diff options
Diffstat (limited to 'content/know/concept/calculus-of-variations/index.pdc')
| -rw-r--r-- | content/know/concept/calculus-of-variations/index.pdc | 8 |
1 files changed, 4 insertions, 4 deletions
diff --git a/content/know/concept/calculus-of-variations/index.pdc b/content/know/concept/calculus-of-variations/index.pdc index 26c5753..9cae283 100644 --- a/content/know/concept/calculus-of-variations/index.pdc +++ b/content/know/concept/calculus-of-variations/index.pdc @@ -251,7 +251,7 @@ meaning they do not depend on any derivatives of any $f_n(x)$: $$\begin{aligned} \phi_m(f_1, ..., f_N, x) = 0 - \qquad + \qquad \qquad \int_{x_0}^{x_1} \phi_m(f_1, ..., f_N, x) \dd{x} = C_m \end{aligned}$$ @@ -261,12 +261,12 @@ by simply redefining the constraint as $\phi_m^0 = \phi_m - C_m = 0$. To solve this constrained optimization problem for $f_n(x)$, we introduce [Lagrange multipliers](/know/concept/lagrange-multiplier/) $\lambda_m$. -In the former case $\lambda_m(x)$ is a function of all $x$, while in the +In the former case $\lambda_m(x)$ is a function of $x$, while in the latter case $\lambda_m$ is constant: $$\begin{aligned} - \int \lambda_m(x_i) \: \phi_m(\{f_n\}, x) \dd{x} = 0 - \qquad + \int \lambda_m(x) \: \phi_m(\{f_n\}, x) \dd{x} = 0 + \qquad \qquad \lambda_m \int \phi_m(\{f_n\}, x) \dd{x} = \lambda_m C_m \end{aligned}$$ |