Talk:Vector-valued function

From Wikipedia, the free encyclopedia

Superscript text==Alternate Representations?==

The article for Inverse function theorem uses this function in an example:

F(x,y)= \begin{bmatrix} {e^x \cos y}\\ {e^x \sin y}\\ \end{bmatrix}

This current article (Vector-valued function) uses this notation:

\mathbf{r}(t)=f(t)\mathbf{{\hat{i}}}+g(t)\mathbf{{\hat{j}}}

The Inverse function theorem and Jacobian articles seem to be using a kind of shorthand.

If they are accurate, it could be helpful in deciphering the column-vector notation to have something like the following in this current article:

\mathbf{r}(t)=f(t)\mathbf{{\hat{i}}}+g(t)\mathbf{{\hat{j}}} = \begin{bmatrix} {f(t)}\\ {g(t)}\\ \end{bmatrix}
\mathbf{r}(x,y)=f(x,y)\mathbf{{\hat{i}}}+g(x,y)\mathbf{{\hat{j}}} = \begin{bmatrix} {f(x,y)}\\ {g(x,y)}\\ \end{bmatrix}

Ac44ck 17:28, 18 May 2007 (UTC)

The column-vector notation seems to be shorthand for matrix multiplication an implied row-vector (which consists of unit spacial-vectors) by the column-vector:

\mathbf{r}(x,y)=f(x,y)\mathbf{{\hat{i}}}+g(x,y)\mathbf{{\hat{j}}} = \begin{bmatrix} \mathbf{{\hat{i}}} ~~ \mathbf{{\hat{j}}} \end{bmatrix} * \begin{bmatrix} {f(x,y)}\\ {g(x,y)}\\ \end{bmatrix}

Ac44ck 19:38, 18 May 2007 (UTC)

[edit] Show a vector in the graph?

The graph shows the curve traced through space by the end of a vector rather than showing a vector. Interpreting the graph of the given vector-valued function could be helped by showing a vector for some value of 't'. Ac44ck 23:33, 19 May 2007 (UTC)