# Vector-valued function is smooth over an interval

• MHB
• A.Magnus
In summary, the parametrization of a curve represented by a vector-valued function is smooth on an open interval when the derivatives of the function are continuous and the tangent vector is not equal to zero for any value of t in the interval. This ensures that the curve does not slow down or backtrack on itself, making it a regular curve of order 1 and smooth of class C^1.
A.Magnus
I am reading Larson's Calculus textbook and come across this paragraph about vector-valued function:

The parametrization of the curve represented by the vector-valued function

$$\textbf{r}(t) = f(t)\textbf{i} + g(t)\textbf{j} + h(t)k$$

is smooth on an open interval $I$ when $f'$, $g'$ and $h'$ are continuous on $I$ and $\textbf{r}'(t) \neq \textbf{0}$ for any value of $t$ in the interval $I$.

Can somebody please tell me why is that $\textbf{r}'(t)$ has to be not equal to zero? Thank you beforehand for your time and gracious helping hand. ~MA

MaryAnn said:
I am reading Larson's Calculus textbook and come across this paragraph about vector-valued function:

The parametrization of the curve represented by the vector-valued function

$$\textbf{r}(t) = f(t)\textbf{i} + g(t)\textbf{j} + h(t)k$$

is smooth on an open interval $I$ when $f'$, $g'$ and $h'$ are continuous on $I$ and $\textbf{r}'(t) \neq \textbf{0}$ for any value of $t$ in the interval $I$.

Can somebody please tell me why is that $\textbf{r}'(t)$ has to be not equal to zero? Thank you beforehand for your time and gracious helping hand. ~MA

Hey MaryAnn! (Smile)

The vector $\mathbf{r}'(t)$ is the tangent of the curve.
If it is zero somewhere, it means that the curve slows to a stop or even backtracks on itself.

The $\mathbf{r}'(t) \ne \mathbf 0$ restriction is not generally part of the definition of smooth though.
Instead it's part of the definition of a regular curve (of order $1$).
So we're really talking about a regular smooth curve (regular of order $1$ and smooth of class $C^1$).

## What is a vector-valued function?

A vector-valued function is a mathematical function that takes in an input and returns a vector as an output. It can also be defined as a function that maps a scalar input to a vector output.

## What does it mean for a vector-valued function to be smooth?

A vector-valued function is considered smooth if it has continuous partial derivatives of all orders. This means that the function is differentiable and its derivatives exist at all points within the given interval.

## Can a vector-valued function be smooth over any interval?

No, a vector-valued function can only be smooth over a specified interval if it meets the criteria of being differentiable and having continuous partial derivatives of all orders within that interval.

## What is the importance of a vector-valued function being smooth over an interval?

A vector-valued function being smooth over an interval is significant because it allows for the function to be easily manipulated and analyzed using calculus. It also means that the function is well-behaved and can be used to model real-world phenomena accurately.

## How can one determine if a vector-valued function is smooth over a given interval?

To determine if a vector-valued function is smooth over a given interval, one can check if the function is differentiable and has continuous partial derivatives of all orders within that interval. This can be done by taking the partial derivatives of the function and checking for continuity at all points within the interval.

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