- #1
mliuzzolino
- 58
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Homework Statement
For a Gaussian landscape, the log-fitness change caused by a mutation of size r in chemotype element i is
[itex] Q_i(r) = -\vec{k} \cdot S \cdot \hat{r_i}r - \dfrac{1}{2} \hat{r_i} \cdot S \cdot \hat{r_i}r^2 [/itex].
To find the largest possible gain in log-fitness achievable by mutating chemotype element i, maximize [itex] Q_i(r) [/itex] with respect to r.
Homework Equations
The solution is:
[itex] \Theta _i = \dfrac{|\vec{k} \cdot S \cdot \hat{r_i}|^2}{2\hat{r_i} \cdot S \cdot \hat{r_i}} [/itex]
The Attempt at a Solution
[itex] Q_i(r)' = -\vec{k} \cdot S \cdot \hat{r_i} - \hat{r_i} \cdot S \cdot \hat{r_i}r = 0 [/itex]
[itex] \hat{r_i} \cdot S \cdot \hat{r_i} r = -\vec{k} \cdot S \cdot \hat{r_i} [/itex]
[itex] r = \dfrac{-\vec{k} \cdot S \cdot \hat{r_i}}{\hat{r_i} \cdot S \cdot \hat{r_i}} [/itex]
It's been forever since I've dealt with vector calculus so I know that I'm approaching this entirely the wrong way. Any points in the right direction will be greatly appreciated!