Why are curves in the plane of the form R -> R^2?

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In summary, curves in the plane are represented by functions of one variable (R -> R^2), while curves in space are represented by functions of one variable (R -> R^3). Functions of two variables (R^2 -> R) represent surfaces in the plane, and functions of three variables (R^3 -> R) represent surfaces in space. Vector fields in the plane are represented by functions of two variables (R^2 -> R^2). This is because curves can be defined by one parameter, while surfaces require two parameters.
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Simfish
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So here are some functions of the following types...

f: R -> R^2 (curves in the plane)
f: R -> R^3 (curves in space)
f: R^2 -> R (functions f(x,y) of 2 vars)
f: R^3 -> R: (functions f(x,y,z) of 3 vars)
f: R^2 -> R^2 (vector fields v(x,y) in the plane)

The question is - why are curves in the plane of the form R -> R^2? My intuition tells me R^2 -> R^2 (since after all, curves in the plane are based on x and y coordinates...). And R^2 is a cartesian product of two sets. For any curve, I'd expect x AND y input values...
 
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Simfish said:
The question is - why are curves in the plane of the form R -> R^2? My intuition tells me R^2 -> R^2 (since after all, curves in the plane are based on x and y coordinates...). And R^2 is a cartesian product of two sets. For any curve, I'd expect x AND y input values...

Hi simfish! :smile:

Because a curve can be defined by one parameter - it's one-dimensional.

The parameter could be length, or angle, or anything convenient.

Usually, it's the length, s.

Then f(s) is the position (on a plane or in space) of the point whose distance along the curve is s.

So f maps the real numbers (R) into the plane or space.

You could use two parameters, but they wouldn't be independent.

Essentially, using (x,y) to define a curve in R2 would be using a function from s to (x,y) and then from (x,y) to R2! :frown:

A surface is two-dimensional, and needs two parameters. For example, points on a sphere are specified by latitude and longitude, so the "function for a sphere" in space would be a map from R2 to R3, specifying a point (x,y,z) for every point (theta,phi).
 

What is a domain?

A domain is the set of all possible input values for a function or mathematical relation. It is typically represented by the x-axis on a graph.

What is a range?

A range is the set of all possible output values for a function or mathematical relation. It is typically represented by the y-axis on a graph.

How do you determine the domain and range of a function?

The domain can be determined by looking at the input values that are allowed for the function. The range can be determined by looking at the corresponding output values for each input in the domain.

Why is it important to understand domains and ranges?

Understanding domains and ranges is important because it helps us to identify the set of values for which a function or relation is valid. This can help us to make accurate predictions and solve real-world problems.

What are some common mistakes when determining domains and ranges?

One common mistake is forgetting to consider any restrictions on the input values, such as dividing by zero or taking the square root of a negative number. Another mistake is incorrectly identifying the minimum and maximum values for the domain and range.

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