If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let:
the 1st vector in the cross product = your index finger = in red below
the 2nd vector in the cross product = your middle finger = in purple below
Then $\mathbf{\text{ first vector } \times \text{second vector } } =$ your thumb = in green below.
My picture confirms this.
Yet what about with your left hand? Regarding the picture below, the blue vector is supposed to denote the final cross product. I work backwards to find that the left hand must satisfy:
the 1st vector in the cross product $= \partial_z \, \mathbf{r} =(0, 0, 1) $ = in black = the thumb
the 2nd vector in the cross product$= \partial_\phi \, \mathbf{r} =$ in green = the index finger ?
Then $\mathbf{n} = \partial_z \, \mathbf{r} \times \partial_\phi \, \mathbf{r} =$ in blue = the middle finger.
Best Answer
The correct application for the right-hand rule (the way I learned it) is: First vector = thumb; second vector = index finger; third vector (i.e result) = middle finger. And of course it involves the correct gesture of the hand: thumb and index finger fully stretched off, middle finger only half. I guess the gesture is clear because it is the one most easily performed (ergonomically). And the assignment to the fingers at least for me feels more straightforward this ways: first/second/third finger = first/second/third vector. There seem to be (culture-specific?) variants on the market, thus cf. https://commons.wikimedia.org/wiki/File:RHR.svg vs. https://commons.wikimedia.org/wiki/File:Right_hand_rule_cross_product.svg
If you prefer to use your left hand, the rule is easily adapted by either changing the gesture (not recommended) or using a differnt vextor-finger assignment, hopefully with suitable mnemonics. For example: Thumb = first vector; middle finger = the second (i.e. middle of the three vectors) vector; index finger = indicates the result.