I’m reading the Chapter 8 of *“Geometric Algebra for Computer Science”*, and trying to find the derivative of a function like this f(B) = \parallel \! B \wedge \mathbf{v}_0 \! \parallel^2, with \parallel \! \mathbf{v}_0 \! \parallel^2 = 1.

Intuitively, I would like to know how rotating a plane represented by a *bivector* B would affect the *trivector* volume.

And I think I should further write the

bivectorB as a function of arotorR (maybe two rotors), e.g. B(R) =RB_0R^{-1}, by following the saying:

“Elements of geometric algebra should only be perturbed by rotors”:

e^{-\delta P/2} X e^{\delta P/2} = X + X \times \delta P + O(\delta^2) \quad\quad (8.4)

(page 217 of“Geometric Algebra for Computer Science”).

- I’m a little confused in how to apply the chain rule to a function that involves one or two rotor variables.
- I get stuck and am not sure how to go further after doing the following for \partial_B f(B).

Here comes the part for f(B) and \partial_B f(B):

For any two multivectors X and Y, and a k-blade M, we have

X Y = X \wedge Y + X \rfloor Y + X \times Y \quad and \quad MM = M \rfloor M,

Then we have \parallel \! M \! \parallel^2 = M * \tilde M = M\rfloor \tilde M = (-1)^{\frac{1}{2}k(k-1)} MM = -M^2 (for k=2),

and f(B) = \parallel \! B \wedge \mathbf{v}_0 \! \parallel^2 = -(B \wedge \mathbf{v}_0 )^2 = - \frac{1}{4}(B\mathbf{v}_0 + \mathbf{v}_0 \hat B)^2 = - \frac{1}{4}(B\mathbf{v}_0 + \mathbf{v}_0 B)^2 .Then for the derivative,

\partial_B f(B) \overset 0 = -\frac{1}{4} ( \overset \backprime \partial_B ( \overset \backprime B\mathbf{v}_0 + \mathbf{v}_0 \overset \backprime B)(B\mathbf{v}_0 + \mathbf{v}_0 B) + \overset \backprime \partial_B (B\mathbf{v}_0 + \mathbf{v}_0 B)( \overset \backprime B\mathbf{v}_0 + \mathbf{v}_0 \overset \backprime B) )

\quad\quad\quad\quad \!\! \overset 1 = -\frac{1}{4} (2\mathbf{v}_0 (B\mathbf{v}_0 + \mathbf{v}_0 B) + 2(B\mathbf{v}_0 + \mathbf{v}_0 B) \mathbf{v}_0)

\quad\quad\quad\quad \!\! \overset 2 = - (B + \mathbf{v}_0 B \mathbf{v}_0)

- In the step “\overset 1 =”, I treated the part without ^\backprime , i.e., \mathbf{v}_0 and (B\mathbf{v}_0 + \mathbf{v}_0 B), as constants. Is there anything wrong?

I do not feel confident in:

- when we can treat such parts as “constants” in taking derivative using product rule and “Table 8.1 / 8.2”, and
- when we need to apply a projection \mathsf{P}[\ ] (though this can become apparent when one take derivative using definition).

Any hints / corrections? Thank you!