parametrization of curves. With , the position vector with as parameter was:

    This function takes in a variable and maps it onto a vector in .

    Alternatively, the function's domain can contain points in any dimension , making it a vector valued function of variables:

    These function often appear in the context of change of variables, with . For example, the variable transformation to polar coordinates is a function

    defined by

    Level curves

    The level curves of a map

    You stumble upon a function in on your desk. It has a cumbersome shape, its surface all hilly and wobbly. You'd like to bring it in your pocket. Enter: level curves.

    Level curves let's you bring a function in down onto a -plane. Many concepts in math have esoteric applications, hard to appreciate for most of us. Level curves is a stark exception, at least if you ever laid your eyes upon a map.

    The level curves of a map are the curvy, curly lines showing equal height. Walking along a level curve means you stay at the same height all the time.

    Level curves formalized

    We can interpret the level above the sea as a function of the surface coordinates. Calling the surface coordinates and , then abracadabra! we have a function in two variables , where is the level above the sea.

    Letting , with some constant, we find the level curves for , by varying the and the as much as we can while still satisfying the equation for the level curve:

    An intuitive interpretation of the level curves concept is: imagine cutting a graph in with planes parallel to the -plane. The cuts are level curves: along the whole curve cut out by the plane, the -value is the same.

    Example

    The level curves to the function

    are given by the intersection with the plane

    The intersection can be written as

    Thus, the level curves can be expressed as concentric circles with radius .

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