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Multivariable calculus

Course: Multivariable calculus > Unit 1

Lesson 6: Visualizing multivariable functions (articles)
Math
Multivariable calculus
Thinking about multivariable functions
Visualizing multivariable functions (articles)
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Reduce reliance on graphs

Google Classroom
Although graphs are a great way to think about single variable functions, they don't always work for multivariable functions.

Background

Graphs are not the only way

If you have a single-variable function, like the one below, it's common to visualize it using a graph.
f, left parenthesis, x, right parenthesis, equals, minus, start fraction, 1, divided by, 2, end fraction, x, squared, plus, 2, x, plus, 3
However, it's important to remember that graphs are not the same thing as functions. This might seem obvious, but graphs are so useful for representing single-variable functions that people often hold on to the idea of graphs a little too tightly as they shift focus to multivariable functions.
For example, do you remember the derivative? At some point you may have seen the formal definition, which looks like this:
f′(x)=lim⁡h→0f(x+h)−f(x)h←Formal definition of the derivative.\begin{aligned} \quad f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h} \quad \leftarrow \small{\gray{\text{Formal definition of the derivative.}}} \end{aligned}
Be honest, though, how often did you actually think about this limit while doing exercises, learning how to compute derivatives, and interpreting the meaning of derivatives?
It's much simpler to think of the derivative as representing the slope of the graph of f. And there's nothing wrong with that! At least, for single-variable calculus there's not.
Graph of minus, start fraction, 1, divided by, 2, end fraction, x, squared, plus, 2, x, plus, 3
In multivariable calculus, we will not always visualize functions with graphs. As a result, when we extend the idea of a derivative, you cannot always think about it as a slope. But that doesn't mean we won't visualize it! It's just that the visualization might be somewhat different at times.
And, in the same way, the understanding of an integral as computing the signed area under a curve is so useful that students in single-variable calculus rarely think about it differently. Why would you? Don't fix what isn't broken, right?
Mastering multivariable calculus requires the flexibility to think of functions differently—still visually, but differently. It also requires incorporating fundamental notions like derivatives and integration into those new ways of thinking.
For example, the derivative is fundamentally asking how the output of a function changes as you slightly tweak its input. If a function has a multidimensional output, interpreting this as "slope" doesn't really make sense. Instead, you might have to visualize how a small change to the input influences each coordinate of the output.
Similarly, integration is fundamentally adding up a bunch of tiny little values—well, infinitely many infinitely tiny values—but this does not always mean area. In physics, for example, it's common to compute the "work" done on an object by some force using an integral, but there's not always a clear way to view that "work" as any kind of area.

Five different visualizations

In the next several articles, we'll go through five different ways to visualize multivariable functions. Here we'll just get a taste for each one.
In each of the following descriptions, "input space" and "output space" refer to where the input and output of a function live. For example, if a function takes in an ordered pair left parenthesis, x, comma, y, right parenthesis, like left parenthesis, 2, comma, 5, right parenthesis, and outputs a single number, like 5, the input space would be the x, y-plane and the output space would be the real number line.
  • Graphs, our old friend. Graphs have the benefit of showing both the input space and the output space at once, but as a result, they are highly limited by dimension. For this reason, they are only really useful for single-variable functions and multivariable functions with a two-dimensional input and a one-dimensional output.
  • Contour maps. Contour maps only show the input space and are useful for functions with a two-dimensional input and a one-dimensional output.
  • Parametric curves/surfaces. Parametric curves and surfaces only show the output space and are used for functions whose output space has more dimensions than the input space.
  • Vector fields. These apply to functions whose input space and output space have the same number of dimensions. For example, functions with two-dimensional inputs and two-dimensional outputs, or three-dimensional inputs and three-dimensional outputs can be used with vector fields.
  • Transformations. These have the benefit of applying to any function, no matter the dimension of the input and output space. However, the downside is that they can only be represented using an animation or a schematic drawing. As such, they are most useful for gaining a conceptual understanding of what a function is doing, but are impractical for representing the function precisely.
With each new topic and definition that you learn, a good way to test your understanding is to see if you can make sense of it in the context of functions that you visualize in each of these different ways. For example, the derivative indicates slope in the context of graphs, but the multivariable version of a derivative might mean something entirely different for parametric functions, vector fields, and contour maps.

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