# Definition of Smooth & Piecewise Smooth Curve

• gotjrgkr
In summary, a smooth curve is a continuous function from an interval [a,b] into a set of real numbers for which all derivatives exist and are continuous and smooth. A piecewise smooth curve is a curve that is smooth on a set of intervals within the original interval [a,b]. This means that while the curve may not be smooth on the entire interval, it is still smooth on each of the smaller intervals within it. Additionally, it is required that the tangent vector is never equal to zero. In other words, the curve cannot have any sharp corners or breaks in its slope.
gotjrgkr
Hi!
I want to know the precise definition of smooth curve and piecewise smooth curve if the curve indicates a continuous function from an interval [a,b] into a set of real numbers.

A smooth curve is a curve for which all of its derivatives exists, which gives that all derivatives must be continuous and smooth. A piecewise smooth curve is a curve for which there exists a set of intervals where the curve is smooth on each of those intervals. As an example, a curve might not be smooth on [a,b] but it is still piecewise smooth if it is smooth on both [a,c] and (c,b].

It would be better to say "tangent vector" than "all of its derivatives". Also, we require that the tangent vector never be 0.

Klockan3 said:
A smooth curve is a curve for which all of its derivatives exists, which gives that all derivatives must be continuous and smooth. A piecewise smooth curve is a curve for which there exists a set of intervals where the curve is smooth on each of those intervals. As an example, a curve might not be smooth on [a,b] but it is still piecewise smooth if it is smooth on both [a,c] and (c,b].

In the statement 'the curve is smooth on each of those intervals', can i interpret it as 'each restriction of the curve to a subinterval of the interval is smooth'??

## 1. What is a smooth curve?

A smooth curve is a continuous mathematical function that has no sharp corners or breaks. This means that the slope of the curve changes gradually and continuously at all points.

## 2. How is a smooth curve defined mathematically?

A smooth curve is defined as a function that is infinitely differentiable, meaning that it has derivatives of all orders. This ensures that the curve is continuous and has no sharp corners or breaks.

## 3. What is the difference between a smooth curve and a non-smooth curve?

A smooth curve has derivatives of all orders, while a non-smooth curve may have sharp corners or breaks and does not have derivatives at these points. This means that a smooth curve is continuously differentiable, while a non-smooth curve is not.

## 4. What is a piecewise smooth curve?

A piecewise smooth curve is a curve that is composed of multiple smooth segments, with each segment having a different mathematical definition. This means that the curve may have sharp corners or breaks at the points where the segments meet, but each individual segment is smooth.

## 5. How is a piecewise smooth curve different from a smooth curve?

A piecewise smooth curve may have sharp corners or breaks at the points where the segments meet, while a smooth curve is continuous and has no sharp corners or breaks. Additionally, a piecewise smooth curve is composed of multiple smooth segments, while a smooth curve is defined by a single mathematical function.

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