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Diffstat (limited to 'source/know/concept/legendre-transform/index.md')
| -rw-r--r-- | source/know/concept/legendre-transform/index.md | 16 |
1 files changed, 7 insertions, 9 deletions
diff --git a/source/know/concept/legendre-transform/index.md b/source/know/concept/legendre-transform/index.md index d09613f..0d168aa 100644 --- a/source/know/concept/legendre-transform/index.md +++ b/source/know/concept/legendre-transform/index.md @@ -11,9 +11,8 @@ layout: "concept" The **Legendre transform** of a function $$f(x)$$ is a new function $$L(f')$$, which depends only on the derivative $$f'(x)$$ of $$f(x)$$, and from which the original $$f(x)$$ can be reconstructed. -The point is that $$L(f')$$ contains the same information as $$f(x)$$, -just in a different form, -analogously to e.g. the [Fourier transform](/know/concept/fourier-transform/). +The point is that $$L(f')$$ contains the same information as $$f(x)$$ +in a different form, like e.g. a [Fourier transform](/know/concept/fourier-transform/). Let us choose an arbitrary point $$x_0 \in [a, b]$$ in the domain of $$f(x)$$. Consider a line $$y(x)$$ tangent to $$f(x)$$ at $$x = x_0$$, @@ -23,18 +22,17 @@ $$\begin{aligned} y(x) &= f'(x_0) (x - x_0) + f(x_0) \\ - &= f'(x_0) \: x - C + &= f'(x_0) \: x - C(x_0) \end{aligned}$$ -Where $$C \equiv f'(x_0) \: x_0 - f(x_0)$$. +Where $$C(x) \equiv f'(x) \: x - f(x)$$. We now define the *Legendre transform* $$L(f')$$, -such that for all $$x_0 \in [a, b]$$ we have $$L(f'(x_0)) = C$$ -(some authors use $$-C$$ instead). -Renaming $$x_0$$ to $$x$$: +such that for all $$x_0 \in [a, b]$$ we have $$L(f'(x_0)) = C(x_0)$$ +(some authors use $$-C$$ instead): $$\begin{aligned} L(f'(x)) - &= f'(x) \: x - f(x) + &\equiv f'(x) \: x - f(x) \end{aligned}$$ We want this function to depend only on the derivative $$f'$$, |