Parametrics and vector valued functions

In summary, the problem of finding the area under a parametric curve can be solved by using the integral of the product of the parametric equations. However, when considering the same curve represented by a velocity vector, the result is not related to the area under the curve but rather the arc-length. This is because the integral of the velocity vector gives a vector as a result, not a number.
  • #1
Considering the problem of finding the area under a parametric curve, I thought,
y=y(t), x=x(t)
That result seems straighforward.

I also thought, what if I let the VVF v=<x(t),y(t)> represent the same curve. To find the area, under the curve (I have in the back of my mind the concept of velocity and position), I would solve the integral, r=[inte]vdt.

Should these two results be related? I think they should, but the math shows they aren't. Should the magnitidue of the latter equal the absolute value of the former? Looks like no. Why not?
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  • #2
Thinking of t as time, integrating the velocity vector give the position as a vector with tail at the original point. r= &int;v dt is a vector not a number. Since ||v|| is the speed, &int; ||v||dt gives arc-length, not area under the curve.
  • #3

Thank you for sharing your thoughts on parametrics and vector valued functions. I agree that finding the area under a parametric curve can be straightforward using the method you mentioned. However, when considering a vector valued function, it is important to note that the integral represents the displacement vector rather than the area under the curve. This is because the vector valued function represents both the position and velocity of the curve at different points in time.

In terms of the relationship between the two results, they are related in the sense that they both use the same curve. However, the first integral represents the area under the curve at a specific point in time, while the second integral represents the total displacement of the curve over a given time interval. So while they may not be equal, they are both important in understanding the behavior of the curve.

I hope this helps clarify the differences between the results and why they may not be related in the way that you initially thought. Both approaches have their own significance and can provide valuable insights into the curve.

What is a parametric function?

A parametric function is a mathematical representation of a curve or surface where the variables are expressed in terms of one or more parameters. This allows for a more versatile and flexible way of defining curves and surfaces.

What is the difference between a parametric function and a regular function?

The main difference between a parametric function and a regular function is that a regular function has only one independent variable, while a parametric function has multiple independent variables (parameters).

What is a vector valued function?

A vector valued function is a mathematical function that takes in one or more parameters and outputs a vector. This type of function is often used to represent curves or surfaces in three-dimensional space.

How do you graph a parametric function?

To graph a parametric function, plot points using the parameter as the input and the resulting coordinates as the output. Then, connect these points to create a smooth curve or surface.

What are some real-life applications of parametric and vector valued functions?

Parametric and vector valued functions have many real-life applications, such as in computer graphics, physics, engineering, and economics. They are used to model and analyze various phenomena, including motion of objects, population growth, and economic trends.

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