I've been studying vector functions and one thing that I don't quite understand is finding the surface that some vector function r(t) lies on.
For instance, in my textbook (Calculus Early Transcendentals, 8E, Stewart) if r(t) = cost(i) + sint(j) + t(k), then the surface r(t) lies on is a cylinder. I understand why, but I don't really see how the book got there.
Some problems I've encountered were like this: Let r(t) = tcost(i) + t(j) + tsint(k). Find the surface on which r(t) lies.
Since I don't understand how to the textbook concluded on a cylinder for a helix, I don't really see how to approach these types of problems.
Any explanations on how the book got to a cylinder (textbook explanation wasn't quite adequate), or general tips on how to find surfaces on which a vector function lies would be greatly helpful. Thank you!
I suppose, the best answer would be to consider a general surface. Every surface is going to have to have some sort of tangent vectors associated to every point. You can see this by the blue vectors and the grid lines on this particular made up surface:
Any vector which is 'confined' to this surface is going to have to be a linear combination of some integral of the surface vectors, which I will denote by $\textbf{S}_\alpha$ (as the surface vectors would highlight the tangential direction of the path). So let us consider two dimensional surface, $\Sigma$, embedded upon a three dimensional euclidean space, $\textbf{R}_3$. We recognize that this is a surface being 'embedded' into ambient coordinates.
You may be familiar with $(x,y,z)$, or $(r,\theta,z)$, etc. But these can change based on one's needs and so there may be an infinite number of coordinate systems, especially curvilinear ones. So we will denote the coordinates by $(Z^1,Z^2,Z^3)$ which can be indexed as such: $Z^i$. In this coordinate system, the basis vectors are denoted by $\textbf{Z}_i$
Also, there may be many different surface coordinates chosen $(s,t)$, $(r,\theta)$ if youre specifying the surface $f(r,\theta)$ or otherwise. We will keep it general and state the variables as $(S^1,S^2)$ which can be abbreviated as $S^\alpha$.
Suppose that we keep things general and say that the parametrization of the surface is going to be in the form: $$\Sigma:\left\{\begin{array} \\ Z^1=Z^1(S^1,S^2) \\ Z^2=Z^2(S^1,S^2) \\ Z^3=Z^3(S^1,S^2)\end{array}\right.$$
The surface vectors can be calculated by the following formula: $$\textbf{S}_\alpha=\sum_{i=1}^3\frac{\partial Z^i}{\partial S^\alpha}\textbf{Z}_i$$
We define the partial derivative of the ambient coordinates as $Z^i_\alpha$. In some scenarios, some books will omit the summation sign and just use the following notation:
$$\textbf{S}_\alpha=Z^i_{\phantom{.}\alpha}\textbf{Z}_i$$
Therefore. Any vector, $\vec{\eta}$ on this surface will be a linear combination of the surface vectors. A vector path function $\vec{\eta}(t)$ will be a linear combination of the integralx of the surface vectors with respect to the each's variable, with the resultant integrals parametrised by the vector function's argument. The coefficients can be described by the symbol $C^a$. Then we may say: $$\vec{\eta}=C^a\int\textbf{S}_\alpha dZ^\alpha$$
To give an example with the cylinder. Let us parametrize it as: $$\Sigma:\left\{\begin{array} \\ x=R\cos\theta \\ y=R\sin\theta \\ z=z\end{array}\right.$$
In this case, the ambient coordinates are $Z^i=(Z^1,Z^2,Z^3)=(x,y,z)$ where $\textbf{Z}_i$ will correspond to $(\hat{x},\hat{y},\hat{z})$.
The surface coordinates are $S^\alpha=(S^1,S^2)=(\theta,z)$, where $\textbf{S}_1$ will correspond to $\hat{\theta}$ and $\textbf{S}_2$ will correspond to $\hat{z}$ Then the surface vectors are:
$$\hat{\theta}=\left[\begin{array}\\ -R\sin\theta \\ R\cos\theta \\ 0\end{array}\right]\text{, and, }\hat{z}=\left[\begin{array}\\ 0 \\ 0 \\ 1\end{array}\right]$$
So any vector path function on this surface will be specified by:
$$\vec{\eta}=C^1\left[\begin{array}\\ R\cos\theta(t) \\ R\sin\theta(t) \\ c_1\end{array}\right]+C^2\left[\begin{array}\\ c_2 \\ c^3 \\ z(t)\end{array}\right]$$
We notice we can modify these coefficients and permit a $\theta(t)$ and $z(t)$ such that we obtain a helix.
I guess to summarize, any vector path must be a linear combination of the surface vectors and crossing any of the surface vectors with the vector must result in the parametrization again.