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?

$$\begin{bmatrix} \frac{dx}{dr}&\frac{dx}{d\theta}& \frac{dx}{d\phi} \\ \frac{dy}{dr}&\frac{dy}{d\theta}& \frac{dy}{d\phi} \\ \frac{dz}{dr}&\frac{dz}{d\theta}& \frac{dz}{d\phi} \end{bmatrix} \begin {bmatrix} dx\\dy\\dz \end {bmatrix}$$

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?

$$\begin{bmatrix} \frac{dx}{dr}&\frac{dx}{d\theta}& \frac{dx}{d\phi} \\ \frac{dy}{dr}&\frac{dy}{d\theta}& \frac{dy}{d\phi} \\ \frac{dz}{dr}&\frac{dz}{d\theta}& \frac{dz}{d\phi} \end{bmatrix} \begin {bmatrix} dr\\d\theta\\d\phi \end {bmatrix} = \begin{bmatrix} \frac{dx}{dr}dr + \frac{dx}{d\theta}d\theta + \frac{dx}{d\phi}d\phi \\ \frac{dy}{dr}dr + \frac{dy}{d\theta}d\theta + \frac{dy}{d\phi}d\phi \\ \frac{dz}{dr}dr + \frac{dz}{d\theta}\theta + \frac{dz}{d\phi}d\phi \end{bmatrix} = \begin{bmatrix} dx \\ dy \\ dz \end{bmatrix}$$