Understanding Image of a Curve - Ask Luca

• pamparana
In summary, the conversation discusses the term "image of a curve" in relation to parametrized curves and level curves. The author explains that the image of a parametrized curve is the image of the map defining it, and a parametrization of a curve is a parametrized curve whose image is the level curve. The questioner expresses confusion and the author clarifies the concept.
pamparana
Hello,

Just started reading a beginners book on elementary differential geometry and have a small question about the term "image of a curve". It says that a parameterized curve whose image is contained in a level curve is called a parametrization of C.

I am a bit confused with this statement. What does the image of a parameterized curve mean?

Would much appreciate someone clarifying this doubt for me.

Thanks,
Luca

Surely you know what it means when we speak of the image of a map f:A-->B? It means simply the set f(A). Well, a parametrized curve is a map $\gamma:(\alpha,\beta)\rightarrow\mathbb{R}^n$. So the image of this paramatrized curve is the image of this map; namely $\gamma((\alpha,\beta))$.

What the author is saying here is that if you have a curve C defined as the level set of some function (i.e. a level curve), and if you find a parametrized curve $\gamma:(\alpha,\beta)\rightarrow\mathbb{R}^n$ whose image is that level curve (i.e. $\gamma((\alpha,\beta))$=C), then said parametrized curve is called a parametrization of C.

That makes sense! Thanks.

1. What is the image of a curve in mathematics?

The image of a curve in mathematics is the set of all points that the curve maps to on a coordinate plane. It represents the shape of the curve and is often visualized as a graph.

2. How is the image of a curve different from the range?

The range of a curve is the set of all possible output values, while the image is the set of actual points that the curve maps to. In other words, the range includes all possible y-values, while the image only includes the y-values that the curve maps to.

3. What is the importance of understanding the image of a curve?

Understanding the image of a curve is important in many areas of mathematics and science, including calculus, geometry, and physics. It allows us to analyze and manipulate curves, as well as make predictions and interpretations based on their shape and behavior.

4. How do you find the image of a curve?

To find the image of a curve, you can use a graphing calculator or plot points manually on a coordinate plane. Alternatively, you can use algebraic methods such as substitution and elimination to solve for the x and y values of points on the curve.

5. Can the image of a curve be infinite?

Yes, the image of a curve can be infinite if the curve is continuous and extends infinitely in one or both directions. This is often the case with exponential, logarithmic, and trigonometric curves.

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