Talk:Vector-valued function

^{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)

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)