Why Does the Matrix Transformation Not Equal dxdydz?

[Iraser]

Hey everyone, could anyone help me with this proof?
| dx/dr dx/d0 dx/d@ |
| dy/dr dy/d0 dy/d@ | dot [drd0d@] = dxdydz
| dz/dr dz/d0 dz/d@ |

d0 is d[theta] and d@is d[phi]

i cannot get this to equal, after solving it many ways i always get 3dxdydz - 3dxdydz??

Any help would be appreciated

[EDIT]

Well i have attempted this multiple times, and verified this with Wolfram Alpha as well. This can not be proved to equal dxdydz

 

[Stephen Tashi]

Are asking about the product of a matrix times a vector?

\begin{bmatrix} <br /> \frac{dx}{dr}&amp;\frac{dx}{d\theta}&amp; \frac{dx}{d\phi}<br /> \\ \frac{dy}{dr}&amp;\frac{dy}{d\theta}&amp; \frac{dy}{d\phi}<br /> \\ \frac{dz}{dr}&amp;\frac{dz}{d\theta}&amp; \frac{dz}{d\phi}<br /> \end{bmatrix} <br /> \begin {bmatrix}<br /> dx\\dy\\dz<br /> \end {bmatrix}

If you're asking something about the dot product of vectors, you need to explain how the thing on the left side of the expression is a vector.

My guess is that you're trying to prove something involving calculus by applying simple algebra to the differential expressions. That might not work. For example, if

x = f(r,\theta,\phi) some books might write

dx = (\partial x/ \partial r) dr + (\partial x/ \partial \theta) d\theta + (\partial x/ \partial \phi) d\phi to indicate how to approximate a small change in x.

I suppose some books might use dx/dr instead of \partial x/ \partial r Will some kind person please explain this distinction?

 

[I like Serena]

You didn't explain what your symbols represent exactly.
So I'm going to make an educated guess.
Did you mean the following?

<br /> \begin{bmatrix} <br /> \frac{dx}{dr}&amp;\frac{dx}{d\theta}&amp; \frac{dx}{d\phi}<br /> \\ \frac{dy}{dr}&amp;\frac{dy}{d\theta}&amp; \frac{dy}{d\phi}<br /> \\ \frac{dz}{dr}&amp;\frac{dz}{d\theta}&amp; \frac{dz}{d\phi}<br /> \end{bmatrix} <br /> \begin {bmatrix}<br /> dr\\d\theta\\d\phi<br /> \end {bmatrix} <br /> =<br /> \begin{bmatrix} <br /> \frac{dx}{dr}dr + \frac{dx}{d\theta}d\theta + \frac{dx}{d\phi}d\phi<br /> \\ \frac{dy}{dr}dr + \frac{dy}{d\theta}d\theta + \frac{dy}{d\phi}d\phi<br /> \\ \frac{dz}{dr}dr + \frac{dz}{d\theta}\theta + \frac{dz}{d\phi}d\phi<br /> \end{bmatrix} <br /> =<br /> \begin{bmatrix} <br /> dx<br /> \\ dy<br /> \\ dz<br /> \end{bmatrix} <br />